SOLVING BOLTZMANN EQUATION, PART I: GREEN S FUNCTION

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1 Bulletin of the Institute of Mathematics Academia Sinica New Series Vol. 6, No., pp SOLVING BOLTZMANN EQUATION, PART I: GREEN S FUNCTION TAI-PING LIU,a AND SHIH-HSIEN YU,b Institute of Mathematics, Academia Sinica, Taipei. a liu@math.stanford.edu Department of Mathematics, National University of Singapore. b matysh@nus.edu.sg Abstract We present an approach for solving the Boltzmann equation based on the explicit construction of the Green s functions. The Green s function approach has been useful for the study of nonlinear interior waves and the boundary waves. In this Part I we study the Green s functions for the initial and initial-boundary value problems. In the forthcoming Part II, we will study the general solutions of the Boltzmann equation. Our presentation is self-contained, and, besides the synthesis of the existing ideas, we also established the detailed analysis of the leading terms of the Green s function.. Introduction The kinetic theory starts with the density distribution function fx, t, where x = x,x,x 3 R 3 is the space variables, t is the time variable, and =,, 3 R 3 is the microscopic velocity. Thus the kinetic theory differs from the fluid dynamic description of the gases in the inclusion of the microscopic velocity as an independent variables. With the knowledge of the density distribution function fx, t, one can derive the macroscopic Received March 3, and in revised form June,. AMS Subject Classification: 76P5, 35L65, 8C4. Key words and phrases: Boltzmann equation, Green s function, Mixture Lemma, Fluid-like waves, Particle-like waves. The research of the first author is supported in part by National Science Foundation Grant DMS 7948 and National Science Council Grant M--. The research of the second author is supported by the National University of Singapore start-up research grant R

2 6 TAI-PING LIU AND SHIH-HSIEN YU [June variables, functions of the space and time variables x, t through computing the moments: ρx,t R fx,t,d, density, 3 ρvx,t R 3 fx,t,d, momentum, v = v,v,v 3 fluid velocity, ρex,t v R 3 fx, t, d, internal energy, ρex,t R 3 fx,t,d = ρe + ρ v, total energy, p ij x,t R i v i j v j fx,t,d,p = p ij 3 i,j 3, stress tensor, q i x,t R i v i 3 v fx,t,d, heat flux.. The most important equation for the kinetic theory for the gases is the Boltzmann equation, [], t fx,t, + x fx,t, = Qf,fx,t,,. k The left hand side of the equation, t f+ x f, is the transport term, measuring the time rate of change along the particle path. The Boltzmann equation says that the change is due to the collision operator Qf,f, which takes the form of the binary collision: Qg,h R 3 S Ω gh hg + g h +h g B,Ωd dω;.3 { = [ Ω] Ω, = + [ Ω] Ω..4 The cross section B,Ω depends on the inter-molecular forces between the particles. For the hard sphere models B,Ω = Ω..5 The Maxwellian, thermo-equilibrium states have the defining property that QM,M =

3 ] SOLVING BOLTZMANN EQUATION, PART I: GREEN S FUNCTION 7 and are of the form M = M ρ,v,θ ρx, t vx,t e Rθx,t, πrθx,t 3/ determined by the macroscopic variables. Here θ is the temperature defined by p/ρ = Rθ. For simplicity, in most part of our presentation, we will set the mean free path k =. The main purpose of this Part I and the forthcoming Part II of the present paper is to present the Green s function approach for the study of the solutions of the Boltzmann equation. The Green s function approach aims at pointwise, more quantitative understanding of the solutions of the Boltzmann equation. This is necessary for the qualitative understanding of the rich phenomena that the kinetic theory can model, particularly the nonlinear interior waves and the behavior of the solution near the boundary. There are studies based on physical considerations and asymptotic theory for the Boltzmann equation, see [4], [43], [8], [9], []. We start with the construction and analysis of the Green s function in this Part I. The construction involves explicit inversion of the Fourier transform for the fluidlike waves and Picard iterations for particle, particle-like and other singular waves. The Green s functions are for the linearized Boltzmann equation. The simplest situation is to consider the Boltzmann equation linearized around a global Maxwellian M: f = M + Mg so that the linearization yields g t + x g = Lg, linearized Boltzmann equation, Lg = Q Mg,M, linearized collision operator. M.6 The most basic Green s function is for the initial value problem for the above linearized Boltzmann equation. Because we are considering the Boltzmann equation linearized around a fixed Maxwellian, the equation is of constant coefficients in x, t. Thus the equation is both time and space translational invariant and so the Green s function has the property: Gx,x,t,s,; = Gx x,t s,;

4 8 TAI-PING LIU AND SHIH-HSIEN YU [June and satisfies { Gt + x G = LG, Gx,,; = δxδ..7 The Green s function Gx x,t s,; describes the propagation of the perturbation over the Maxwellian M when at time s the perturbation consists of particles concentrated at the location x = x and with the microscopic velocity concentrated at. The basic reason for the study of the Green s function is that it can be used to study the general solutions. The simplest situation is to consider the initial value problem for the linear Boltzmann equation with a given source h = hx,t,: { gt + x g = Lg + h, gx,, = g x,..8 Multiply the equation with the Green s function and integrate to yield the solution representation: gx,t, = Gx y,t,; g y, d dy R 3 R 3 t + Gx y,t s,; hy,s, d dyds..9 R 3 R 3 We will construct this Green s function first for the -dimensional case, x R, and then for the 3-dimensional case, x R 3. In preparation for these constructions, we make the following preliminaries. In Section, we will first review the basics for the Boltzmann equation and in Section 3 for the linearized Boltzmann equation. For the study of the fluid aspects of the Boltzmann equation, we consider in Section 4 the Euler and Navier-Stokes equations in gas dynamics. The construction of the Green s function is done first for the case of plane waves, i.e. the case when the space variable is -dimensional, x R, { Gt + x G = LG, Gx,,; = δ xδ ;.

5 ] SOLVING BOLTZMANN EQUATION, PART I: GREEN S FUNCTION 9 and the solution formula for the general initial value problem becomes g t + x g = Lg, gx,, = g x,, gx,t, =. R R Gx y,t,; 3 g y, d dy. We will consider the fluid-like waves in Section 5 and the particle-like and other singular waves in Section 6. The fluid-like waves are studied by the Fourier transform and the spectral analysis near the origin. The particle-like waves are constructed by Picard iterations and their regularity properties are studied by the Mixture Lemma. To combine these two types of waves for the complete picture of the Green s function requires intricate analysis. In Section 7 we construct the Green s function for the initial value problem for the general waves in 3-dimensional space. There are Huygens waves and other fluid-like waves. After the Green s function for a global Maxwellian is constructed for the initial value problem, we will use it to construct the Green s function for the initial-boundary value problem in Section 8. These construction are pointwisely explicit and will be used in the forthcoming Part II for the study of nonlinear waves and boundary layers for the Boltzmann equation. Our presentation is essentially self-contained. The existing materials, [3], [3], for the initial value problem, [33], for initialboundary value problem, are reorganized with new perspectives. Some issues are clarified and new materials are added. We raise open problems from time to time. For the Green s function approach aimed at the understanding of the dissipation effects of the boundary, see [34] and [47].. Boltzmann Equation In this section we review two basic properties of the Boltzmann equation.: the conservation laws and the H-Theorem... Conservation laws The collision operator Qf,f,.3, redistributes the density function

6 TAI-PING LIU AND SHIH-HSIEN YU [June fx, t, as a function of the microscopic velocity. Nevertheless, on the macroscopic level, the collision operator conserves the mass, momentum and energy: R 3 Qg,hd =, mass momentum.. energy The above is proved by simple change of variables, noting that the transformation.4 has Jacobian one: R 3 ΨQg,hd = 4 The key property is thus that the functions R 3 Ψ + Ψ Ψ Ψ Qg,hd. Ψ, Ψ i i, i =,,3, Ψ 4 /, Ψ = Ψ j, j =,...,4, are collision invariants in the following sense...3 Definition.. A function Ψ of the microscopic velocity is a collision invariant if under the transformation.4. Ψ + Ψ = Ψ + Ψ.4 In fact, the relation.4 is equivalent to the fact that the functions,, / are collision invariant. Thus any function in the span of these functions are also collision invariant and has the property that the integration of them times the collision operator vanishes. There is the basic theorem of Boltzmann saying that these are all the collision invariants; its proof is omitted, [8]. Theorem. Boltzmann. Any collision invariant is a linear combination of,, /. Multiply,, / with the Boltzmann equation. and integrate

7 ] SOLVING BOLTZMANN EQUATION, PART I: GREEN S FUNCTION to yield the conservation laws for the macroscopic variables of.: t ρ + x ρv =, conservation of mass, t ρv + x ρv v + P =, conservation of momentum, t ρe + x ρve + P v + q =, conservation of energy..5 Remark.3. There are 5 conservation laws: for the mass, 3 for the momentum, and for the energy. On the other hand, there are 4 unknowns: for the density ρ, 3 for the fluid velocity v, 6 for the symmetric stress tensor P, 3 for the heat flux q, and for the internal energy e. Thus the conservation laws are not a system of self-contained partial differential equations. The macroscopic quantities are computed as moments of the density function fx, t,. The stress tensor consists of second moments and the heat flux consists of third moments. To derive equations for these would require the fourth moments and so on. For instance, one may multiply the Boltzmann equation by i v i j v j and integrate to derive the evolutionary equation for the stress tensor p ij, a second moment: t p ij + x i v i j v j fd = i v i j v j Qf,fd. R 3 R 3 This would involve the third moment, the integration of i v i j v j f. The evolution of the third moments then involves the fourth moments and so on. Also the right hand side is not zero anymore, as i v i j v j is not a collision invariant. Thus the Boltzmann equation is equivalent to a system of infinite partial differential equations. Another way to view the kinetic theory is to consider the distribution function fx, t,, for each fixed space and time variables x,t, as a function of the microscopic variables. The space of functions in is infinite dimensional. For the fluid dynamics, the conservation laws are closed by making assumptions on the dependence of the stress tensor and heat flux on the conserved quantities and their differentials. For the Euler equations in gas dynamics, p ij E = pδ ij, q E =, Euler stress and heat flux;.6

8 TAI-PING LIU AND SHIH-HSIEN YU [June and for the Navier-Stokes equations, p ij NS = pδ ij µ[ vi + vj 3 x j x i 3 k= vk δ x k ij ] µ 3 B k= vk δ x k ij, Navier-Stokes stress; q NS = κ x θ, Navier-Stokes heat flux..7 We will derive the viscosity µ and the heat conductivity κ later. They are functions of the temperature µ = µθ, κ = κθ. For the in-depth study of the fluid dynamics from the point of view of the kinetic theory, the readers are referred to [4], [43]... H-Theorem and Maxwellians The Boltzmann equation has the so-called molecular chaos hypothesis built in the collision operator in that the loss term ff and the gain term f f in the collision operator Qf,f are supposed to be the two particles distributions before the collision, taken here as the products of the single particle distributions f. This is the hypothesis of no correlation before collision. An important consequence is the H-Theorem on the irreversibility of the Boltzmann process: R 3 log fqf,fd = 4 R 3 R 3 S + log ff f f [f f ff ]BdΩd d..8 The above is proved by simple change of variables, again using fact that the transformation.4 has Jacobian one,.. From.8, the Boltzmann Theorem yields that R 3 log fqf,fd =, if and only if log f is a collision invariant, in the span of,, /..9 In other words, log f, as a function of the microscopic velocity, is quadratic, or, equivalently, f is gaussian in. Direct calculations show that the distribution function is explicitly given in terms of the macroscopic variables:

9 ] SOLVING BOLTZMANN EQUATION, PART I: GREEN S FUNCTION 3 fx,t, = ρx, t vx,t e Rθx,t M πrθx,t 3/ [ ρ,v,θ].. Here we have introduced the macroscopic quantity θ, the temperature, through the ideal gas relation between the temperature θ, the pressure p and the density ρ: p = Rρθ.. The states M ρ,v,θ are called the Maxwellian distributions. They are the thermo-equilibrium states in that QM,M =.. The H-Theorem is obtained by integrating the Boltzmann equation. times log f t H + x H, H f log fd, H f log fd..3 R 3 R 3 Thus a Boltzmann solution has the tendency to reach the thermo-equilibrium and the Boltzmann process is irreversible. Remark.4. From the kinetic theory point of view, the fluid dynamics and thermodynamics are the study around the Maxwellians. The fact that the Maxwellian distribution. is determined by only two state variables ρ and θ, and is consistent with the basic axiom in thermodynamics that there are exactly two independent state variables. One can choose any two state variables and all the other state variables are functions, the constitutive relations, of the two chosen state variables. With the given distribution., this thermodynamics axiom clearly holds. For the non-equilibrium situation one can define, symbolically, the pressure p p + p + p 33 3 = 3 R 3 v fx,t,d, pressure, and we have from. that the internal energy and the pressure are related as: p = ρe..4 3

10 4 TAI-PING LIU AND SHIH-HSIEN YU [June This is the relation for the monatomic gases and, when combined with the ideal gas relation., we have e = 3 Rθ..5 In fact, one often takes.5 as the defining property of the temperature θ in terms of the internal energy e. The monatomic gases have the 3 degrees of freedom of translational energy. For the non-monatomic gases, there are other degree of freedom of rotational and vibrational energy, for instance. For a gas with α, α 3, degrees of freedom, we would have e = α Rθ..3. One-dimensional flows For plane waves, the density function fx,t, depends only on -dimensional space variable x, x = x,x,x 3. We will write x = x. The Boltzmann equation becomes t fx,t, + x fx,t, = Qf,fx,t,,.6 Note that the microscopic velocity =,, 3 must remain 3-dimensional. The Green s function Gx,t,;, x R, for the initial value problem, c.f..7, becomes { Gt + x G = LG for x R,.7 Gx,,; = δ xδ 3. Here we have highlighted the dimension of the delta functions. For convenience, we assume that the density distribution fx,t, is even in and 3..8 It is easy to see that if this assumption is made for the initial data, it remains so for later time. With this, the fluid velocity v = v,v,v 3 becomes and we will write the Maxwellians as v = v,, v,,,.9 M [ρ,v,θ] M [ρ,v,θ]..

11 ] SOLVING BOLTZMANN EQUATION, PART I: GREEN S FUNCTION 5 The thermo-equilibrium manifold, the collection of all Maxwellians, is then 3-dimensional. There are 3 effective collision invariants:,,, dimensional collision invariants.. 3. Linearized Boltzmann Equation Consider the solution f of the Boltzmann equation. as a perturbation of the Maxwellian M = M [ρ,v,θ] : f = M + Mg, g t + x g = Lg + Γg, Lg = Q Mg,M, linearized collision operator, M Γg = Q Mg, Mg, nonlinear term. M 3. The Boltzmann equation linearized around M with the weight function M is g t + x g = Lg Linear collision operator Any Maxwellian M = M [ρ,v,θ] is a thermo-equilibrium state, Q M, M =. The equilibrium manifold is a 5-dimensional manifold of the Maxwellians, parametrized by ρ, v, θ. As a consequence, the kernel of linearized collision operator Lg = Q Mg,M/ M is the tangent space of the equilibrium manifold at the base Maxwellian M of the linearization. This is found by the differentiation of the explicit form. of the Maxwellian state with respect to the 5 parameters ρ, v, θ. For instance the differentiation with respect to v yields M v = v Rθ M, which is, as a function of the microscopic velocity, in the span of {M, M}. In the form of the linearized variable g with the weight M, 3., it is in the

12 6 TAI-PING LIU AND SHIH-HSIEN YU [June span of { M, M}. The other differentiations yield altogether the span { M, M, M} of the tangent space, and we have { Lψj =, j =,...,4, ψ M, ψ i = i M, i =,,3, ψ 4 M, linear collision kernel. 3.3 There is also the algebraic way of finding the linear collision kernel, cf.., 3.3. Remark 3.. The Boltzmann Theorem says that the linear collision kernel is exactly the above 5-dimensional space, the linear equilibrium states. The study of the Boltzmann solutions contains the following two important considerations. The first is the tendency of the Boltzmann solutions converging to the local Maxwellians, as predicted by the H-Theorem. There is the Cercignani conjecture, [7] related to this efforts, see [], [6], [46], []. Then there is the study of the Boltzmann solutions flowing around the 5-dimensional equilibrium manifold. The study of the later is essential for the understanding of the relation of the kinetic theory with the fluid dynamics. On the linear level, we call the projection, in the function space of the microscopic variable, onto the linear collision kernel the linear macro projection. Its orthogonal projection is called the micro projection. These projections have been used for a long while, see for instance, in the content of energy method, the recent papers, [], [3], [4], and the hypocoercivity theory, [39]; its introduction for the above specific purpose was done in [3]. A good part of the present effort is to illustrate the Boltzmann flow from the perspective of these two basic considerations. For the hard sphere models,.5, and taking the base Maxwellian M for linearization to be M [,,] = e 3.4 π 3/

13 ] SOLVING BOLTZMANN EQUATION, PART I: GREEN S FUNCTION 7 the linearized collision operator has the explicit form, e.g. [9], Lg = ν + Kg νg + R K, 3 g d, ν e + + π e η dη, C + ν C + for some C, C >, K, π exp 8 8 exp The linearized collision operator around a Maxwellian M [ρ,v,θ] is a simple translation and dilation of the above. Here, as in what follows, we have taken, for simplicity, the gas constant R =. The Hilbert space L for functions of the microscopic variables will be used. We will also use the weighted spaces: g L,β sup R 3 g + β, β. 3.6 The integral operator K has some boundedness and smoothing properties. The following lemma follows from direct computations. Lemma 3.. The operator K is compact and the integral operators K, K, defined by the kernel K, K respectively, are bounded in L. For any β > there exist positive constants Cβ and C such that K j L,β+ Cβ j L,β, Kj L, C j L. 3.7 The integral operator K has limited smoothing property in, 3.7. This is due to the singularity of the kernel K, at =, 3.5. We

14 8 TAI-PING LIU AND SHIH-HSIEN YU [June decompose it into K = K + K = K,D + K,D, K i z R K 3 i, z d for i =,, K, = χ Dν K,, K, = χ Dν K,, χ r for r [,], suppχ [,], χ Cc R, χ. 3.8 Here the cutoff parameter D will be chosen to be small. We may write the linearized Boltzmann equation 3. as: t g + x g + νg = K + K g. 3.9 The operator K shares the same smoothing property as K and has strength of the order of the cut-off parameter D: K h L,β+ C β D h L,β for β. 3. The operator K is a smoothing operator because it does not inherit the singular nature of the kernel K, at =. Lemma 3.3. The operator K given in 3.8 is a smoothing operator in : for any h L, i, K h H i = O h L. Here H i are the standard Sobolev spaces on L. Proof. From the definition of K in 3.8, α K h h α χ K, d. R 3 Dν The function K, is smooth for > Dν. Thus, χ Dν K, is a globally smooth function. It is easy to see that for any i α the function α χ Dν K, L and therefore defines a bounded operator from L to L, and the lemma is proved.

15 ] SOLVING BOLTZMANN EQUATION, PART I: GREEN S FUNCTION 9 We have the following basic theorems, [4], [5], [9]. Theorem 3.4. The linearized collision operator L is non-negative self-adjoint in L and the Green s function G is bounded in L x L : G L x L. 3. Proof. The form of the linear collision operator with the weight of M serves the convenient purpose that the collision operator is self-adjoint in the Hilbert space L : Lg,h = g,lh, g,h = g,h L ghd. 3. R 3 This is shown by simple changes of variables in the integrations such as., Lg,h = g,lh = g 6 R R 3 3 S MM [ + g M h [ + h M M M g M g M ] h M h M ]BdΩd d, 3.3 making use of the fact that the Maxwellians have the defining property that log M is collision invariant, or, M M = MM. To prove the boundedness of the Green s function, we use the energy method by integrating the linear Boltzmann equation 3. times g to yield, using the non-negativeness of L: d g,g dt L dx = Lg,g R 3 L dx. 3.4 R 3 Thus the Green s function as an operator in propagating the solution of the initial value problem,.8,.9, is contractive in L xl and so 3. holds. 3.. Macro-micro projections One forms the orthogonal basis for the kernel of the collision operator

16 3 TAI-PING LIU AND SHIH-HSIEN YU [June 3.3: χ M /, χ i i v i M /, i =,,3, χ 4 6 v 3M /. 3.5 The macro projection P and the micro projection P are P g We will write 4 g,χ j χ j, P I P. 3.6 j= g P g, g P g, g = g + g, g,g =. 3.7 The macro projection P consists of the isotropic part P iso and the momentum part P m : P = P iso + P m, P iso g = g,χ χ + g,χ 4 χ 4, P m Clearly, the linear collision operator L satisfies g,χ i χ i. 3.8 i= P L = LP =, P L = LP = L; 3.9 and, from the Boltzmann Theorem, the nonlinear collision term Γg, 3., also satisfies P Γ =, P Γ = Γ. 3. Theorem 3.5. The spectrum σl of the linearized collision operator L has the following property: There is the eigenvalue with corresponding eigenfunctions the 5-dimensional collision invariants, 3.3. Besides the eigenvalues, the maximum value of the spectrum is a negative number ν. In other words, there is a spectrum gap between and the rest of the spectrum.

17 ] SOLVING BOLTZMANN EQUATION, PART I: GREEN S FUNCTION 3 Consequently, for any function g L. Lg,g ν P g,p g, 3. Proof. From 3.5, the collision operator can be viewed as the compact perturbation K of the multiplicative operator ν. From the Weyl s Theorem, the essential spectrum σ ess = {y : y ν } of L is the values taken by ν. As ν is an increasing function of, ν ν = min ν, 3. As L is self-adjoint and nonpositive, the discrete eigenvalues σ discrete L = σl\σ essential L ν,]. Therefore, there are finite eigenvalues in interval [ ν + ε,] for any small positive ε. In particular, the zero eigenvalue is isolated. Thus the maximum of the rest of the spectrum ν is negative, ν > ν >. Finally, 3. follows from the fact that the micro projection P projects to the orthogonal complement of the kernel of L. This establishes 3. and the theorem is proved. The above theorem is due to the efforts of Hilbert, Carleman, and Grad. For constructive estimates of the spectrum gap, see [], [37], [38]. Corollary 3.6. There exist positive constants C, C such that Lg,g C + P g,p g, 3.3 g,g C [g,g + Lg,g]. 3.4 Proof. From the expression L = K ν and the estimates, 3.5, ν D +, K L = D <, for some positive constants D, D, and so we have Lg,g = LP g,p g D + P g,p g + D P g,p g. 3.5 The estimate 3.3 is shown with C = D ν /D + ν by adding 3. times D /ν and 3.5. We have from the Cauchy-Schwarz inequality that g,g = g + g,g + g

18 3 TAI-PING LIU AND SHIH-HSIEN YU [June = O[ + P g,p g + + P g,p g]. Thus 3.4 follows from 3.3 if + P g,p g = OP g,p g, which holds trivially because, after the macro projection P, it is a finite dimensional situation. This completes the proof of the corollary Macroscopic variables The fluid variables for the conservation laws ρ g,m / = g,m /, m i g, i M / = g, i M /, m m,m,m 3, E g, M / = g, M /. 3.6 There is some ambiguity with earlier notations, as, for instance, the ρ here is not the density, but the perturbation of the density. There is no confusion here and we use these for the sake of notational simplicity. We will also use the projection to the orthogonal basis χ i, i =,,,3,4: ρ g,χ = g,χ, m i g,χ i = g,χ i, 3.7 Ē g,χ 4 = g,χ 4, g x,t, = ρx,tχ + m i x,tχ i + Ēx,tχ i= 4. Euler and Navier-Stokes For the discussion of this section, the mean free path k is retrieved to highlight the dependence of the hydrodynamics dissipation parameters, the viscosities and heat conductivity on k.

19 ] SOLVING BOLTZMANN EQUATION, PART I: GREEN S FUNCTION 33 Boltzmann equation has its fluid dynamics aspects. The most basic equations for gas dynamics are the Euler equations. The Euler equations in gas dynamics can be directly derived from the Boltzmann equation and provide the basic information on the propagation of fluid-like waves in the kinetic theory. The Navier-Stokes equations and the Boltzmann equation share the basic property that they are both dissipative. Boltzmann equation is dissipative partly as a consequence of the H-Theorem. The heat conductivity and viscosity coefficients for the Navier-Stokes equations are also the basic dissipation parameters for the Boltzmann solutions. We will derive the compressible Navier-Stokes equations in the spirit of Chapman-Enskog expansion. 4.. Euler equations Each Maxwellian is a constant solution of the Boltzmann equation,.. In general, local Maxwellians do not form a solution of the Boltzmann equation. As the mean free path k tends to zero, for smooth solutions, one expects Qf,f to go to zero: Qf,fx,t, = k[ t fx,t, + x fx,t,]. And so, by the H-Theorem,., f M f, where M f are the Maxwellians determined by the macroscopic variables of the solution f. We call M f t + x M f = 4. the Euler equations in the kinetic form. Assuming that the distribution function is locally Maxwellian f = M f, then, by direct calculations, it is easy to see that stress tensor P = pi, the pressure p; heat flux q =. Here I is the 3 by 3 identity matrix. Thus in this zero mean free path limit, k +, the conservation laws.5 are simplified to the Euler equations in

20 34 TAI-PING LIU AND SHIH-HSIEN YU [June gas dynamics t ρ + x ρv =, t ρv + x ρv v + pi =, t ρe + x ρve + pv =. 4. The state variables are related by the constitutive relations for the monatomic gases: p = Rρθ = 3 ρe = p e As ρ Here the first relation is the ideal gas law,.. We will take the gas constant R =. The second equality comes from the monatomic gases law.4. From these we have the last equality by the thermal dynamics relation de = pd/ρ + θds, with s the entropy, for any choice of positive constants A and p. The Euler equations are self-contained, as the stress tensor is now a given function of the conserved quantity and the heat flux is zero. So the number of the dependent variables is now = 5, the same as the number of equations. Similarly, the linear Euler equations are the projection of the linearized Boltzmann equation 3. to the tangent space of the equilibrium Maxwellian states manifold at the base Maxwellian M = M = M [ρ,v,θ ]. Thus the linear Euler equations in the kinetic form is the macro projection of the linearized Boltzmann equation, 3.9: P g t + x P P g =. 4.4 The resulting conservation laws, in terms of the macroscopic variables are the linear Euler equations in gas dynamics. There are two versions. The first is using the conservative macroscopic variables 3.6. This one is the same as the linearized version of the full Euler equations 4.: ρ t + x m =, m t + 3 xe =, 4.5 E t + 5 x m =.

21 ] SOLVING BOLTZMANN EQUATION, PART I: GREEN S FUNCTION 35 Here we have, for simplicity, assumed that the base state for the linearization is taken to be ρ =, v =, θ =. The second version is using the macroscopic variables through the orthogonal projections 3.7, 3.8: ρ t + v x ρ + x m =, m t + v x m + x ρ + 3 xē =, 4.6 Ē t + v x Ē + 3 x m =. 4.. Euler, Euler flux projections In the linear Euler equations 4.4, the flux operators P i P,i =,,3, is on the 5-dimensional space {χ j, j =,...,4}. For definiteness, we consider P P. By straightforward calculations, we have the following Euler characteristics λ i and Euler characteristic directions E i : P E j = λ j E j, j =,...,5, λ = v c, λ = v, λ 3 = v + c, λ 4 = λ 5 = v, E = 3 χ χ + 5 χ 4, E = 5 χ χ 4, E 3 = 3 χ + χ + 5 χ 4, E 4 = χ, E 5 = χ Here we have taken the base state to be ρ,v,θ and so the sound speed c is c = The Euler characteristic directions are orthogonal: 5θ E j,e k = for λ j λ k. This is seen easily as follows: λ j E j,e k = P E j,e k = E j,e k = E j, E k = E j,p E k = λ k E j,e k.

22 36 TAI-PING LIU AND SHIH-HSIEN YU [June The above Euler characteristic directions have been normalized: E i,e j = δ ij, i,j =,...,5, 4.9 so that the macro projection can also be written as P g = We define the Euler projections B j : 4 g,e j E j. j= B j g = g,e j E j, j =,...,5. 4. For the -dimensional Boltzmann equation,.6, the linearized Euler equations are P g t + P P g x =. 4. The kernel of the linearized collision operator is the span of Lψ j =, j =,,4, ψ M, ψ = M, ψ 4 M, -dimensional linear collision kernel, 4. and the macro and micro projections become, with v = v,,, P g g,χ χ + g,χ χ + g,χ 4 χ 4 ; P I P, χ M /, χ vm /, χ 4 6 v 3M /. The Euler characteristics are P E j = λ j E j, j =,,3, λ = v c, λ = v, λ 3 = v + c, 3 E = χ χ + 5 χ 4, E = 5 χ χ 4, 3 E 3 = χ + χ + 5 χ

23 ] SOLVING BOLTZMANN EQUATION, PART I: GREEN S FUNCTION 37 t dx dt = λ dx dt = λ dx dt = λ 3 x Figure : Euler waves. The Euler projections B j are defined as: B j g = g,e j E j, j =,, The initial value problem for the Euler equations { P g t + P P g x =, P gx,, = ḡx,, is decoupled by the Euler projections: g j g,e j, ḡ j ḡ,e j, g j t + λ j g j x =, g j x, = ḡ j x This can be solve by the characteristic method to yield the Euler waves, Figure : { gx,t, = 3 j= g jx,te j, g j x,t = ḡ j x λ j t. 4.8 The Euler flux projections B j are defined for nonzero Euler characteristic speeds λ j as follows: B j g = g,e j E j, j =,,3, P B j. 4.9 λ j j=

24 38 TAI-PING LIU AND SHIH-HSIEN YU [June It is easy to see that P = B j, P B i = B i P =, j= B i E j = δ ij E i, B i Bj = δ ij λ i Bi, i,j =,,3, P Bi =. 4. Although B i P =, it is not true in general that B i P =. For the study of initial-boundary and the boundary value problems, it is essential to differentiate the direction of the Euler waves. For this, we define the upwind Euler projection B + and the downwind Euler projection B : B + λ i > B i, B λ i <B i. 4. Similarly, the upwind-downwind Euler flux projections B ± are defined as B i. 4. B + λi> B i, B λi< 4.3. Navier-Stokes dissipation parameters The full Euler equations 4. are derived by assuming that the distribution function is locally Maxwellian, f = M f. For the derivation of the Navier-Stokes equations, the distribution function is assumed to deviates slightly from the local Maxwellian. For our purpose of the construction of the Green s function, we will consider the case of linear Navier-Stokes equations only. In the linear case, the Euler equations 4.6 are derived from the linear Boltzmann equation 3. by considering only the macro projection of the macro part g = P g of the Boltzmann solution g, 4.6. The macro projection is the same as considering the conservation laws 4.6 induced by the macro part. We now derive the linear Navier-Stokes equations by the same procedure. First we write the linear Boltzmann equation 3. as: g + g t + x g + g = k Lg, g P g, g P g.

25 ] SOLVING BOLTZMANN EQUATION, PART I: GREEN S FUNCTION 39 The macro and micro projections of this equation are g t + x P g + g =, macro part of linear Boltzmann equation; 4.3 g t + x P g + g = k Lg, micro part of linear Boltzmann equation. 4.4 In the spirit of Chapman-Enskog expansion, we assume that the macro part g dominates the micro part g and that the differential x,t h is dominated by the quantity h itself. Another way of thinking of this is to consider the time-asymptotic dissipation of the Boltzmann solution toward a given Maxwellian. As we will see, the macro part decays slower than the micro part. Thus we will make the simplification of the micro part 4.4 of the linear Boltzmann equation into x P g k Lg, or g kl [ x P g ] Chapman-Enskog relation. 4.5 Plug the Chapman-Enskog relation 4.5 into the macro part 4.3 of the Boltzmann equation: g t + x P g = k x P L [ x P g ]; or, g t + x P g = k x i x jp i L [P j g ], i,j= Navier-Stokes equations in kinetic form. 4.6 The linear Navier-Stokes equations in gas dynamics form are the conservation laws, obtained by integrating the kinetic equation 4.6 by the linear collision invariants, the kernel of the linear collision operator, 3.5, 3.6. The integration of 4.6 times χ yields the usual conservation of mass: ρ t + x m + v ρ =. The integration of 4.6 times χ 4 yields the conservation of energy: E t + v E + 3 x m = κ x E,

26 4 TAI-PING LIU AND SHIH-HSIEN YU [June with the heat conductivity coefficient κ: κ = k P χ 4,L P χ We now compute the integration of 4.6 times χ l, l =,,3, to yield the conservation of momentum and the viscosity coefficients: m l t Observe that i,j,n= = = = = + = k v i m l x i + ρ x l + E 3 x l i= i,j,n= m n x,t x i x j χl,p i L [ P j χ n ]. 4.8 m n x,t x i x j χl,p i L [ P j χ n ] i,j,n= i,j,n= i,j,n= j= + n l + j l m n x,t x i x j P i χ l,l [ P j χ n ] m n x,t x i x j P i [ l v l M,L P j n v ] n M m n x,t x i x j P i l [ M,L P j n ] M m j x,t x l x j P l [ M,L P j ] M i=l, n=j m n x,t x n x l P n l [ M,L P l n ] M j =l, i=n m l x,t x j x j P j l [ M,L P j l ] M n=l, i=j. Due to the rotational invariance of the L and integration, for i j, P i j M = P i M [,L P i j ] M,L [ P j M ]

27 ] SOLVING BOLTZMANN EQUATION, PART I: GREEN S FUNCTION 4 and P i M = = = [,L P i ] M P i + j M,L [ P i + j M ] P i + j M + i j, [ L i + j ] M P + i j P i [ M,L P i ] M + 3 which implies that P i j M P i M = 3 With these, 4.8 becomes m l t + i= P i j M v i m l x i + ρ x l + = 3µ m l x,t x l x l +µ = µ = { m l x,t j= j= x j x j { µ j l,l [ P i j M ], [,L P i ] M E 3 x l,l [ P i j M m j x,t x l x j + n l } + m j x,t x l x j m l x j + m j x l 3 δ jl n= m n x n m n x,t x n x l + j l µδ jl Here the viscosity µ and the bulk viscosity µ B are given as ]. 4.9 n= m l x,t x j x j m n x n µ k P [ M,L P ] M, µ B 5 µ Using 4.7 and 4.3 the linear Navier-Stokes equations in gas dynamics }

28 4 TAI-PING LIU AND SHIH-HSIEN YU [June form are then: ρ t + x m + v x ρ =, m l t + 3 v i m l + ρ + x i= i x l 3 = 3 m l j= x j {µ x j E t + v x E + E x l + m j x l 3 δ jl n= 3 x m = κ x E. m n x n 5 3 µδ jl n= } m n x, l=,,3, n 4.3 Another form of the Navier-Stokes equations that are convenient for the study of fluid waves are obtained by integrating the kinetic equation 4.6 times the Euler characteristic directions. We do this for -D case so that the Navier-Stokes equation in the kinetic form is g t + x P g = k x P L [ P g ]. 4.3 Express g = g j E j j= in terms of the Euler characteristic directions E j, j =,,3, 4.4. This yields the Navier-Stokes equations in gas dynamics form: g i t + λ i g i x = 3 A ij g j xx, i =,,3, j= A ij = k P E i,l [ P E j ] The dissipation parameters matrix A ij are related to the viscosity µ and heat conductivity κ. For instance, from 4.4, k A = P E,L [ ] P E = 5 P 3 χ 5 χ + χ 4,L [ P 3 χ 5 χ + χ 4 ] Note that χ is a linear collision invariant and so P χ =, and that χ 4 is odd in and χ is even in and so P χ,l χ 4 =. Thus

29 ] SOLVING BOLTZMANN EQUATION, PART I: GREEN S FUNCTION 43 it follows from 4.7, 4.3, and 4.9 that A = k 5 = 3 µ + 5 κ. Similarly, we have { 5 P χ,l [ P ] χ + P χ 4,L } P χ 4 A = kp E,L [P E ] 3 = P χ χ + χ 4,L [ P ] 3 χ + χ 4 3 = P χ 4,L 3 P χ 4 = 5 5 κ, A 3 = k P E,L [ P E 3 ] = P χ χ +χ 4,L [ P 3 5 ] χ + χ +χ 4 = { 5 P χ,l [ P ] χ + P χ 4,L } P χ 4 5 = 3 µ + 5 κ, A = k P E,L [ P E ] P = k χ + χ 4,L [ P 3 χ + χ 4 ] = k 3 P χ 4,L 3 P χ 4 = 5 5 κ, A 3 = k P E,L [ P E 3 ] 3 = k P 5 3 χ + χ 4,L [ P 3 5 ] χ + χ + χ 4 3 = k P χ 4,L 3 P χ 4 = 5 5 κ, A 33 = k P E 3,L [ P E 3 ] = k P χ + χ +χ 4,L [ P 3 5 ] χ + χ +χ 4 = k { 5 P χ,l [ P ] χ + P χ 4,L } P χ 4 5 = 3 µ + 5 κ.

30 44 TAI-PING LIU AND SHIH-HSIEN YU [June Due to the symmetry of A ij we have [A ij ] = 3 µ + 5 κ 3 5 κ 3 µ + 5 κ 3 5 κ 3 5 κ 3 5 κ 3 µ + 5 κ 3 5 κ 3 µ + 5 κ 4.34 where the viscosity µ and heat conductivity κ are given by 4.3 and 4.7. The dissipation parameters matrix [A ij ] is non-negative, and but not positive definite. This is typical for systems of viscous conservation laws with physical viscosity matrix, [5]. It follows from the general theory for the system of hyperbolic-parabolic equations that the dominant dissipation waves are given by the diagonal system, [35]: ḡ j t + λ j ḡ j x = A j ḡ j xx, j =,,3, 4.35 with the dissipation parameters A A = k P E,L [ P E ] = 3 µ + 5 κ, A A = k P E,L [ P E ] = 3 5 κ, A 3 A 33 = k P E 3,L [ P E 3 ] 4.36 = A = 3 µ + 5 κ. The Green s function { Ḡ j t + λ j Ḡj x = A j Ḡj xx, j =,,3, Ḡx, = δx, 4.37 consists of dissipation waves for this simplified system are the collection of heat kernels Hx,t;A, Figure, c,f, Figure : Ḡ j x,t = Hx λ j t,t;a j, 4.38 Hx,t;A 4πAt e 4At. x In the kinetic form, 4.3, 4.33, this is translated into Ḡx,t = Hx λ j t,t;a j E j E j, 4.39 j=

31 ] SOLVING BOLTZMANN EQUATION, PART I: GREEN S FUNCTION 45 dx dt = λ dx dt = λ t dx dt = λ 3 x Figure : Navier-Stokes leading waves. containing the operator in the function of the microscopic velocity: E j E j g E j,ge j D Green s Function, Fluid-Like Waves In this and the following two sections, we will construct the Green s function for the initial value problem for the linear Boltzmann equation. This and the next sections will concentrate on the -dimensional linearized Boltzmann equation, cf. 3., g t + x g = Lg. 5. For simplicity, we will take the base Maxwellian for linearization, 3. to be M = M M [,,] so that the explicit expression of the linearized collision operator is given in 3.5 and the Euler characteristic values are 5 λ = 3, λ =, λ 3 = 5 3. The -D Green s function Gx,x,t,s,; = Gx x,t s,; satisfies.7: { Gt + x G = LG for < x <, t >, 5. Gx,,; = δxδ 3. The Green s function Gx y,t s,; describes the propagation of the perturbation over the Maxwellian M when at time σ the perturbation con-

32 46 TAI-PING LIU AND SHIH-HSIEN YU [June sists of particles concentrated at space x = and with microscopic velocity. Consider the initial value problem { gt + x g = Lg, gx,, = g x,. 5.3 Multiply the equation with the Green s function and integrate to yield the solution representation for 5.3: gx,t, = Gx y,t,; g y, d dy locally in x,t,, R R gx,t G t g x locally in x,t and in a semi-group format. 5.5 Here the formula 5.4 is for pointwise expression; 5.5 is to view the Green s function as an operator. Consider the Fourier transform in the space variable x: ĝη,t e iηx gx,tdx, gx,t e iηx ĝηdη. π π R Take the Fourier transform of.7 to obtain Ĝ t + i ηĝ = LĜ, < η <, t >, Ĝx,,; = π δ 3. R 5.6 With this, one can use the isometry property of the Fourier transform in the L x to study the dissipation property of the Boltzmann solutions. Instead, we will invert the Fourier transform for the fluid-like part of Ĝ, to gain quantitative information on the Green s function in the physical space variable x. The formal expression of the Green s function is Gx,t,; = e iηx+ iη+lt δ 3 dη. 5.7 π Plug this into the solution formula 5.4 to yield gx,t, = R R [ ] e iηx y+ iη+lt dηg y, dy. 5.8 π R

33 ] SOLVING BOLTZMANN EQUATION, PART I: GREEN S FUNCTION 47 Thus in the operator expression, c.f. 5.5, we can write Gx,t e iηx+ iη+lt dη. 5.9 π R Remark 5.. The operator i η + L on functions of the microscopic velocity is the focus of our study for this section. As the operator has the one-dimensional Fourier parameter η, we do not have complete information of its spectrum. Nevertheless, the point spectrum for η near origin has explicit expression and correspond to the fluid-like waves, similar to that for the compressible Navier-Stokes equations. These waves dominate the Green s function time-asymptotically. The construction and description of these waves is the main concern of this section. To obtain the complete description of the Green s function, there are two further steps to take. The first is to construct the singular waves through a series of Picard iterations, done in Section 6. These two constructions of waves offers two distinct decompositions of the Green s function, and allows us finally to apply the soft analysis of weighted energy method and Sobolev calculus toward the end of Section 6 for the complete pointwise description of the Green s function. 5.. Spectral near origin From the expression 5.9 for the Green s function, it is natural to consider the spectrum of the operator i η + L. We consider first the situation near the origin, η small. Let σ,e = ση,eη be the eigenvalueeigenfunctions for i η + L: i η + Le = σe. 5. We know from Theorem 3.5 that the zero eigenvalue of the linear collision operator L is isolated and with multiplicity 3, 4.. Thus for η the spectrum of i η + L should be also be 3 eigenvalues near zero. The following Lemma 5.4 provides the spectrum information we need. Statement I III of Lemma 5.4 are true for hard potentials with Grad cutoff. However, the analytic property, IV, is only true for the hard sphere

34 48 TAI-PING LIU AND SHIH-HSIEN YU [June model, see Remark 5.5. Since analyticity is required for the following analysis, we will assume the hard sphere model throughout. Still, for the sake of completeness, we point out the fact that I III are generally true for hard potentials with Grad cutoff. Definition 5.. For any complex φ,ψ L, define the pseudo inner product as [φ,ψ] = φψd. 5. R 3 Remark 5.3. Note that we do not take complex conjugate in 5., so [, ] is not an inner product. Since iη +L is the sum of two multiplicative operators, iη, ν, and an integral operator K, iη + L, is symmetric with respect to [, ]. Namely, [ ] [ ] φ, iη + Lψ = iη + Lφ,ψ. 5. Consequently, if σ i σ k, [e j,e k ] =. Lemma 5.4. Consider the spectrum Specη of the operator i η+l, η R. For hard potentials with Grad cutoff, the following statement I III are true. I For any < δ, there corresponds τ = τδ > such that i For η > δ, Specη {z C : Rez < τ}. ii For η δ, the spectrum within the region {z C : τ Rez} consisting of exactly three eigenvalues σ η,σ η,σ 3 η: Specη {z C : τ Rez} = {σ η,σ η,σ 3 η}. II For η, the eigenvalues σ η,σ η,σ 3 η satisfy σ η = iηλ A η + O η 3, σ η = iηλ A η + O η 3, σ 3 η = iηλ 3 A 3 η + O η 3, 5.3

35 ] SOLVING BOLTZMANN EQUATION, PART I: GREEN S FUNCTION 49 cf. Figure 3, where A j = P E j,l P E j is the Navier-Stokes dissipation coefficient, cf In view of 5.3, for < η, the eigenvalues σ η,σ η,σ 3 η are distinct. After being normalized according to [e j,e k ] = δ jk, cf. Remark 5.3, the eigenvectors e η,e η,e 3 η take the following form: A jk e j η =E j + iη E k + e j + O η, λ k λ j k j e j =L P E j, 5.4 where A jk = P E j,l P E k is the Navier-Stokes dissipation coefficient, cf III For all < δ, the semigroup e iη+lt can be decomposed as e iη +Lt = Π δ + χ { η <δ} e zt dz, z iη + L 5.5 πi Γ where Π δ L = Oe aτt, aτ > depends on τ and therefore on δ, χ { } is the indicator function, and Γ can be any close curve that lies entirely on {Re z > τ} and that encloses the three eigenvalues σ η,σ η,σ 3 η, Figure 3. IV Particularly, for the hard sphere model, σ j η,e j η are holomorphic in η for all η. Consequently, 5.3 and 5.4 hold even for complex η. Proof. The spectrum of i η + L has been studied and this lemma has been proved, [4], [36], [4], [45], by making use of the spectral gap at origin for L, Theorem 3.5. Here we emphasize the computation of eigenvalues and eigenvectors, 5.3 and 5.4, and prove II only. Apply Macro-Micro projection to the equation i η + Le j = σ j e j : iηp P e j + P e j = σ j P e j, 5.6a iηp P e j iηp P e j + LP e j = σ j P e j. 5.6b In view of 5.6a, it is convenient to set σ j = iηγ j. From 5.6b we can solve P e j in terms of P e j as P e j = iη[l iηp iηγ j ] P P e j.

36 5 TAI-PING LIU AND SHIH-HSIEN YU [June Re z = τ Im σ 3 η Γ σ η Re σ η Figure 3: Spectrum near origin. Substituting this back to 5.6a, we obtain the following equation for P e j : P + iηp L iηp iηγ j P P e j = γ j P e j. 5.7 Through Macro-Micro decomposition, we have transformed the original infinite dimensional eigen-equation into the 3 dimensional equation 5.7. Our next step is to solve 5.7 by the implicit function theorem. We will do this only for e, since e,e 3 can be treated similarly. Instead of normalizing e according to [e,e ] =, we temporarily adopt the normalization e to e by E,e =. Set Under this setup, 5.7 becomes P e = E + β E + β 3 E 3. G γ,β,β 3 ;η λ + iη P L iηp iηγ j P E + β E + β 3 E 3,E +γ =, G γ,β,β 3 ;η λ β + iη P L iηp iηγ j P E + β E + β 3 E 3,E +γ β =,

37 ] SOLVING BOLTZMANN EQUATION, PART I: GREEN S FUNCTION 5 G 3 γ,β,β 3 ;η λ 3 β 3 + iη P L iηp iηγ j P E + β E + β 3 E 3,E 3 +γ β 3 =. 5.8 This equation has a solution γ,β,β 3 ;η = λ,,;. Moreover, G,G,G 3 γ,β,β 3 = λ λ,,, λ. λ 3 λ Consequently, by the implicit function theorem, for η, 5.8 has the solution γ η,β η,β 3 η, smooth in η, with γ,β,β 3 = λ,,. Differentiating 5.8 with respect to η yields γ η = i P L P E,E = ia, β η = i P L P E,E = ia, λ λ λ λ β 3 η = i P L P E,E 3 = ia 3. λ 3 λ λ 3 λ Hence σ η = iηλ A η +Oη 3. Moreover, since P e = iη[l iηp iηγ ] P P e, e A η = E + iη E + A 3 E 3 + L P E + Oη. λ λ λ 3 λ We now normalize e : put e = e [e,e ]. Note that [e,e ] is generally a complex number. By some direct computation, [e,e ] = + Oη. We unambiguously refer to [e,e ] as the complex square root closer to. Under this choice, [e,e ] is smooth in η, and, in the case of IV, holomorphic in η. Also we have 5.4: e η = + Oη A E + iη E + A 3 E 3 + L P E λ λ λ 3 λ + Oη

38 5 TAI-PING LIU AND SHIH-HSIEN YU [June A =E + iη E + A 3 E 3 + L P E + Oη. λ λ λ 3 λ Remark 5.5. The linear collision operator L = ν + K has the property that ν + α, where α = for the hard sphere models and < α < for Grad cut-off hard potentials. The analyticity of the eigenvalues σ i η holds only for the hard sphere models because of this. To see that, note that γ + λ iη = E,P [L iηp + iηλ + ρ] P E + b The key is the behavior at = : n η [L iηp + iηλ + ρ] n! n ν n e, for hard sphere model. Hence, γ + λ iη n= nn + n! n +! η n+ iλ η + η n+, n= which converges as η. On the other hand, for hard potentials γ + λ iη n= n! n α e d Γn αη n+, which diverges for any < α < and η. Thus the complex analytic method that we will use to obtain the explicit expression of the inverse Fourier transform for the hard sphere models cannot be applied to the hard potential models. There is the study for the hard potential models, [8], which yields a version of the pointwise estimates for the Green s function weaker than that of Theorem 5.9 for hard sphere models. The real analytic method used in [8] is motivated by [9] for viscous conservation laws and [] for the boundary layers. It would be interesting to obtain, for the hard potential models, similar estimates as for the hard sphere models in Theorem 5.9. n= We now proceed to compute explicitly the second term of 7..

39 ] SOLVING BOLTZMANN EQUATION, PART I: GREEN S FUNCTION 53 Definition 5.6. For φ L, e j [e j φ [φ,e j ]e j. Lemma 5.7. For η, e zt dz z iη + L = e σjt e j [e j. 5.9 πi Γ j= Proof. Define a subspace H η of L as H η {φ : [φ,e j η] =, for j =,,3}. Let be the linear span. Clearly, L = e,e,e 3 H η, 3 e j [e j is the projection onto e,e,e 3 along H η, and 3 e j [e j is the projection onto H η along e,e,e 3. Moreover, since e j is an eigenvector of iη +L and iη + L is symmetric with respect to [, ], 5., both e,e,e 3 and H η are z iη + L invariant: z iη + L e,e,e 3 e,e,e 3, Set S η = iη + L e j [e j, T η = j= z iη + L H η H η. iη + L j= 5. e j [e j. As a consequent of 5., z S η : e,e,e 3 e,e,e 3, z T η : H η H η, and z iη + L = z S η + z T η. S η becomes a 3 by 3 diagonal matrix under the basis {e,e,e 3 }, so by some direct computations: e zt z S η dz = πi Γ e σjt e j [e j. j= It remains only to show Γ ezt z T η dz =. First, since iη +L

40 54 TAI-PING LIU AND SHIH-HSIEN YU [June [ + + Oην + K], for η, iη + L = L + Oη, 5. where Oη symbolizes an operator with Oη L = Oη. Next, for each φ H η, [φ,e j η] = = [φ,e j ] + [φ,oη] = φ,e j + φ Oη = φ,e j = φ Oη. Therefore, P φ = Oη φ. Combine this fact with 5. and Lemma 3.5 to obtain T η φ L = φ L L L + Oη P φ + φ Oη φ L ν P φ L + Oη φ L φ L = ν + Oη. Consequently, Tη is uniformly bounded for η. Since Γ encloses σ,σ,σ 3 and σ j = Oη, in the region enclosed by Γ we may assume z η. Under this assumption, z T η = Tη + zt η is holomorphic in z and therefore Γ ezt z T η dz=. zt η 5.. Fluid-like Waves Plugging 7. and 5.9 to 5.7, we have Gx,t,; = π F Π δ + j= δ e ixη+σjηt e j η [e j η dη, π δ where F symbolizes the inverse Fourier transform. Since F : L η L x is an isometry, F Π L δ x L = t Oe Cδ, 5. for some Cδ > depending on δ.

41 ] SOLVING BOLTZMANN EQUATION, PART I: GREEN S FUNCTION 55 Definition 5.8. Define the fluid-like waves as G L x,t,;δ j= δ e ixη+σjηt e j η [e j η dη. 5.3 π δ To highlight the dependence on t, we will frequently abbreviate G L x,t,; δ as G t L. Our goal of this subsection is to the extract leading terms, the main fluid-like waves, from G L. Theorem 5.9. For any fixed C j > A j, j =,,3, there exists C > and δ > such that jt G L x,t,;δ e x λ 4A j t 4πAj t + E j E j j= e = O t + j= jt x λ 4C j t L + e t C. 5.4 x. Note that here our estimate is pointwise in x, unlike, which is L in Proof. Pick C j > C j > C j > A j. We then fix some δ >, so small such that whenever η < δ: σ j,e j are holomorphic in η. 5.3, 5.4, and Lemma 5.7 hold. Moreover, { } σ j η iλ j η A j η < min A j C j, A j η Because of the analyticity property of the eigenvalues and eigenfunctions, we can apply the complex analysis method to compute explicitly the inverse Fourier transform by the contour integral. This will yield the fluidlike behavior of the Green s function. By Cauchy integral theorem, we can substitute the path [ δ,δ] by any other path, lying entirely in the analytic region { η < δ}, with the same endpoints. We choose the contour

42 56 TAI-PING LIU AND SHIH-HSIEN YU [June Im Imη = c Γ Γ Re η = δ Γ 3 Re η = δ Re Figure 4: Path of integration. Γ + Γ + Γ 3, Figure 4, Γ = Γ c {η : Reη = δ, Im η lies between and c}, Γ = Γ c {η : δ Reη δ, Im η = c}, Γ 3 = Γ 3 c {η : Reη = δ, Im η lies between and c}, 5.6 where c is a constant, specified later. On Γ, put Reη = u so that η = u+ic. Since e ixη+σ jηt = exp [ A j t η i x λ ] jt x λ jt + Oη 3 t, A j t 4A j t it is desirable to set c = x λ jt A j t. However, in order that Γ + Γ + Γ 3 lies entirely in { η < δ}, c cannot be arbitrarily large. For this reason, we will process the integral on Γ in two separated situations. Case : x λ jt A j t < δ In this case, set c = x λ jt A j t. On Γ, By 5.5, e e ixη+σ jηt = e jt x λ 4A j t e A jtu + e x λ j t 4A j t e A jtu e O η 3 t. jt x λ 4A j t e A jtu { O η 3 t, for η 3 t < e O η 3 t, for η 3 t > = Oe jt x λ 4A j t e A jtu e O η 3 t η 3 t }

43 ] SOLVING BOLTZMANN EQUATION, PART I: GREEN S FUNCTION 57 [ { = Oexp x λ jt A j tu + min A j 4A j t C j ] x ta j u λj t + η 3 t A j t [ = Oexp x λ jt A j tu + 4A j t u 3 t + x λ jt 3 = Oe x λjt 4C j t = π t e A j tu 3 u + x λ j t t A j C j. Therefore, e ixη+σjη e j η [e j η dη π Γ δ jt x λ 4A e j t e u ta j E j E j + η O du δ + δ e x λjt 4C π j t δ x λ j t = e 4A j t 4π = +e jt δ x λ 4C j t π δ δ δ e u t A j 3 e u t A j 4 u + x λ j t t e A jtu du E j E j t + Odu, 3 x λ j t 4A j t } ] + A jtu 3 e j η [e j η du x λ j t x λ j t e 4A j t 4πAj t E j E j + e 4C j t δ Aj t O e u du t + t π δ A j t = jt x λ 4A e j t 4πAj t + E j E j +e jt x λ 4A j t x λ j t + e 4C j t t + [ δ Aj t e u du t π δ A j t δ Aj t e u du O t δ A j t t + ] O

44 58 TAI-PING LIU AND SHIH-HSIEN YU [June = jt x λ x λ j t 4A e j t 4πAj t + E j E j + e t + 4C j t O. Case : x λ jt A j t > δ In this case, set c = δ sign x λ jt. Note that in this case we have x λ j t > δa j t. On Γ, by 5.5, e ixη+σ jηt [ exp ix λ j tη A j tη + A j t] 3 η [ = exp x λ j t δ δ + A jt 4 u [ ] δ exp A j t δ = Oe t C. 3 + A jt 3 δ 4 + u ] Consequently, e ixη+σjη e j η [e j η dη = Oe t C. π Γ For both cases, on Γ,Γ 3, Im η takes the same sign as x λ j t. Therefore, e ix λ jtη. Moreover, for both cases, Im η c δ. Combining this fact with 5.5, we have e ix λ jtη η A j t+oη 3 t exp [ A j t = Oe t C. δ δ + A jt 4 3 ] δ + δ 4 This implies e ixη+σjη e j η [e j η dη = Oe t C. π Γ or Γ 3 Remark 5.. The computation of the inverse Fourier transform is an essential part of the study of the Green s function here. The approach goes back to Zeng 994, [48], Liu-Zeng 997, [35] for the viscous conservation laws. This is generalized to the Boltzmann equation, Liu-Yu 4 [3]. For

45 ] SOLVING BOLTZMANN EQUATION, PART I: GREEN S FUNCTION 59 the generalization to the 3-D case, Liu-Yu 6 [3], to be presented later, the inversion of the Fourier transform using the complex analytic method requires additional thinking. There is an essential difference between Green s function for the viscous conservation laws and that for the Boltzmann equation in that the former contains heat kernel with singularity, c.f. 4.39, while the later s singularity resides not in the heat kernels, 5.4. Instead, the singularity for the Boltzmann equation is contained in the essential kinetic waves that will be constructed in the next section Scale separations Boltzmann equation has much richer wave phenomena than the equations for the fluid dynamics. We illustrate here a basic separation of scales property, that the micro part decays at a faster rate than the macro part. Theorem 5.. For any fixed C j > A j, j =,,3, there exists C and δ such that jt e x λ 4C j t G L x,t,;δp L, P G L x,t,;δ L =O + e t C, t + j= 4C e j t P G L x,t,;δp L =O t + 3 j= jt x λ 5.7a + e t C. 5.7b Proof. Since P E j E j = E j E j P =, Theorem 5.9 immediately implies 5.7a. For 5.7b, after referring to the proof of Theorem 5.9, one finds that: In Case, e ixη+σjη e j η [e j η dη = Oe t C, π Γ or Γ 3 In Case, e ixη+σjη e j η [e j η dη = Oe t C. π Γ +Γ +Γ 3

46 6 TAI-PING LIU AND SHIH-HSIEN YU [June Now that P Oe t/c P = Oe t/c, it remains only to consider Case : e ixη+σjη P e j η [e j η P dη. π Γ P e j = Oη implies P e j η [e j η P = Oη. Therefore, e ixη+σjη P e j η [e j η P dη = e ixη+σjη η dηo π Γ Γ δ x λj t = = e = e e x λjt 4C j t δ jt δ x λ 4C j t x λ j t 4C j t t + 3 δ O. e u t A j 3 u + A t j t + e u 4 duo t duo 6. -D Green s Function, Particle-Like Waves The Boltzmann equation is the meso-scopic equation, being between the microscopic interacting particle systems and the macroscopic fluid dynamics equations. In the last section we have shown that the fluid-like waves for the Boltzmann solution have as their leading terms closely related to the Euler and Navier-Stokes waves. In this section we finish the construction of the Green s function by considering the short waves in the Green s function. However, because the spectrum the short waves represent is not explicit, the construction is indirect and elaborate, starting with the construction of the singular waves. These waves contains particle wave, particle-like waves, essential kinetic waves and a series of increasingly smooth waves. We now outline the definition of these waves. For this, we need the following solution operators, c.f. 3.8: the solution operators of the equa- Definition 6.. Denote by S t and O t D tions { ht + x h + νh =, hx, = h x, x R, 6.

47 ] SOLVING BOLTZMANN EQUATION, PART I: GREEN S FUNCTION 6 hx,t S t h x, and { jt + x j + νj = K j, jx, = j x, x R, jx,t O t Dj x. 6. These operators are studied in the first subsection. For the complete construction of the singular waves, the first step is to use the damped transport operator 6. and the essential kinetic operator 6. to extract the particle-like waves from the Green s function. Recall the Green s function,.7, G t + x G + νg = KG, x R, R 3, Gx,,; = δxδ 3, x R. The first term is the particle waves defined by { h t + x h + νh =, h x, = δxδ We then define the particle-like waves h j, j =,,..., as follows: We have G h t + x G h + νg h = KG h + Kh, G h x,,; =. Thus we define the second term h by { h t + x h + νh = Kh = KS t δxδ 3, h x, =. 6.4 In general, the particle-like waves are defined through the Picard iterations as: { h j t + x h j + νh j = Kh j, h j x, =, j =,,

48 6 TAI-PING LIU AND SHIH-HSIEN YU [June The Green s function minus these particle-like waves k g k G h i 6.6 i= satisfies { gkt + g kx = Lg k + Kh k, g k x,, =. 6.7 The source Kh k is bounded for k, Lemma 6.9; and as it turns out, this is so also for the 3 D case. We next use the essential kinetic operator O t D, 6., 3.8, to construct an essential kinetic wave h: { ht + hx + ν h = K h + Kh, hx,, =. 6.8 The function h is named the essential kinetic wave as it is localized. Moreover, the new source K h is now smooth in microscopic velocity, Lemma 3.3: G h h h h t + G h h h h x = LG h h h h + K h, 6.9 G h h h hx,, =. With the source smooth in microscopic velocity, we use the Picard iterations, c.f. 6.5, again: gt + x g + νg = K h, g j+ t + x g j+ + νg j+ = Kg j, 6. g j x, =, j =,,,.... This is the final step in the construction of the singular waves. It is shown by a Mixture Lemma that the smoothness of the source in induces an increasingly more smooth functions g j, as j increases. The construction in the last section of the fluid-like waves and the construction of singular waves in this section offer two decompositions of the Green s function. This allows us to apply the energy method and Sobolev

49 ] SOLVING BOLTZMANN EQUATION, PART I: GREEN S FUNCTION 63 calculus to gain global pointwise analysis of the Green s function. 6.. Essential kinetic operators The operator S t, 6., is a damped transport equation and is solved by the characteristic method: S t h x, = e νt h x t,. 6. From this we have immediately the following lemma. Lemma 6.. For any β, the operator S t satisfies S t L x L,β e νt, S t L x L e νt. 6. The operator O t D, 6., captures the singular part of the linear Boltzmann operator. Lemma 6.3. There exist positive constants D and C such that for any D,D the operator O t D satisfies O t D L x L C e ν t/. Proof. First, we regard O t D as an operator on L x L, and consider the initial value problem { jt + j x + νj K j = jx, g x. Take the Fourier transform to result in FO t D g η = ĵη,t = e i η ν+k tĝ η. Since the operator K, 3.8, is symmetry and K L = OD, the real part of the spectrum of the operator i η ν + K is in {z C : Rez ν + OD}. Thus we may choose D sufficiently small so that

50 64 TAI-PING LIU AND SHIH-HSIEN YU [June ν + OD < ν /. This implies that there exist positive constants C, D such that, for D,D From this e iη ν+k t L C e ν t/ for any η R. O t Dg L xl = FOt Dg L ηl C e ν t ĝ L ηl =C e ν t g L x L. This lemma results in the existence of the operator O t D in the functional space L xl and global decaying rate in time. We next use the Picard s iteration to analyze the operator O t D in the sup norm. Lemma 6.4. The operator O t D is a bounded operator on L x L,β for any β. There exist positive constants D, C such that, for any D,D, O t D g L x L,β C e ν t/ g L x L,β. Proof. From Lemma 3. This and Lemma 6.3 yield t K h L,β+ C β D h L,β for β. sk S t s K S s s K S s s 3 K S s k s k+ K S s k+ ds k+ ds L x L,β C β k+ D k+ e νt/. Thus, Picard s iteration gives a convergent geometric sequence in L x L,β for sufficiently small D > : O t D = S t + + t t S t s K S s ds + sk t s S t s K S s s K S s ds ds + S t s K S s s K S s s 3 K S s k s k+ K S s k+ ds k+ ds +, 6.3

51 ] SOLVING BOLTZMANN EQUATION, PART I: GREEN S FUNCTION 65 and the lemma follows. The following lemma yields the significant hyperbolic property of the operator S t and O t D. We will use the crucial property that, for the hard sphere model, 3.5, ν ν as and so, for some positive constant ν, ν = O +, ν t νt for R Lemma 6.5. For any given β, there exists sufficiently small D > such that S t g x L,β [ Oe ν t/3 O t Dg x L,β [ Oe ν t/ max g y L y x <t,β + max e ν y x /3 g y L y x >t,β max g y L y x <t,β + max e ν y x /4 g y L y x >t,β where ν is given in 3. and ν in 6.4. ], 6.5 ], 6.6 Proof. We use the representation 6. for S t. For, we have from 6.4, S t g x, e νt + β g x t, L,β From 6.4, for > e ν t/3 + β max y x <t g y L,β. 6.7 S t g x, e νt/3 ν t/3 + β g x t, L,β e ν t /3 ν t/3 + β g x t, L,β max e ν x y /3 ν t/3 + β g y L y x >t,β. 6.8 The estimate 6.5 follows from 6.7 and 6.8. From the construction of O t D in Lemma 6.4, one can view Ot D as a small perturbation of St. From

52 66 TAI-PING LIU AND SHIH-HSIEN YU [June 3. and 6.5, one has that t sk L S t s K S s s K S s s 3 K S s k s k+ K S s k+ g ds k+ ds [ ] C β D k+ e ν t/ max g y L y x <t,β + max y x >t e ν y x /4 g y L,β.,β 6.9 Thus, the Picard s iteration in 6.3 converges for sufficiently small D > and 6.6 follows. 6.. Particle-like waves In this section we compute and estimate the particle-like waves h,h,h, 6.3, 6.4. First, by 6., we have h x,t,; = e νt δ x tδ From 6. we compute h as the following: Lemma 6.6. Set t t x t. We have e νt t ν t K, h x,t,; =, for x t x t <,, otherwise. 6. In particular, h e ν t ν x x,t,; = O K,, for x t t,, otherwise. 6. Proof. By the Duhamel s principle, 6.4, and 6., we have h x,t;, = t e ν s e νt s K, δx t s sds. 6.3 The above integral is zero unless the characteristic line through x, t and

53 ] SOLVING BOLTZMANN EQUATION, PART I: GREEN S FUNCTION 67 s dy ds = x, t dy dt = s = t y Figure 5: Characteristics. with speed intersects with the source δx s at some time s = t during the time period,t, Figure 5, x t t = t. This yields the expression 6.. Note that we have x = t t + t t t + t, and so the estimate 6. follows from the linear growth ν + of the function ν, 3.5. Lemma 6.7. Kh x,t, = Oe ν t+ x e 3 + log + log where χ { } is the indicator function. x t t χ j x t t ff <, 6.4 Proof. From the explicit expression 3.5, we obtain K, = O e

54 68 TAI-PING LIU AND SHIH-HSIEN YU [June Combining this with the estimate 6. for h, we have Kh x,t, = Oe ν t+ x > x t t = Oe ν t+ x e + 3 x > t t = Oe ν t+ x e 3 > x t t 9 e 9 6 e e = Oe ν t+ x e D + Oe ν t+ x e 3 I + I, D e 9 d e 9 6 e d d where D { > > x t }, D { t > }. We further split I into three parts: I = A + A + I + I + I 3, A 3 e 9 6 e 9 6 d where A { D : < & < }, A { D : < & > }, A 3 D \ A A.

55 ] SOLVING BOLTZMANN EQUATION, PART I: GREEN S FUNCTION 69 For I, I d A π x t t x t t 4π z r z + r dθdrdz = log z dz = 4π z log where we use the cylindrical coordinates: x t t d A x t t, 4π z r z + r drdz r = + 3 3, z =, θ = tan 3 3. Next, we compute I under the following three different conditions: <, < <, and <. For >, since A =, I =. For < <, I A d. 6.6 Without lost of generality, assume < <. We consider the following cylindrical coordinates Thus, r = + 3 3, z =, θ = tan 3 3. I x t t x t t 4π π z r z r + z drdθdz dz 4π log x t t 4π + log x t t. 6.7

56 7 TAI-PING LIU AND SHIH-HSIEN YU [June For a =, we have I d π A π x t t x t t r z r + z a drdθdz A d log z a dz. 6.8 z We shall estimate the integral on x t t, a, and a,a, a, separately. In the first interval, log z a a x t t z In the second interval, we let τ = z a. Thus, a a a dz + log a x t z dz t + log a x log t t C + log a + log x t. t log z a log a τ dz = dτ z τ log τ log log a + dτ τ C + log a. For the last interval, log z a dz log a a z a z dz C log a. 6.9 Therefore, we can conclude I = O + log + log x t t. 6.3

57 ] SOLVING BOLTZMANN EQUATION, PART I: GREEN S FUNCTION 7 Similarly, I 3 = O e 5 44 e 5 44 d A 3 = O x t t dz = O log x t z t + log x t. 6.3 t In D, note that > >. Thus, I D e 5 44 d = O. 6.3 Combine all the inequalities above we can conclude 6.4. Apply the Duhamel principle to 6.5, we obtain a integral representation of h : Lemma 6.8. h x,t, = t e νt s Kh x t s,s,ds h x,t, = Oe ν 3 t+ x e 3 + log Proof. Using the estimate 6.4, we obtain h x,t, t Oe νt s e ν s+ x t s e 3 + log + log x t + s s χ j x t+ ff s s < ds e ν t+ x e 3 t + log + log x t + s s

58 7 TAI-PING LIU AND SHIH-HSIEN YU [June χ j x t+ ff s s < ds For simplicity, put β =, α = x t. The problem is now reduced to the estimate of the following integral R = = t t log x t + s s χ j x t+ s ds s <ff log α βs s χ { α βs s <} ds In order to prove the lemma, it is enough to show R = Ot + log β. Since we are considering 6.36, in order that χ, we require s < α βs < s Express 6.37 in seven different cases, classified by the signs of α,β, Figure 6:. < s <, for α =, < β <,. α β+ < s < α β, for α >, β >, α 3. β+ < s <, for α >, < β <, α 4. β+ < s <, for α >, < β <, α 5. β+ < s < α β, for α <, β <, α 6. β+ < s <, for α <, < β <, α 7. β+ < s <, for α <, < β <. For case, we can evaluate the integral 6.36 directly and obtain R = log β t. Case 5, 6, and 7 are similar to case, 3, and 4 respectively, so, without lost of generality, we consider only case, 3, and 4.

59 ] SOLVING BOLTZMANN EQUATION, PART I: GREEN S FUNCTION 73 s α α βs α β+ α β s s Figure 6: range of s. For case, since β >, α β < α β+. Therefore, t R α β log α βs s α β t log β + α β t log β + χ n t> α β log α β =t log β + χ n o t> α β t ds log β + log α α βs ds β t χ n o + χ o n t> α t> α β β log α βs o z dz + α β t α β O αβ + t log ds log α βs ds = Ot + log β. For case 3, since < β <, R = t α log α s β ds min n t, α β α o α + β α β χ n o + t> α β t α β χ n t> α β o log α s β ds

60 74 TAI-PING LIU AND SHIH-HSIEN YU [June min n t, α β α o log α s β ds + Ot + log β. For α < s < α β, < β < α s β <. Therefore, R min n t, α β α o log α s β ds + Ot + log β t log + log β + Ot + log β = Ot + log β. For case 4, Therefore, R α β < s = β < β + α s <. t α β log β + α. ds t log β s The next lemma shows that the source Kh for the next Picard iteration 6.5 is finite. That the source is gaussian in microscopic velocity is an interesting fact, which follows from the detailed estimates in the above lemma. Lemma 6.9. Kh x,t, Ce ν 3 t+ x e Proof. Applying the estimate for h, Lemma 6.8, and the estimate 6.5 for K, we obtain Kh x,t, = O R 3 e + log d = Oe 8 e ν 3 t+ x 9 e ν t+ x 3 e 3 + log e 64 d. R 3 e 9 64

61 ] SOLVING BOLTZMANN EQUATION, PART I: GREEN S FUNCTION 75 By Cauchy-Schwarz inequality, R 3 e log e 64 d e 9 64 d + log e 3 d. R 3 R 3 This completes the proof of the lemma Essential kinetic waves With the estimate of the source Kh in Lemma 6.9, we now study the essential kinetic wave h, 6.8. Lemma 6.. For any β, there exists D > such that h L,β = Oe ν x +t D Proof. With the operator O t D of 6., we have by Duhamel principle that h = t O t s D Kh sds = t O t s ν D Oe s+ x 3 e 8 ds, where we have used Lemma 6.9 for Kh. The estimate 6.39 now follows from Lemma 6.5 on the exponential decay in time and the hyperbolicity property of the operator O D. We omit the details. With this we have { gt + g x + νg = Kg + K h, gx,, = ; 6.4 g G [h + h + h ] h. 6.4 We have the following estimate on the smoothness in microscopic velocity for the source K h.

62 76 TAI-PING LIU AND SHIH-HSIEN YU [June Lemma 6.. For any fixed natural number l, l K h L x L = Oe ν t 3D. 6.4 Proof. From Lemma 3.3, we have K h H i = O h L The lemma now follows from Lemma 6.. For the singular waves g j of 6., we have the following estimate. Lemma 6.. For each fixed β > and j =,,..., there exists positive constants D j such that g j L,β = Oe ν x +t D j Proof. This is proved by induction on j and uses Lemma 6.5. The procedure is similar to the proof of Lemma 6.; details are omitted. We have, c.f. 6.9, G i= hi h j i= gi t + G i= hi h j = LG i= hi h j i= gi + Kg j, G i= hi h j i= gi x,, =. i= gi x 6.44 Using the notion of damped transport operator S and the integral operator K, the Picard iteration 6. has the representation: g j+ = t S t s Kg j s ds = and so inductively we have t s S t s KS s s Kg j s ds ds. g j = t s sj S t s KS s s KS s s 3 K S s j s j+ K hsj+ ds j+ ds, j =,,, A major point is that the above repeated convolutions of the damped transport operator S and the integral operator K yield the smoothness in x,t as

63 ] SOLVING BOLTZMANN EQUATION, PART I: GREEN S FUNCTION 77 a consequence of the smoothness of the source K h in, 6.4. This is the Mixture Lemma we study in the next subsection Mixture Lemma We rewrite the linearized Boltzmann equation as follows: g t + g x + νg = Kg This form indicates that there are two essential mixing mechanisms: The mixing mechanism in x is due to particles travelling in different velocity. This is represented by the operator S t, the LHS of the equation, which represents the transport as well as part of the loss term in the collision operator. The mixing mechanism in is due to the rest of the collision of particles. This is represented through the integral operator K. We introduce a sequence of mixture operators M t i as follows. Definition 6.3. For any g L x L, k-th degree Mixture operator Mt k is given as follows: M t k g t s sk S t s KS s s KS s s 3 K S s k s k KS s k g ds k ds The Picard iteration for solving the initial value problem t h + x h Lh =, hx, h x, can be rewritten as hx,t = S t h x + + t t S t s KM s j h dsx + j= S t s KS s h dsx + M t h x t t S t t KS t t KM t j h dt dt x This series is an asymptotic expansion, useful for identifying the singular parts of the solution.

64 78 TAI-PING LIU AND SHIH-HSIEN YU [June Remark 6.4. The Mixture Lemma states that the mixture of the two operators S and K in M t k transports the regularity in the microscopic velocity to the regularity of the space time x,t. The Mixture Lemma is similar in spirit as the well-known Averaging Lemma, [5], [6], [3], [7], see also [4] for the gliding regularity for the non-dissipative Landau damping. These two Lemmas have been introduced independently and used for different purposes. Lemma 6.5 Mixture Lemma. There exist C k >, k =,,..., such that x k Mt k g C ke ν t L x L k l L g x L Proof. In order to explain the main ideas, we will go to some details for the cases of k = and k =. The proof uses the characteristic method, [3], [6]. For k =, we write down the explicit form of the Mixture operator by spelling out the operator S t hx, = e νt hx t,, 6.: t s M t g x, = e νt s ν s s ν s K, K, R 3 R 3 g x t s s s s, d d ds ds. 6.5 Set A,,,t,s,s e νt s ν s s ν s K, K,, 6.5 z x t s s s s, and we have x M t g x, = t s R 3 l= R 3 A,,,t,s,s x g z, d d ds ds. Next, we use the change of variables and the chain rule: V, V, 6.5 V g z, = s x g z, g z,. 6.53

65 ] SOLVING BOLTZMANN EQUATION, PART I: GREEN S FUNCTION 79 By the switching the order of differentiations and applying the integration by parts, we obtain x M t g x, t s { = A, V, V V,t,s,s R 3 R 3 s g z, g z, V A, V, V V,t,s,s } dv dv ds ds To estimate the L norm of x M t g x,, we only need the integrability. Note that, as the result of the above switching of the of s A + V A order of differentiation, it yields the integrability t s s ds ds = t. If we were to switch the differentiation with respect to x with the differentiation with respect to V, then we would get the non-integrable factor s, instead of s. This would implies that the characteristic method only works for the short time scale. On the other hand, we now have the partial differentiation of the function A. The second point is then to show the integrability of V A. We have from 6.5 V A e 3 4 ν t K, K, From the Hőlder inequality and Fubini Theorem, we conclude x M t g L x L C e ν t g L x L + g L x L For k =, there are more changes of variables that have to be done in the right order. We have xm t g x, = where t s s s3 B,,, 3, 4,t,s,s,s 3,s 4 R 3 R 3 R 3 R 3 x g w, 4 d 4 d 3 d d ds 4 ds 3 ds ds 6.57 B,,, 3, 4,t,s,s,s 3,s 4 K, K, K, 3 K 3, 4

66 8 TAI-PING LIU AND SHIH-HSIEN YU [June e νt s ν s s ν s s 3 ν 3 s 3 s 4 ν 4 s w x t s s s s s 3 3s 3 s 4 4s Again, we change the variables and make the switching of the differentiations: V =, V =, V 3 = 3, V 4 = 3 4, V g w, 4 = s x g w, 4 g w, 4, V 3 g w, 4 = s 3 x g w, 4 g w, 4, xg = { } s s V 3 V 3 g + V g + V 3 g + g. 6.6 The key observation is that we switch the second derivative with respect to x evenly to V and V 3. There are two reasons for doing this. The first is that the resulting factor s s 3 is integrable. Another is that B keeps the integrability after integration by parts. We set s t, s 5 so that B = 4 i= {e ν P i j= V js i s i+ } { [ 4 π Vi exp i j= V j i j= V ] j 8 V i= i V i 8 V i exp i j= V j + i j= V } j With this and the boundedness of ν, we have V B { [ ] = Oe 3 4 ν t Kǫ,ǫ K ǫ,ǫ K ǫ,ǫ 3 K ǫ 3,ǫ 4 } +Kǫ,ǫ K ǫ,ǫ K ǫ,ǫ 3 K ǫ 3,ǫ 4, V 3 B { [ ] = Oe 3 4 ν t Kǫ,ǫ K ǫ,ǫ 3 K ǫ,ǫ 3 K ǫ 3,ǫ 4 } +Kǫ,ǫ K ǫ,ǫ K ǫ,ǫ 3 K ǫ 3,ǫ 4, V 3 V B {[ ] [ ] = Oe 3 4 ν t Kǫ,ǫ Kǫ,ǫ 3 K ǫ,ǫ 3 K ǫ 3,ǫ 4

67 ] SOLVING BOLTZMANN EQUATION, PART I: GREEN S FUNCTION 8 [ ] + Kǫ,ǫ K ǫ,ǫ K ǫ,ǫ 3 K ǫ 3,ǫ 4 [ ] +Kǫ,ǫ K ǫ,ǫ 3 K ǫ,ǫ 3 K ǫ 3,ǫ 4 +Kǫ,ǫ K ǫ,ǫ K ǫ,ǫ 3 K ǫ 3,ǫ 4 }, 6.6 for some < ǫ <. From these we obtain V BdV 4 dv 3 dv dv = Oe 3 4 νt, V 3 BdV 4 dv 3 dv dv = Oe 3 4 νt, 6.63 V 3 V BdV 4 dv 3 dv dv = Oe 3 4 νt. We have x Mt g x, = t s s s3 R 3 R 3 g V B + V 3 B g V R 3 3 V B + B g }dv 4 dv 3 dv dv ds 4 ds 3 ds ds R 3 s s 3 { Apply 6.63 to the above, we conclude from the Hőlder inequality and Fubini Theorem that x M t g L x L C e ν t g L x L + g L x L + L g. x L 6.65 This completes the proof of the Mixture Lemma. Lemma 6.6. The wave g j of 6.45 satisfy, for some positive constant D k, l xg k L x L = Oe ν t D k, k =,,, l k Proof. This is direct consequence of the smoothness of K h in microscopic velocity, Lemma 6. and the Mixture Lemma, 6.49.

68 8 TAI-PING LIU AND SHIH-HSIEN YU [June 6.5. Global wave structure of the Green s function We now put together the previous pieces of the waves in the Green s function to gain global understanding of the Green s function. There is a missing information on the pointwise description outside the finite Mach region. This missing information is obtained through the following process. First, the regularity resulting from Picard iterations through the Mixture Lemma allows use to apply the Sobolev calculus if we have pointwise information on the Sobolev norms. The pointwise description of these Sobolev norms is obtained by a weighted energy estimate in the proof of the following main theorem for the -D Green s function. In this theorem, we describe only the Green s function itself. We will be interested in the differential of the Green s function. For that we need to include other singular waves besides the kinetic-like waves. Lemma 6.7. The Green s function minus the singular waves satisfies l x G h j h j= k j= g j L x L = O, k =,,, l k Proof. The function G j= h j h k j= gj satisfies the linearized Boltzmann equation with the source Kg k, From Lemma 6.6, we know the smoothness and exponential decay in time of the source. Thus the lemma follows from the boundedness of the solution operator for the linearized Boltzmann equation, 3.4, and the Duhamel s principle: l x G h j h k j L t g = G t s K l x gj sds j= j= xl t G t s L x L K L xg l j ds L x L = O t L x L e t s C ds = O Theorem 6.8. The Green s function G = G F + G K + G R, 6.69

69 ] SOLVING BOLTZMANN EQUATION, PART I: GREEN S FUNCTION 83 consists of the main fluid-like part: G F j= jt 4A e x λ j t E j E j, 6.7 4πAj t + where λ j,e j, j =,,3, are Euler characteristics, 4.4, and A j is the Navier-Stokes dissipation parameters, 5.3; the particle-like part: G K h + h + h, , 6., 6., 6.33, 6.34; and the remainder G R, which is a bounded function satisfying, for any fixed constants C j > A j, j =,,3, and some positive constant C, G R L = O j= x λ j t t + e 4C j t + Oe x +t C. 6.7 Moreover, the Green s function as an operator has the following properties: GP L, P G L = O j= x λ j t t + e 4C j t + Oe x +t C 6.73 P GP L = O e t + 3 j= jt x λ 4C j t + Oe x +t C Proof. The study of the structure of the Green s function is done according to the following two cases: Case. Within finite Mach region, x Mt, for some constant M > λ j, j =,,3, and sufficiently large. By, G G L L x L decays exponentially in time. As the essential kinetic waves h j,j =,,, and h, 6. Lemmas 6.6, Lemmas 6.8, Lemma 6., and g j, j =,,..., Lemma 6., also decay exponentially in time, we have G G L h j h g j L = Oe Ct j= j= xl

70 84 TAI-PING LIU AND SHIH-HSIEN YU [June The fluid-like wave G L is smooth and bounded, Theorem 5.9. This fact and Lemma 6.7, applied here with k =, yield G G L h j h j= j= g j H xl = O By Sobolev inequality, we have from 6.75 and 6.76 that for some constant C >, G G L h j h j= j= g j L x L = Oe t C Thus in the finite Mach region, for some constant C >, G G L h j h g j L = Oe t+ x C j= j= The theorem then follows from Theorem 5.9 and Theorem 5.. Case. Outside finite Mach region, x > Mt. Outside of the finite Mach region, 6.75 still holds. Therefore, it remains to obtain the pointwise estimate in the space variable x. The Green s function minus the essential kinetic waves satisfies, 6.44: R t + R x = LR + S, Rx,, =, R G G K h S Kg. j= gj, 6.79 We apply the weighted energy method to this equation with a weight wx,tr: wx,t e x Mt N, 6.8 for some large constants M, N to be chosen later. Multiply 6.79 with

71 ] SOLVING BOLTZMANN EQUATION, PART I: GREEN S FUNCTION 85 wx,t and integrate to obtain d dt R wr,rdx+ x wr,m R+wR, LRdx= wr, Sdx. R N x R 6.8 There are terms on the left hand side of 6.8 that are of good sign: w[ R,MR + R, LR]dx = w[ M R,Rdx + R, LR]dx. N N R From 3.3 and 3.4, R Thus we may choose and yield x wr, N x Rdx C R R M N = C N =, w[r,r + LR,R]dx. N d dt R wr,rdx + wr,rdx wr, Sdx. 6.8 R R We conclude from the Cauchy-Schwarz inequality and the estimate of the source S by 6.43 that d wr, Rdx+ wr,rdx=o ws,sdx=o we ν x +t D dx. dt R R R R 6.83 By further restriction of the choice of the weight function wx,t, 6.8, we have d wr, Rdx+ dt R R ν D N, and we obtain the desired energy estimate R wr, Rx, tdx+ wr,rdx = O e x Mt N e ν x +t D dx=oe 4t, R 6.84 t R wr, Rx, sdxds R wr, Rx, dx+o

72 86 TAI-PING LIU AND SHIH-HSIEN YU [June = O Similarly, since the function R is in HxL, 6.76, we may apply the above weighted energy method to the equation for x R and obtain w x R, x Rx,tdx = O R By the Sobolev inequality and with M chosen sufficiently large, we have from 6.85 and 6.86 that, for some positive constant C, R L = Oe x Mt N = Oe C x +t, for x > Mt This completes the proof of the theorem D Green s Function In this section we consider the 3-D Green s function for the initial-value problem,.7: { Gt + x G = LG for x R 3, Gx,,; = δxδ. 7. Here δ was denoted by δ 3 in the last two sections for -D case to emphasize that it is a 3-dimensional delta function. Here both δx and δ are 3-dimensional delta functions. We will derive the explicit form of the Green s function and give pointwise estimate of the remaining terms. As in the -D case, we construct the essential kinetic waves and fluid-like waves separately. However, there are essential differences with the -D case. For instance, there is Huygens principle for 3-D case. The complex analytic method for inverting the Fourier transform is also more sophisticated than the -D case. 7.. Kinetic waves As with the -dimensional case, we construct the kinetic waves using

73 ] SOLVING BOLTZMANN EQUATION, PART I: GREEN S FUNCTION 87 the Picard s iteration,c.f. 6.3, 6.5: { h t + x h + νh =, h x, = δxδ ; { h j t + xh j + νh j = Kh j, h j x, =, j =,, The difference here is that the delta function δx = δ 3 x is now 3- dimensional. Meanwhile, there is a stronger dispersion for the 3-D case. As a consequence, the same number of iterations will result in a bounded source. The first term is the same as before, 6., an exponentially decaying delta function along the characteristic direction: h x,t,, S t δxδ = e νt δx tδ. 7.4 The second term is: h x,t,, = = = t t t t S t s Kh x,s,, ds e R νt s K, h x t s,s,, d ds 3 R 3 e νt s ν s K, δx t s sδ d ds e νt s ν s K, δx t s sds 7.5 which is still a generalized function, though the generalized part is of lower dimension. From direct calculations, h x,t,, = t t S t s Kh x,s,, ds s e νt s ν s τ ντ K, K, R 3 δx t s s τ τdτd ds

74 88 TAI-PING LIU AND SHIH-HSIEN YU [June = t s [ exp νt s ν x t s τ s τ ν τ s τ K, x t s τ s τ K x t s τ s τ ], s τ 3dτds.7.6 From the explicit expression in 3.5, and hence K, = O exp 8 K, x t s τ s τ s τ = O t τ + τ x exp t τ + τ x 8 s τ, K x t s τ, s τ s τ = O x t s s exp x t s s 8 s τ. For x,t,, R 3 R + R 3 R 3 satisfying we have K, x t s τ s τ We denote min x t s s >, s t K x t s τ s τ, s τ 3 = O, τ s t. S {x,t,, : x t = x t = x 3 3 t t}. 3 Then for x,t,, R 3 R + R 3 R 3 \S, we have h x,t,, <. Moreover, we have νt s + ν x t s τ s τ + ν τ s τ 3

75 ] SOLVING BOLTZMANN EQUATION, PART I: GREEN S FUNCTION 89 ν + t s + ν + x t s τ s τ + ν + τ s τ ν t + x for x,t,, / S. Therefore, there exists a C > such that h x,t,, L Oe t+ x C. This implies Kh L Oe t+ x C and so h 3 x,t,, L Oe t+ x C. As with the -D case, we can stop at h and construct h as well as g j, j =,,... The Mixture Lemma can also be generalized easily to the present 3-D case. We omit the details. 7.. Euler waves Before we study the fluid-like waves for the Boltzmann equation, we consider the Euler waves. We use the conservative form of the linearized Euler equations 4.5: ρ t + x m =, m t + 3 xe =, E t + 5 x m =. This is the linearization with base state ρ =,v =,θ = and so the sound speed c is given as c = 5θ 3 = 5 3. From this we derive the wave equation for ρ t : 5 ρ E =, t

76 9 TAI-PING LIU AND SHIH-HSIEN YU [June ρ t tt = x 3 xe 5 = x 3 xρ = c x ρ t. 7.7 The wave equation can be solved explicitly by the Kirchhoff s formula and exhibits the Huygens principle for the 3-D situation we consider here. With ρ t thus determined, the other variables can also be explicitly constructed: ρx,t = ρx, + t ρ tx,sds, Ex,t = Ex, + 5 t ρ tx,sds, 7.8 mx,t = mx, + t 3 xex,sds. Note that, although ρ t satisfies the Huygens principle, the formula for the other variables involves time integration and so in general are not zero inside the acoustic cone and decay algebraically there. The viscous version is studied in Lemma 7.3 later. To draw the comparison with the Boltzmann equation, we take Fourier transform of the Euler equations ˆρ η t ˆρ ˆm + i 3 η ˆm = ; 7.9 Ê 5 ηt Ê ˆρη, t isinc η t c η η t +cosc η t 3c ˆmη, t = I+ cosc η t η η t i sinc η t ˆρη, η 3c η η ˆmη,. Êη, t 5i sinc η t c η η t cosc η t Êη, 7. We therefore will be interested in the well-known inversion of the Fourier transform of the following types: Theorem 7. Kirchhoff. The inversion of the Fourier transform of ĝŵ and ĝŵ t, where are ŵ = sinc η t, ŵ t = cosc η t, c η w gx = t gx + ctydsy, 7. 4π y =

77 ] SOLVING BOLTZMANN EQUATION, PART I: GREEN S FUNCTION 9 w t gx = gx+ctydsy + ct 4π y = 4π y = gx+cty ydsy. 7. The Kirchhoff s formula for the Green s function of the Euler equations involves the function g as δ functions. For study of the dissipative Navier- Stokes and the Boltzmann equation, we consider g as the heat kernel. The following lemma gives estimates of the viscous version of Huygens principle. Lemma 7.. For any positive integer l, w e x Ct+ x ct t + l = Oe Ct+, 7.3 t + l w t e x Ct+ x ct t + l = Oe Ct t + l+ Proof. By the Kirchhoff formula 7., There are two cases. J w e x Ct+ t + l Case. x ct O + t. J = Ot + y = e Ot y t + l = t 4π y = e x cty Ct+ t + l dsy. dsy = Ot + t + l + t = O t l Case. x ct O + t. In this case, min x cty = x ct, y =

78 9 TAI-PING LIU AND SHIH-HSIEN YU [June and so e x cty Ct+ e x cty Ct+ x ct Ct+ ; 7.6 and J = t + y = t + e x ct Ct+ e x cty Ct+ x cty Ct+ t + l y = e x cty Ct+ t + l dsy x ct dsy e Ct t + l This proves 7.3; 7.4 is shown similarly. The following will be needed for the viscous version of the procedure 7.8 in yielding the algebraic decaying rate inside the acoustic cone. Lemma 7.3. For x < ct t { τ y = and for x > ct t } e x cτy Ct dsy dτ + t 5/ { τ y = C + t x + t + ; 7.8 } e x cτy x ct Ct dsy dτ + t 5/ Ce Ct + t. 7.9 Proof. From 7.6, t { τ t y = } e x cτy Ct dsy dτ + t 5/ { τ y = } e x cτy Ct e x cτ Ct + t 5/ dsy dτ 7. There are three cases.

79 ] SOLVING BOLTZMANN EQUATION, PART I: GREEN S FUNCTION 93 Case. x O + t. t { τ y = +t } e x cτy Ct dsy dτ + t 5/ { t + τ +t y = } e x cτy Ct e x cτ Ct + t 5/ dsy dτ +t τ t dτ + O + t 5/ +t τ t + 5 te x cτ Ct τ dτ = O. + t 3 Case. + t x ct + + t. t { τ y = +t } e x cτy Ct dsy dτ + t 5/ { t + τ +t y = } e x cτy Ct e x cτ Ct + t 5/ dsy dτ +t τe x cτ Ct t dτ + O + t 5/ +t τ t + 5 te x cτ Ct τ dτ = O + t x. Case 3. x ct + + t. t { τ y = t + } e x cτy Ct dsy dτ + t 5/ t t { τ y = } e x cτy Ct e x cτ Ct + t 5/ dsy dτ t τe c t 4C e x cτ Ct t + t 5/ dτ + O t Oe x ct Ct = + t. τ t + 5 te x cτ Ct τ dτ This completes the proof of the lemma.

80 94 TAI-PING LIU AND SHIH-HSIEN YU [June 7.3. Spectrum near origin Like the one-dimensional case, we have Lemma 7.4. Consider the spectrum Specη of the operator i η+l, η R 3. I For any < δ, there corresponds τ = τδ > such that i For η > δ, Specη {z C : Rez < τ}. ii For η δ, the spectrum within the region {z C : τ Rez} consisting of exactly five eigenvalues σ η, σ η, σ 3 η, σ 4 η, σ 5 η: Specη {z C : τ Rez} = {σ η,σ η,σ 3 η,σ 4 η,σ 5 η}. II For all < δ, the semigroup e i η+lt can be decomposed as e i η+lt = Π δ + χ { η <δ} e zt dz, z i η + L 7. πi Γ where Π δ L = Oe aτt, aτ > depends on τ and therefore on δ, and Γ can be any close curve that lies entirely on {Re z > τ} and that encloses the five eigenvalues σ η,σ η,σ 3 η,σ 4 η,σ 5 η. We now compute the eigenvalues and eigenfunctions: i η + Lψ j η = λ j ηψ j η, η R To simplify 7., we invoke a symmetric property of L. Consider the action of a three dimensional orthonormal transformation Q O3 on L : O3 L L Q, f fq. 7.3 Lemma 7.5. The collision operator Q and the linearized collision operator L are invariant under orthonormal transformation. More precisely, for any

81 ] SOLVING BOLTZMANN EQUATION, PART I: GREEN S FUNCTION 95 Q O3: Q Qf,Qg = QQf,g, and LQf = QLg. Proof. QQf, g = Qf,gQ Ω S [ ] = fq g fqg BQ, Ωd Ωd. R 3 We change variables Ω = QΩ and = Q and observe the following relations Q = Q [Q Ω] Ω = Q [Q Q QΩ]QΩ = Q [ Ω]QΩ = Q, = + [Q Ω] Ω = Q + [Q Q QΩ]QΩ = Q + [ Ω]QΩ = Q, Therefore, QQf, g = Q R 3 QΩ S [ ] fq gq fqgq BQ Q,QΩdQΩdQ [ ] = fq gq fqgq R 3 Ω S B,ΩdΩd = QQf,Qg. The invariance of L under Q can be proven similarly. We omit the details. Let g O3 be an orthonormal transformation that sends η η to,,. Apply g to 7., we have, by Lemma 7.5, i g η g ψ j + g Lψ j = i g g gη + L g ψ j = i gη + L g ψ j = i η + L g ψ j = λ j g ψ j. 7.4 Therefore, the original equation 7. is reduced to i η + L e j η = σ j η e j η, 7.5

82 96 TAI-PING LIU AND SHIH-HSIEN YU [June with λ j = σ j, ψ j = ge j. Apply Macro-Micro projection to the equation i η + Le j = σ j e j : i η P P e j + P e j = σ j P e j, 7.6a i η P P e j i η P P e j + LP e j = σ j P e j. 7.6b Set i η = ǫ and σ j = ǫγ j. From 7.6b we can solve P e j in terms of P e j : P e j = ǫ[l ǫp ǫγ j ] P P e j. 7.7 Substituting this back to 7.6a, we obtain the following equation for P e j : P + ǫp L ǫp ǫγ j P P e j = γ j P e j. 7.8 For notation simplicity, put I ǫ,γ = P + ǫp P L ǫp ǫγ. When ǫ =, 7.8 has degeneracy: γ = γ 4 = γ 5 =, so implicit function theorem does not apply directly. However, as it turns out, Lemma 7.6. I ǫ,γe j,e k =, if j k and j,k = 4 or 5, for any ǫ,γ. Consequently, under the basis {E,E,E 3,E 4,E 5 }, the matrix representation of I ǫ, γ becomes: 3 3. Proof. Consider the reflection J O3 :,, 3,, 3. Clearly, P,P,L,,γ j,ǫ, as linear operators, commute with J. Therefore, I commutes with J. Consequently, since E 4 = M is an odd function of, I E 4 is also an odd function of. Now that E,E,E 3,E 5 are even

83 ] SOLVING BOLTZMANN EQUATION, PART I: GREEN S FUNCTION 97 function of, I E j,e k R = I E j E k d =, for j = 4, k =,,3, The case j = 5 can be proven similarly. We omit the details. With the aid of Lemma 7.6, we can compute the eigenvalues and eigenfunctions. By 7.9, P e,p e,p e 3 E,E,E 3 and P e 4,P e 5 E 4,E 5. Therefore, the problem of solving for γ j,p e j for j =,,3 is reduced to the -dimensional case. For j = 4,5, γ 4 ǫ,γ 5 ǫ can be solved from I ǫ,γ 4 E 4,E 4 = γ 4, γ 4 ǫ= =, I ǫ,γ 5 E 5,E 5 = γ 5, γ 5 ǫ= =. 7.3 Consider E O3 :,, 3, 3,. I E 4,E 4 = EI E 4,EE 4 = I EE 4,EE 4 = I E 5,E 5. This, together with 7.3, imply γ 4 = γ 5. Since the eigenspace for γ 4 = γ 5 always degenerates with multiplicity, we gain extra freedom in specifying e 4,e 5. However, the choice must obey [e j,e k ] = δ jk. As it turns out see the proof of the lemma below, we can choose P e 4 E 4, P e 5 E 5. Lemma 7.7. View ǫ as a real variable. For ǫ, γ j ǫ, j =,...,5, are real and analytic functions with γ j ǫ = λ j + A j ǫ + Oǫ, for j =,,3, γ 4 ǫ = γ 5 ǫ = A 4 ǫ + Oǫ, 7.3a 7.3b where A j is the Navier-Stokes dissipation coefficients. Note that A 4 = A 5. Moreover, P e j ǫ =β j ǫe + β j ǫe + β j3 ǫe 3, for j =,,3, P e 4 ǫ =β 44 ǫe 4, P e 5 ǫ =β 55 ǫe 5,

84 98 TAI-PING LIU AND SHIH-HSIEN YU [June where β jk ǫ is a real and analytic function with β jj = + Oǫ for j =,,3,4,5, β jk = Oǫ, for j k. β 44 = β 55. Proof. Since I E j,e k is real ǫ is thought of as a real variable here, σ j ǫ, β jk ǫ are real. 7.3b follows at once after we differentiate 7.3. It remains only to compute P e 4,P e 5. Consider the ansatz: P e 4 = β 44 E 4, P e 5 = β 55 E 5, β 44,β 55 R. 7.3 Recall that P e j = ǫ[l ǫp ǫγ j ] P P e j, 7.7. Therefore e 4,e 5 are real and E [e 4,e 5 ]=e 4,e 5 =ǫ β 44 β 55 L ǫp E ǫγ 4 4, L ǫp ǫγ 5 5. This is zero, since L ǫp ǫγ 4 E 4 is odd in while L ǫp ǫγ 5 E 5 is even in. The validity of the ansatz 7.3 is justified. Subsequently, β 44,β 55 can be solved from the normalization condition [e j,e j ] = e j,e j = : β 44 = E + ǫ L ǫp ǫγ 4 4 L = E + ǫ L ǫp ǫγ 5 5 =β 55 = + Oǫ. L We continue with more detailed analysis of σ j,e j. Consider the reflection J :,,,, 3. Note that ǫ + LJ e j ǫ = J ǫ + Le j ǫ = J ǫ + Le j ǫ = σ j ǫj e j ǫ = ǫ γ j ǫ J e j ǫ. Thusǫ γ j ǫ,j e j ǫ is an eigenvalue-eigenfunction pair. Note also that, for j =,,3, P J e j = J P e j E,E,E 3, and this implies γ ǫ, γ ǫ, γ 3 ǫ {γ ǫ,γ ǫ,γ 3 ǫ}. Since γ ǫ,γ ǫ,γ 3 ǫ are distinct for small ǫ, by the fact that γ = c = γ 3,γ =

85 ] SOLVING BOLTZMANN EQUATION, PART I: GREEN S FUNCTION 99 and by continuity, we conclude γ ǫ = γ 3 ǫ, γ ǫ = γ ǫ, J e ǫ = e 3 ǫ, J e ǫ = e ǫ, for all ǫ small. For γ 4 = γ 5, a similar argument yields γ 4 ǫ = γ 4 ǫ. As for the eigenvectors, P J e 4 ǫ,p J e 5 ǫ E 4 5,E. Moreover, P J e 4 ǫ is odd in. Therefore, P J e 4 ǫ E 4 and J e 4 ǫ = e 4 ǫ. Similarly, J e 5 ǫ = e 5 ǫ. Summarizing, we have σ ǫ = σ 3 ǫ, σ j ǫ = σ j ǫ, for j =,4,5, J e ǫ = e 3 ǫ, J e j ǫ = e j ǫ, for j =,4, a 7.33b 7.33c 7.33d Now we switch back to the variable i η = ǫ. Although so far we have been working with real ǫ, since σ j,e j are analytic in ǫ, 7.33 holds for all complex ǫ, as long as ǫ. For simplicity, set ip L j = + L ip η λ j η η Lemma 7.8. For η R 3 with η, λ η = i η c + Λ η A η + O η 4, λ η = A η + O η 4 λ 3 η =i η c + Λ η A η + O η 4, λ 4 η = A 4 η + O η 4, λ 5 η = A 4 η + O η 4, A j = P E j,l P E j, 7.35 for some analytic function Λ : R R. Furthermore, there exists analytic

86 TAI-PING LIU AND SHIH-HSIEN YU [June functions a j k,l : R R such that ψ η = L [a, η + i η a, η χ + a, η + i η a, η 3 k= ] + a 4, η + i η a 4, η χ 4 ψ η = L [a, η χ + ia, η η k χ k η, 7.36a η j χ j + a 4, η χ 4 ], 7.36b j= ψ 3 η = L 3 [a, η i η a, η a, η i η a, η 3 k= ] + a 4, η i η a 4, η χ 4 χ η k χ k η, 7.36c [ ψ 4 η = L 4 a 4, η gχ ], 7.36d [ ψ 5 η = L 4 a 4, η gχ 3 ], 7.36e with 3 a, =, a, =, a4, = 3 a 4, = 5, a 4, =. Proof. From 7.3a, we have σ i η = i η c+a i η + 5, a, = 5, η k k! k= +i η η k k +! k= i η c + A i η Λ η +i η η k k+! k= d k γ dǫk d k+ γ dǫ k+ d k+ γ dǫ k+ = i η c + Λ η A η + O η 4.

87 ] SOLVING BOLTZMANN EQUATION, PART I: GREEN S FUNCTION By 7.33, σ 3 i η = σ i η = i η c A i η Λ η i η η k k +! k= = i η c + Λ η A η + O η 4. Similarly, for j =,4,5, by 7.33, σ j i η =i η A j i η + i η = A j η + O η 4. η k k +! k= d k+ γ dǫ k+ d k+ γ j dǫk+ This proves follows by similar computations. We omit the details. Note that g P η = P η g = g = + + ip L ip η λ j η η L ip η λ j η ip η g, g M = g M = k= k η k M. η 7.4. Fluid-like waves, I This and next subsections are for the study of the fluid-like waves. For the 3-D case considered here, there is a geometrically richer class of fluid-like waves as hinted by the study of the Euler waves. In this subsection, we will explicitly constructed the leading fluid-like waves. The next subsection will be concerned with the remaining fluid-like waves, which decay at faster rate than the leading fluid-like waves considered here. As the one dimensional case, we have e zt dz 5 z iη + L = e λjt ψ j [ψ j. πi Γ j=

88 TAI-PING LIU AND SHIH-HSIEN YU [June This implies G = π F Π δ + 5 j= π 3 e ix η+λjηt ψ j [ψ j dη, η <δ with F Π L δ x L = Oe Cδ. t We again define the fluid-like waves in the Green s function as: G L x,t 5 π 3 e ix η+λjηt ψ j η [ψ j η dη j= η <δ = e ix η[ ] e ληt ψ η [ψ η + e λ3ηt ψ 3 η [ψ 3 η dη η <δ + + η <δ η <δ e ix η e λ ηt ψ η [ψ η dη [ ] e ix η e λ 4ηt ψ 4 η [ψ 4 η + ψ 5 η [ψ 5 η dη We arrange the pairing in 7.37 to define the following pairings: Ĝ L η,t = ĜHη,t+ĜCη,t+ĜRη,t+ĜPR η,t+ĝpr η,t, 7.38 where Huygens Pairing Ĝ H η,t = Contact Pairing, j=,3 j=,3 e λ jη ψ j η [ψ j η Ĝ C η,t = e λ η ψ η [ψ η, e λ jη L j P m ψ j η [L j P m ψ j η, Rotational Pairing, Ĝ R η,t = e λ 4η ψ j η [ψ j η j=4,5

89 ] SOLVING BOLTZMANN EQUATION, PART I: GREEN S FUNCTION 3 t t t ct ct R 3 R 3 a Huygens wave. b Solid Huygens wave. Figure 7: Huygens wave and solid Huygens wave. + st Riesz Pairing, Ĝ PR η,t = j=,3 j=,3 L j P m ψ j η [L j P m ψ j η, e λ jη L j P m ψ jη [L j P m ψ jη e A η j=,3 nd Riesz Pairing, Ĝ PR η,t = e A η e λ 4η L j P m ψ jη [L j P m ψ jη, j=,3 L j P m ψ j η [L j P m ψ j η, 7.39 We define the leading fluid-like waves to be those induced by the quadratic terms, in the Fourier variable η, in the eigenvalues and the constant term in the eigenfunctions. This is made definite in the proof of the following theorem, There are richer wave patterns for the 3-D case. The Huygens wave for the Euler equations is now a dissipative version G H, with essential support of width t around the acoustic cone, Figure. As we have seen for the Euler equations, there are time integrations of the waves around the acoustic cone. These appear in G PR + G PR, Figure. The contact, thermal waves G C as well as the rotational waves G R are supported as the heat kernel, Figure 8.

90 4 TAI-PING LIU AND SHIH-HSIEN YU [June t t R 3 Figure 8: Diffuse wave. Theorem 7.9. The leading fluid-like waves are G = G H + G C + G R + G PR + G PR G ct H x,t = 4π + ct 4π y = y = H x + ctydsy + 4π H x + cty ydsy, H = 4πA + t 3/ e x 5 j= y = 4A t x j H x + ctydsy j [ M M + [ M j M, H = 5 4πA + t 3/ e x 4A t [ M M, G C x,t = 4πA + t 3/ e x 4A t 5 [ M 5 M, t G c τ PR x,t = H 3 x + cτydsydτ 4π H 3 = j,k= y = 4πA + t 3/ e x [ 4A t x j x kj M k M,