Asymptotic Stability of the Relativistic Boltzmann Equation for the Soft Potentials

Transcription

1 Commun. Math. Phys. Digital Object Identifier (DOI) 1.17/s Communications in Mathematical Physics Asymptotic Stability of the Relativistic Boltzmann Equation for the Soft Potentials Robert M. Strain Department of Mathematics, University of Pennsylvania, David Rittenhouse Lab, 9 South 33rd Street, Philadelphia, PA , USA. strain@math.upenn.edu; URL: strain/ Received: 8 October 9 / Accepted: June 1 Springer-Verlag 1 Abstract: In this paper it is shown that unique solutions to the relativistic Boltzmann equation exist for all time and decay with any polynomial rate towards their steady state relativistic Maxwellian provided that the initial data starts out sufficiently close in L l. If the initial data are continuous then so is the corresponding solution. We work in the case of a spatially periodic box. Conditions on the collision kernel are generic in the sense of Dudyński and Ekiel-Jeżewska (Commun Math Phys 115(4):67 69, 1985); this resolves the open question of global existence for the soft potentials. Contents 1. Introduction.... Statement of the Main Results Linear L Bounds and Decay Linear L Bounds and Slow Decay Nonlinear L Bounds and Slow Decay Linear L Rapid Decay Nonlinear L Rapid Decay... Appendix: Derivation of the Compact Operator... References Introduction The relativistic Boltzmann equation is a fundamental model for fast moving particles; it can be written with appropriate initial conditions as p μ μ F = C(F, F). The authors research was partially supported by the NSF grant DMS

2 R. M. Strain The collision operator [4,9]isgivenby C( f, h) = c R 3 dq q R3 dq R3 dp W (p, q p, q )[ f (p )h(q ) f (p)h(q)]. The transition rate, W (p, q p, q ), can be expressed as q p W (p, q p, q ) = sσ(g,θ)δ (4) (p μ + q μ p μ q μ ), where σ(g,θ)is the differential cross-section or scattering kernel; it measures the interactions between particles. The speed of light is the physical constant denoted c. Standard references in relativistic Kinetic theory include [8,9,1,48,54]. The rest of the notation is ined in the sequel A brief history of relativistic kinetic theory. Early results include those on derivations [4], local existence [3], and linearized solutions [14,18]. DiPerna-Lions renormalized weak solutions [13] were shown to exist in 199 by Dudyński and Ekiel-Jeżewska [19] globally in time for large data, using the causality of the relativistic Boltzmann equation [16,17]. See also [1,57] and [37,38]. In particular [1] proves the strong L 1 convergence to a relativistic Maxwellian, after taking a subsequence, for weak solutions with large initial data that is not necessarily close to an equilibrium solution. There are also results in the context of local [7] and global [49] Newtonian limits, and near vacuum results [,35,49] and blow-up [] for the gain term only. We also mention a study of the collision map and the pre-post collisional change of variables in [3]. For more discussion of historical results, we refer to [49]. We review in more detail the results most closely related to those in this paper. In 1993, Glassey and Strauss [4] proved for the first time global existence and uniqueness of smooth solutions which are initially close to a relativistic Maxwellian and in a periodic box. They also established exponential convergence to the Maxwellian. Their assumptions on the differential cross-section, σ, fell into the regime of hard potentials as discussed below. In 1995, they extended that result to the whole space case [] where the convergence rate is polynomial. More recent results with reduced restrictions on the cross-section were proven in [36], using the energy method from [9 3]; these results also apply to the hard potentials. For relativistic interactions when particles are fast moving an important physical regime is the soft potentials ; see [15] for a physical discussion. Despite their importance, prior to the results in this paper there were no existence results for the soft potentials in the context of strong nearby relativistic Maxwellian global solutions. In 1988 a general physical assumption was given in [18]; see (.7) and (.8). In this paper we will prove global existence of unique L near equilibrium solutions to the relativistic Boltzmann equation and rapid time decay under the full physical assumption on the cross-section from [18]. Our main focus is of course the soft potentials; and we do not require any angular cut-off (although the angular singularity will not be worse than just barely integrable). 1.. Notation. Prior to discussing our main results, we will now ine the notation of the problem carefully. In special relativity the momentum of a particle is denoted by p μ, μ =, 1,, 3. Let the signature of the metric be ( +++). Without loss of generality, we set the rest mass for each particle m = 1. The momentum for each particle is restricted to the mass shell p μ p μ = c with p >.

3 Asymptotic Stability for Soft Potentials Further with p R 3,we may write p μ = (p, p)and similarly q μ = (q, q). Then the energy of a relativistic particle with momentum p is p = c + p.weare raising and lowering indices with the Minkowski metric p μ = g μν p ν, where (g μν ) = diag( 1 111). We use the Einstein convention of implicit summation over repeated indices. The Lorentz inner product is then given by p μ q μ = p q + 3 p i q i. Then also q = c + q >. Note our convention for raising and lowering indices; we only use it in this paragraph and in the Appendix. Now the streaming term of the relativistic Boltzmann equation is given by i=1 p μ μ = p t + p x. We thus write the relativistic Boltzmann equation as t F + ˆp x F = Q(F, F). (1.1) Here Q(F, F) = C(F, F)/p, with C ined at the top of this paper. Above we consider F = F(t, x, p) to be a function of time t [, ), space x T 3 and momentum p R 3. The normalized velocity of a particle is denoted ˆp = c p p = p 1+ p /c. (1.) Steady states of this model are the well known Jüttner solutions, also known as the relativistic Maxwellian. They are given by J(p) = exp ( cp /(k B T )) 4πck B TK (c /(k B T )), where K ( ) is the Bessel function K (z) = z 1 e zt (t 1) 3/ dt, T is the temperature and k B is Boltzmann s constant. In the rest of this paper, without loss of generality but for the sake of simplicity, we will now normalize all physical constants to one, including the speed of light to c = 1. So that in particular we denote the relativistic Maxwellian by J(p) = e p 4π. (1.3) Henceforth we let C, and sometimes c denote generic positive inessential constants whose value may change from line to line. We will now ine the quantity s, which is the square of the energy in the center of momentum system, p + q =, as s = s(p μ, q μ ) = (p μ + q μ )(p μ + q μ ) = ( p μ q μ +1 ). (1.4) The relative momentum is denoted g = g(p μ, q μ ) = (p μ q μ )(p μ q μ ) = ( p μ q μ 1). (1.5)

4 R. M. Strain Notice that s = g + 4. We warn the reader that this notation, which is used in [9], may differ from other author s notation by a constant factor. Conservation of momentum and energy for elastic collisions is expressed as The angle θ is ined by p μ + q μ = p μ + q μ. (1.6) cos θ = (p μ q μ )(p μ q μ )/g. (1.7) This angle is well ined under (1.6), see [1, Lemma ]. We now consider the center of momentum expression for the collision operator below. An alternate expression for the collision operator was derived in [4]; see [49] for an explanation of the connection between the expression from [4] and the one we give just now. One may use Lorentz transformations as described in [9] and [5] to reduce the delta functions and obtain Q( f, h) = dq dω v ø σ(g,θ)[ f (p )h(q ) f (p)h(q)], (1.8) R 3 S where v ø = v ø (p, q) is the Møller velocity given by p v ø = v ø (p, q) = q p q p q p q = g s. (1.9) p q The post-collisional momentum in the expression (1.8) can be written: p = p + q + g ( ) (p + q) ω ω + (γ 1)(p + q) p + q, q = p + q g ( ) (1.1) (p + q) ω ω + (γ 1)(p + q) p + q, where γ = (p + q )/ s. The energies are then p = p + q + g ω (p + q), s q = p + q g (1.11) s ω (p + q). These clearly satisfy (1.6). The angle further satisfies cos θ = k ω with k = k(p, q) and k =1. This is the expression for the collision operator that we will use. For a smooth function h(p) the collision operator satisfies dp 1 p Q(h, h)(p) =. R 3 p By integrating the relativistic Boltzmann equation (1.1) and using this identity we obtain the conservation of mass, momentum and energy for solutions as d dx dp 1 p F(t) =. dt T 3 R 3 p

5 Asymptotic Stability for Soft Potentials Furthermore the entropy of the relativistic Boltzmann equation is ined as H(t) = dx dp F(t, x, p) ln F(t, x, p). T 3 R 3 The celebrated Boltzmann H-Theorem is then formally d H(t), dt which says that the entropy of solutions is increasing as time passes. Notice that the steady state relativistic Maxwellians (1.3) maximize the entropy which formally implies convergence to (1.3) in large time. It is this formal reasoning that our main results make mathematically rigorous in the context of perturbations of the relativistic Maxwellian for a general class of cross-sections.. Statement of the Main Results We are now ready to discuss in detail our main results. We ine the standard perturbation f (t, x, p) to the relativistic Maxwellian (1.3) as F = J + Jf. With (1.6) we observe that the quadratic collision operator (1.8) satisfies Q(J, J) =. Then the relativistic Boltzmann equation (1.1) for the perturbation f = f (t, x, p) takes the form t f + ˆp x f + L( f ) = Ɣ( f, f ), f (, x, p) = f (x, p). (.1) The linear operator L( f ), as ined in (.), and the non-linear operator Ɣ( f, f ), ined in (.5), are derived from an expansion of the relativistic Boltzmann collision operator (1.8). In particular, the linearized collision operator is given by L(h) = J 1/ Q(J, Jh) J 1/ Q( Jh, J) = ν(p)h K (h). (.) Above the multiplication operator takes the form ν(p) = dq dωv ø σ(g,θ) J(q). (.3) R 3 S The remaining integral operator is K (h) = dq dωv ø σ(g,θ) { J(q) J(q ) h(p ) + J(p ) h(q )} R 3 S dq dωv ø σ(g,θ) J(q)J(p) h(q) R 3 S = K (h) K 1 (h). (.4)

6 R. M. Strain The non-linear part of the collision operator is ined as Ɣ(h 1, h ) = J 1/ Q( Jh 1, Jh ) = dq dωv ø σ(g,θ) J(q)[h 1 (p )h (q ) h 1 (p)h (q)]. (.5) R 3 S Without loss of generality we can assume that the mass, momentum, and energy conservation laws for the perturbation, f (t, x, p), hold for all t as dx dp 1 p J(p) f (t, x, p) =. (.6) T 3 R 3 p We now state our conditions on the collisional cross-section. Hypothesis on the collision kernel. For soft potentials we assume the collision kernel in (1.8) satisfies the following growth/decay estimates: σ(g,ω) g b σ (ω), ( ) g σ(g,ω) g b σ (ω). s (.7) We also consider angular factors such that σ (ω) sin γ θ with γ>. Additionally σ (ω) and σ (ω) should be non-zero on a set of positive measure. We suppose further that < b < min(4, 4+γ). For hard potentials we make the assumption ( σ(g,ω) g a + g b) σ (ω), ( ) g σ(g,ω) g a σ (ω). s (.8) In addition to the previous parameter ranges we consider a +γ and also b < min(4, 4+γ)(in this case we allow the possibility of b = ). This hypothesis includes the full range of conditions which were introduced in [18] as a general physical assumption on the kernel (of course we add the corresponding necessary lower bound in each case); see also [15] for further discussions. Prior to stating our main theorem, we will need to introduce the following mostly standard notation. The notation A B will imply that a positive constant C exists such that A CB holds uniformly over the range of parameters which are present in the inequality and moreover that the precise magnitude of the constant is unimportant. The notation B A is equivalent to A B, and A B means that both A B and B A. We work with the L norm h = ess sup x T 3, p R3 h(x, p). If we only wish to take the supremum in the momentum variables we write h = ess sup p R 3 h(p).

7 Asymptotic Stability for Soft Potentials We will additionally use the following standard L spaces: h = T 3 dx R 3 dp h(x, p), h = R 3 dp h(p). Similarly in the sequel any norm represented by one set of lines instead of two only takes into account the momentum variables. Next we ine the norm which measures the (very weak) dissipation of the linear operator h ν = T 3 dx R 3 dp ν(p) h(x, p). The L (R n ) inner product is denoted,.weuse(, ) to denote the L (T n R n ) inner product. Now, for l R, we ine the following weight function: { lb/ w l = w l (p) = p, for the soft potentials: (.7) p l, for the hard potentials: (.8). (.9) For the soft potentials w 1 (p) 1/ν(p) (Lemma 3.1). We consider weighted spaces h,l = w l h, h,l = w l h, h ν,l = w l h ν. Here as usual L l (T3 R 3 ) is the Banach space with norm,l, etc. We will also use the momentum only counterparts of these spaces h,l = w l h, h,l = w l h, h ν,l = w l h ν. We further need the following time decay norm: f k,l = sup (1+s) k f (s),l. (.1) s We are now ready to state our main results. We will first state our theorem for the soft potentials which is the main focus of this paper: Theorem.1 (Soft Potential). Fix l>3/b. Given f = f (x, p) L l (T3 R 3 ) which satisfies (.6) initially, there is an η> such that if f,l η, then there exists a unique global mild solution, f = f (t, x, p), to Eq.(.1) with soft potential kernel (.7). For any k, there is a k = k (k) such that f,l (t) C l,k (1+t) k f,l+k. These solutions are continuous if it is so initially. We further have positivity, i.e. F = J + Jf if F = J + Jf. We point out that k () = in the above theorem, and k (k) k can in general be computed explicitly from our proof. Our approach also applies to the hard potentials and in that case we state the following theorem which can be proven using the same methods.

8 R. M. Strain Theorem. (Hard Potential). Fix l>3/. Given f = f (x, p) L l (T3 R 3 ) which satisfies (.6) initially, there is an η> such that if f,l η, then there exists a unique global mild solution, f = f (t, x, p), to Eq.(.1) with hard potential kernel (.8) which further satisfies for some λ> that f,l (t) C l e λt f,l. These solutions are continuous if it is so initially. We further have positivity, i.e. F = J + Jf if F = J + Jf. Previous results for the hard potentials are as follows. In 1993 Glassey and Strauss [4] proved asymptotic stability such as Theorem. in L l with l>3/. They consider collisional cross-sections which satisfy (.8) for the parameters b [, 1/), a [, b) and either γ or { γ < min a, 1 } b a b,, 3 which in particular implies γ > 1 if b = say. They further assume a related growth bound on the derivative of the cross-section σ g. In [36] this growth bound was removed while the rest of the assumptions on the cross-section from [4] remained the same. These results also sometimes work in smoother function spaces, and we note that we could also include space-time regularity to our solutions spaces. However for the relativistic Boltzmann equation in (1.1) the issue of adding momentum derivatives is more challenging. In recent years many new tools have been developed to solve these problems. A method was developed in non-relativistic kinetic theory to study the soft potential Boltzmann equation with angular cut-off by Guo in [3]. This approach makes crucial use of the momentum derivatives, and Sobolev embeddings to control the singular kernel of the collision operator. Yet in the context of relativistic interactions, high derivatives of the post-collisional variables (1.1) create additional high singularities which are hard to control. Worse in the more common relativistic variables from [4], derivatives of the post-collisional momentum exhibit enough momentum growth to preclude hope of applying the method from [3]; these growth estimates on the derivatives were known much earlier in [3]. Notice also that the methods for proving time decay, such as [1,5,53], require working in the context of smooth solutions. We would also like to mention recent developments on Landau Damping [44] proving exponential decay with analytic regularity. Furthermore we point out very recent results proving rapid time decay of smooth perturbative solutions to the Newtonian Boltzmann equation without the Grad angular cut-off assumption as in [5 7]. In this paper however we avoid smooth function spaces in particular because of the aforementioned problem created by the relativistic post-collisional momentum. Other recent work [33] developed a framework to study near Maxwellian boundary value problems for the hard potential Newtonian Boltzmann equation in L l. In particular a key component of this analysis was to consider solutions to the Boltzmann equation (.1) after linearization as ( t + ˆp x + L ) f =, f (, x, p) = f (x, p), (.11) with L ined in (.). The semi-group for this equation (relativistic or not) will satisfy a certain A-Smoothing property which was pioneered by Vidav [56] about 4

9 Asymptotic Stability for Soft Potentials years ago. This A-Smoothing property which appears at the level of the second iterate of the semi-group, as seen below in (4.4), has been employed effectively for instance in [1,4,33,34]. Related to this a new compactness connected to a similar iteration has been observed in the mixing lemma of [41 43]. The key new step in [33] wasto estimate the second iterate in L rather than L and then use a linear decay theory in L which does not require regularity and is exponential for the hard potentials case that paper considered. Note further that a method was developed in [5,53] to prove rapid polynomial decay for the soft potential Newtonian Boltzmann and Landau equations; this is related to the articles [5,6,1,55], all of which make use of smooth function spaces. In the present work we adapt the method from [53] to prove rapid L polynomial decay of solutions to the linear equation (.11) without regularity, and we further adapt the L estimate from [33] to control the second iterate. This approach, for the soft potentials in particular, yields global bounds and slow decay, O(1/t), of solutions to (.11) inl l. The details and complexity of this program are however intricate in the relativistic setting. And fortunately, this slow decay is sufficient to just barely control the nonlinearity and prove global existence to the full non-linear problem as in Theorem.1. It is not clear how to apply the above methods to establish the rapid almost exponential polynomial decay from Theorem.1 in this low regularity L l framework. To prove the rapid decay in Theorem.1 our key contribution is to perform a new high order expansion of the remainder term, R 1 ( f ), from(4.15). This is contained in Proposition 6.1 and its proof. This term, R 1 ( f ), is the crucial problematic term which only appears to exhibit first order decay. More generally, the main difficulty with proving rapid decay for the soft potentials is created by the high momentum values, where the time decay is diluted by the momentum decay. This results in the generation of additional weights, typically one weight for each order of time decay. At the same time the term R 1 ( f ) only allows us to absorb one weight, w 1 (p), and therefore only appears to allow one order of time decay. In our proof of Proposition 6.1 we are able to overcome this apparent obstruction by performing a new high order expansion for k as R k ( f ) = G k ( f ) + D k ( f ) + N k ( f ) + L k ( f ) + R k+1 ( f ). (The expansion from R 1 ( f ) to R ( f ) requires a slightly different approach.) At every level of this expansion we can peel of each of the terms G k ( f ), D k ( f ), N k ( f ), and L k ( f ) which will for distinct reasons exhibit rapid polynomial decay to any order. In particular we use an L l estimate for L k( f ) which crucially makes use of the bounded velocities that come with special relativity. On the other hand the last term R k+1 ( f ) will be able to absorb k + 1 momentum weights, and therefore it will be able to produce time decay up to the order k + 1. By continuing this expansion to any finite order, we are able to prove rapid polynomial decay. We hope that this expansion will be useful in other relativistic contexts. The rest of this paper is organized as follows. In Sect. 3 we prove L l decay of solutions to the linearized relativistic Boltzmann equation (.11). Then in Sect. 4 we prove global L l bounds and slow decay of solutions to (.11). Following that in Sect. 5 we prove nonlinear L l bounds using the slow linear decay, and we thereby conclude global existence. In the remaining Sections 6 and 7 we prove linear and non-linear rapid decay respectively. Then in the Appendix we give an exposition of a derivation of the kernel of the compact part of the linear operator from (.4).

10 R. M. Strain 3. Linear L Bounds and Decay It is our purpose in this section to prove global in time L l bounds for solutions to the linearized Boltzmann equation (.11) with initial data in the same space L l. We will then prove high order, almost exponential decay for these solutions. We begin by stating a few important lemmas, and then we use them to prove the desired integral bounds (3.5) and the decay it implies in Theorem 3.7. We will prove these lemmas at the end of the section. Lemma 3.1. Consider (.3) with the soft potential collision kernel (.7). Then ν(p) p b/. More generally, R 3 dq S dωv ø σ(g,θ) J α (q) p b/ for any α>. We will next look at the compact part of the linear operator K. The most difficult part is K from (.4). We will employ a splitting to cut out the singularity. The new element of this splitting is the Lorentz invariant argument: g. Givenasmallɛ >, choose a smooth cut-off function χ = χ(g) satisfying χ(g) = { 1 if g ɛ if g ɛ. (3.1) Now with (3.1) and (.4) we ine K 1 χ (h) = dq dω (1 χ(g)) v ø σ(g,θ) J(q) J(q ) h(p ) R 3 S + dq dω (1 χ(g)) v ø σ(g,θ) J(q) J(p ) h(q ). (3.) R 3 S Define K 1 χ 1 (h) similarly. We use the splitting K = K 1 χ + K χ. A splitting with the same goals has been previously used for the Newtonian Boltzmann equation in [53]. The advantage for soft potentials, on the singular region, is that one has exponential decay in all momentum variables. Then on the region away from the singularity we are able to extract a modicum of extra decay which is sufficient for the rest of the estimates in this paper, see Lemma 3. just below. In the sequel we will use the Hilbert-Schmidt form for the non-singular part. The following representation is derived in the Appendix: K χ i (h) = dq k χ R 3 i (p, q) h(q), i = 1,. We will also record the kernel k χ i (p, q) below in (3.9) and (3.1). We have Lemma 3.. Consider the soft potentials (.7). The kernel enjoys the estimate k χ (p, q) C χ (p q ) ζ (p + q ) b/ e c p q, C χ, c >, with ζ = min { γ, 4 b, } /4 >. This estimate also holds for k χ 1.

11 Asymptotic Stability for Soft Potentials We remark that for certain ranges of the parameters γ and b the decay in Lemma 3. could be improved somewhat. In particular the term k χ 1 in (3.9) clearly yields exponential decay. However what is written above is sufficient for our purposes. We will use the decomposition given above and the decay of the kernels in Lemma 3. to establish the following lemma. Lemma 3.3. Fix any small η>, we may decompose K from (.4) as K = K c + K s, where K c is a compact operator on L ν. In particular for any l, and for some R = R(η) > sufficiently large we have w l K ch 1, h C η 1 R h 1 1 R h. Above 1 R is the indicator function of the ball of radius R. Furthermore w l K sh 1, h η h 1 ν,l h ν,l. This estimate will be important for proving the coercivity of the linearized collision operator, L, away from its null space. More generally, from the H-theorem L is nonnegative and for every fixed (t, x) the null space of L is given by the five dimensional space [1]: { } N = span J, p1 J, p J, p3 J, p J. (3.3) We ine the orthogonal projection from L (R 3 ) onto the null space N by P. Further expand Ph as a linear combination of the basis in (3.3): 3 Ph = ah (t, x) + b h j (t, x)p j + c h (t, x)p J. (3.4) j=1 We can then decompose f (t, x, p) as With this decomposition we have f = P f + {I P} f. Lemma 3.4. L. Lh= if and only if h = Ph, and δ > such that Lh, h δ {I P}h ν. This last statement on coercivity holds as an operator inequality at the functional level. The following lemma is a well-known statement of the linearized H-Theorem for solutions to (.11) which was shown in the non-relativistic case in [33]. Lemma 3.5. Given initial data f L ( l T 3 R 3) for some l, which satisfies (.6) initially, consider the corresponding solution, f, to (.11) in the sense of distributions. Then there is a universal constant δ ν > such that 1 ds {I P} f ν (s) δ ν 1 ds P f ν (s).

12 R. M. Strain We will give just one more operator level inequality. Lemma 3.6. Given δ (, 1) and l, there are constants C, R > such that w l Lh, h δ h ν,l C 1 Rh. Notice that Lemma 3.6 follows easily from Lemma 3.3. With these results, we can prove the following energy inequality for any l : f,l (t) + δ l t ds f ν,l (s) C l f,l, δ l, C l >, (3.5) as long as f,l is finite. We will prove this first for l =, and then for arbitrary l>. In the first case we multiply (.11) with f and integrate to obtain f (t) + t ds (Lf, f ) = f. First suppose that t {1,,...}. By Lemma 3.4 we have t t 1 1 ds (Lf, f ) = j= t 1 1 = δ + δ j= t 1 1 j= t 1 1 ds (Lf, f )(s + j) δ j= ds {I P} f ν (s + j) ds {I P} f ν (s + j). ds {I P} f ν (s + j) Then by Lemma 3.5 the second term above satisfies the lower bound δ t 1 1 j= ds {I P} f ν (s + j) δ δ ν t 1 1 j= ds P f ν (s + j). This follows in particular because f j (s, x,v) = f (s + j, x, p) satisfies the linearized Boltzmann equation (.11) on the interval s 1. Collecting the previous two estimates yields t ds (Lf, f ) δ t δ δ ν δ j= t 1 1 j= t 1 1 j= ds {I P} f ν (s + j) ds P f ν (s + j) ds f ν (s + j) = δ t ds f ν (s),

13 Asymptotic Stability for Soft Potentials { } with δ = 1 min δ δ ν, δ >. Plugging this estimate into the last one establishes the energy inequality (3.5) forl = and t {1,,...}. For an arbitrary t >, we choose m {, 1,,...} such that m t m + 1. We then split the time integral as [, t] =[, m] [m, t]. For the time interval [m, t] we have t f (t) + ds (Lf, f ) = f (m). (3.6) m Since L by Lemma 3.4, we see that f (t) f (m). We then have t f (t) + ds (Lf, f ) + δ m ds f ν m (s) C k f. Furthermore with Lemma 3.6 for C δ > independent of t we obtain t m ds (Lf, f ) δ δ t m t m ds f ν (s) C δ ds f ν (s) C δ t m ds 1 R f (s) sup f (s). (3.7) m s t However we have already shown that sup m s t f (s) f (m) C f. Collecting the last few estimates for any t > wehave(3.5) forl =. For l>, we multiply Eq. (.11) with wl f and integrate to obtain t f,l (t) + ds (wl Lf, f ) = f,l. In this case using Lemma 3.6 we have t t ds (wl Lf, f ) δ ds f ν,l (s) C δ,l Adding this estimate to the line above it, we obtain t f,l (t) + δ ds f ν,l (s) f,l + C δ,l t t ds f ν (s). ds f ν (s). Now the just proven integral inequality (3.5) in the case l = (as an upper bound for the integral on the right-hand side) establishes the claimed energy inequality (3.5) for any l>. In fact we prove a more general time decay version of this inequality in (3.51). These will imply the following rapid decay theorem. Theorem 3.7. Consider a solution f (t, x, p) to the linear Boltzmann equation (.11) with data f,l+k < for some l, k. Then f,l (t) C l,k (1+t) k f,l+k. This allows almost exponential polynomial decay of any order. Theorem 3.7 is the main result of this section. We now proceed to prove each of Lemma 3.1, Lemma 3., Lemma 3.3, Lemma 3.4, Lemma 3.5, and then Theorem 3.7 in order. These proofs will complete this section on linear decay.

14 R. M. Strain Proof of Lemma 3.1. We will use the soft potential hypothesis for the collision kernel (.7) to estimate (.3). For α>we more generally consider π ν α (p) = dq v ø J α (q) dθ σ(g,θ) sin θ. R 3 Initially we record the following pointwise estimates: p q g p q, and g p q, (3.8) p q see [4, Lemma 3.1]. However notice that in [4] their g is actually ined to be 1/ times our g. With the Møller velocity (1.9), we thus also have s = 4+g p q, v ø 1. For b (1, 4), these estimates including (3.8) yield s s v ø g b = g 1 b (p q ) (b 1)/ p q p q p q b 1 (p q ) (b )/ p q b 1. We thus obtain R3 ν α (p) dq (p q ) (b )/ π p q b 1 J α (q) dθ sin 1+γ θ p (b )/ R 3 dq J α/ (q) p q b 1 π p (b )/ p 1 b p b/. We note that the angular integral is finite since γ> : π dθ sin 1+γ θ = C γ <. dθ sin 1+γ θ In this case above and the cases below a key point is that dq J α/ (q) R 3 p q β p β, β <3. For b (, 1), with (3.8) we alternatively have s v ø g b = g 1 b (p q ) (1 b)/ (p q ) b/. p q p q Now in a slightly easier way than for the previous case we have ν α (p) p b/. For the lower bound with b (, 4), weuse(.7), (3.8) and the estimate ( ) g s v ø g b = g b (p q ) ( b)/ (p q ) b/. p q p q Alternatively for b (, ), weuse(3.8) to get the estimate ( ) g s v ø g b = g b 1 ( ) p q b (p q ) (b 4)/ p q b. p q p q p q We use these estimates to obtain ν α (p) p b/ as for the upper bound.

15 Asymptotic Stability for Soft Potentials Now that the proof of Lemma 3.1 is complete, we develop the necessary formulation for the proof of Lemma 3.. It is trivial to write the Hilbert-Schmidt form for the cut-off (3.1) part of K 1 from (.4) as k χ 1 (p, q) = J(q)J(p) χ(g) dωv ø σ(g,θ). (3.9) S Furthermore, we can write the Hilbert-Schmidt form for the cut-off (3.1) part of K from (.4) in the following somewhat complicated integral form k χ s 3/ (p, q) = c χ(g) gp q y dy e l ( ) 1+y g σ sin (ψ/),ψ (1+ ) 1+y 1+y I ( jy). (3.1) Here c >, and the modified Bessel function of index zero is ined by I (y) = 1 π We are also using the simplifying notation π e y cos ϕ dϕ. (3.11) Additionally sin (ψ/) = g [g 4+(g +4). (3.1) 1+y ] 1/ p q l = (p + q )/, j =. g This derivation for a pure k operator appears to go back to [9,11], where it was done in the case of the alternate linearization F = J(1+ f ). The author gave a similar derivation with many details explained in full in [5], including the explicit form of the necessary Lorentz transformation; in particular Eq. (5.51) in this thesis. For the benefit of the reader, we have provided the derivation of k χ in the case of the linearization F = J + Jfin the Appendix to this paper. Note that it is elementary to verify that (3.9) under (.7) satisfies the estimate in Lemma 3.. In the proof below we focus on the more involved estimate for (3.1). Proof of Lemma 3.. We consider K from (.4) with Hilbert-Schmidt form represented by the kernel (3.1). From (.7), we have the bound ( ) ( ) g sin (ψ/) b σ sin (ψ/),ψ sin γ ψ g ( ) sin (ψ/) b+γ g γ cos γ (ψ/). (3.13) g We have just used the trigonometric identity sin ψ = sin(ψ/) cos (ψ/).

16 R. M. Strain We estimate these angles in three cases. In each of the cases below we will repeatedly use the following known [4, p.317] estimates y cos (ψ/) 1. (3.14) 1+y The estimates above and below are proved for instance in [4, Lemma 3.1]: 1 sin (ψ/) s(1+y ) 1/4 g Notice that in general from (3.1) and (3.13) we have the bound 1 g(1+y. (3.15) ) 1/4 k χ s3/ (p, q) g γ χ(g) dy e l 1+y yi ( jy) gp q ( ) sin (ψ/) b+γ cos γ (ψ/). (3.16) g We will estimate this upper bound in three cases. Case 1. Take γ = γ < and b + γ = b γ <. Then we have ( ) sin (ψ/) b+γ ( ) ( cos γ (ψ/) s (b γ )/ 1 b γ y g (1+y ) 1/4 1+y s (b γ )/ y γ (1+y (b 3 γ )/4 ) s (b γ )/ y γ (1+y ) (b+3γ)/4. Above we have used (3.14) and (3.15). In this case from (3.16) wehave k χ s3/ (p, q) gp q s (b γ )/ g γ s (3+ γ b)/ C ɛ χ(g) p q χ(g) ) γ dy e l 1+y y 1+γ I ( jy)(1+y ) (b+3γ)/4 dy e l 1+y y 1 γ I ( jy)(1+y ) (b 3 γ )/4. Note that we have just used the ɛ>from(3.1). From (.7), b (, 4 γ ) and γ (, ). We have in this case b (, γ ). Hence b 3 γ ( 6, ). We evaluate the relevant integral above as dy e l 1 1+y y 1 γ I ( jy)(1+y ) (3 γ b)/4 = dy + dy 1 1 dy e l 1+y y 1 γ I ( jy) + dy e l 1+y yi ( jy)(1+y ) ( γ b)/4. 1 (3.17) For the unbounded integral we have used the estimate y γ (1+y ) γ /4. In this case γ b (, ) since < b < γ.

17 Asymptotic Stability for Soft Potentials To estimate the remaining integrals above we use the precise theory of special functions, see e.g. [39,46]. We ine K α (l, j) = dy e l 1+y yi ( jy)(1+y ) α/4. Then for α [, ] from [4, Cor. 1 and Cor. ] it is known that K α (l, j) Cl 1+α/ e c p q. (3.18) We also ine 1 Ĩ η (l, j) = dy e l 1+y y 1 η I ( jy). Then for η [, ) from [4, Lemma 3.6] we have the asymptotic estimate: Ĩ η (l, j) Ce c l j Ce c p q /. (3.19) We will use these estimates in each of the cases below. Thus in this Case 1 by (3.19) and (3.18) the integral in (3.17) is e c p q / + l 1+( γ b)/ e p q / l 1+( γ b)/ e c p q /. We may collect the last few estimates together to obtain k χ (p, q) C s (3+ γ b)/ ɛ χ(g) (p + q ) 1+( γ b)/ e c p q /. p q Note that for any l R we have s l e c p q / C l e c p q /4. This follows trivially from (3.8) and s = 4+g 4+ p q. Furthermore, for any l [, ],weclaim the following estimate (p + q ) l e c p q /4 (p q ) l/ 1 e c p q /8. (3.) p q Using (3.) with l = 1+ γ /, in this Case 1, we have the general estimate k χ (p, q) C ( ) γ / 1 ɛ p q (p + q ) b/ e c p q. This is the desired estimate in the current range of exponents for Case 1. Before moving on to the next case, we establish the claim. Suppose that 1 q p q. In this case (3.) isobvious.if 1 q p, then we have p q q p 1 q. (3.1)

18 R. M. Strain Whence (p + q ) l e c p q /4 C l (p q ) 1 e c p q /8 e c q /64. p q In the last splitting p q, then we alternatively have Similarly in this situation p q p q 1 p. (3.) (p + q ) l e c p q /4 C l (p q ) 1 e c p q /8 e c p /64. p q These last two stronger estimates establish (3.). We move on to the next case. Case. We still consider γ = γ < but now b + γ = b γ. We have ( ) sin (ψ/) b+γ ( ) ( cos γ (ψ/) g b+ γ 1 b γ y g (1+y ) 1/4 1+y g (b γ ) y γ (1+y (b 3 γ )/4 ) C ɛ y γ (1+y ) (b 3 γ )/4. ) γ We have used g ɛ on the support of χ(g) in (3.1). We also used (3.14) and (3.15). Then again from (3.16) wehave s 3/ k χ (p, q) C ɛ dy e l 1+y y 1 γ I ( jy)(1+y ) (b 3 γ )/4. p q From (.7), we have in this case that γ (, ) and b [ γ, 4 γ ). Hence for the exponent above b 3 γ ( 4, 4). Then the relevant integral above is bounded by dy e l 1 1+y y 1 γ I ( jy)(1+y ) (b 3 γ )/4 = dy + dy 1 1 dy e l 1+y y 1 γ I ( jy) + dy e l 1+y yi ( jy)(1+y ) ( γ b)/4. 1 Here we used the same estimates as in Case 1. In this case γ b ( 4, ). Let ζ = max(, γ b). Then in this case by (3.19) and (3.18) the above is Ce c p q / + Cl 1+ζ / e p q / Cl 1+ζ / e c p q /. We may collect the last few estimates together to obtain k χ (p, q) C s 3/ ɛ (p + q ) 1+ζ/ e c p q / (p + q ) 1+ζ/ C ɛ e c p q /4. p q p q We will further estimate the quotient.

19 Asymptotic Stability for Soft Potentials If ζ = γ b, then 1 + γ / [1, ) and (3.) implies (p + q ) 1+ζ / e c p q /4 = (p + q ) 1+ γ / (p + q ) b/ e c p q /4 p q p q (p q ) ( γ )/4 (p + q ) b/ e c p q /8. Alternatively, if ζ = then 1 + ζ / = and (3.) implies (p + q ) 1+ζ / e c p q /4 = (p + q ) b/ (p + q ) b/ e c p q /4 p q p q (p q ) (b 4)/4 (p + q ) b/ e c p q /8. In either situation k χ (p, q) C ɛ (p q ) ζ (p + q ) b/ e c p q /8, with ζ = min { γ, 4 b} /4 >. Case 3. In this last case γ = γ and b + γ. From (3.14) and (3.15): ( ) sin (ψ/) b+γ ( cos γ (ψ/) g b γ g 1 (1+y ) 1/4 ) b+ γ g (b+ γ ) (1+y (b+ γ )/4 ) C ɛ (1+y ) (b+ γ )/4. We have again used g ɛ on the support of χ(g) in (3.1). In this case from (3.16) we have k χ (p, q) C s 3/ g γ ɛ dy e l 1+y yi ( jy)(1+y ) (b+ γ )/4. p q From (.7), b [, 4) and b + γ [, 4+ γ ).Letζ 3 = min(, b + γ ). Again using (3.18) wehave dy e l 1+y yi ( jy)(1+y ) (b+ γ )/4 Cl 1 ζ 3/ e c p q /. Hence k χ (p, q) C s 3/ g γ ɛ (p + q ) 1 ζ3/ e c p q / (p + q ) 1 ζ3/ C ɛ e c p q /4. p q p q If ζ 3 = this estimate can be handled exactly as in Case. If ζ 3 = b + γ : (p + q ) 1 ζ 3/ e c p q /4 = (p + q ) 1 γ / (p + q ) b/ e c p q /4 p q p q (p + q ) (p + q ) b/ e c p q /4 p q (p q ) 1/ (p + q ) b/ e c p q /16. We have again used the estimate (3.). In all of the cases we see that k χ (p, q) satisfies the claimed bound from Lemma 3. with ζ = min { γ, 4 b, } /4 >. We could obtain a larger ζ in Case 3 if it was needed.

20 R. M. Strain With the estimate for the Hilbert-Schmidt form just proven in Lemma 3., we will now prove the decomposition from Lemma 3.3. Proof of Lemma 3.3. We recall the splitting K = K 1 χ + K χ from (3.). For K χ we have the kernel k χ = k χ kχ 1 from (3.9) and (3.1). For a given R 1, choose another smooth cut-off function φ R = φ R (p, q) satisfying φ R 1, if p + q R/, φ R 1, (3.3) supp(φ R ) {(p, q) p + q R }. We will use this cut-off with several different R s in the cases below. Now we split the kernels k χ (p, q) of the operator K χ into We further ine k χ (p, q) = k χ (p, q)φ R (p, q) + k χ (1 φ R ) = kc χ (p, q) + kχ s (p, q). K s = K 1 χ + Ks χ, where Ks χ (h) = R 3 dq ks χ (p, q) h(q). Then the compact part is given by K c = Kc χ, where Kc χ (h) = R 3 dq kc χ (p, q) h(q). Note that the compactness of K c (h) is evident from the integrability of the kernel. In the following we will show that the operators K c and K s satisfy the estimates claimed in Lemma 3.3. Firstoff,forK c, from the Cauchy-Schwartz inequality we have wl K c(h 1 ), h dq dp w R 3 R 3 l (p) k χ c (p, q) h 1 (q)h (p) ( dqdp wl (p) ) 1/ kc χ (p, q) h 1 (q) ( dqdp w l (p) k χ c (p, q) h (p) ) 1/. From the inition of k χ c (p, q) and Lemma 3., we see that w l (p) k χ c (p, q) CR e c p q 1 R (p) 1 R (q), where 1 R is the indicator function of the ball of radius R centered at the origin as ined in Lemma 3.3. By combining the last few estimates we clearly have the claimed estimate for K c from Lemma 3.3. In the remainder of this proof we estimate K s = K 1 χ + Ks χ. For Ks χ we have wl K s χ (h 1), h dq dp w R 3 R 3 l (p) k χ s (p, q) h 1 (q)h (p). With the inition of ks χ (p, q) and Lemma 3., we obtain w l (p) k χ s (p, q) = w l (p) k χ (p, q) (1 φr ) w l (p) R ζ (p + q ) b/ e c p q.

21 Asymptotic Stability for Soft Potentials Furthermore, we claim that w l (p)e c p q w l (p)w l (q)e c p q /. (3.4) By combining the last few estimates including (3.4), with Lemma 3.1, wehave wl K s χ (h 1), h 1 R R ζ dq 3 R3 dp w l (p)w l (q) e c p q (p q ) b/4 h 1(q)h (p) h 1 ν,l h ν,l R ζ. Since ζ>we conclude our estimate here by choosing R > sufficiently large. Notice that the size of this R above clearly depends upon ɛ>from(3.1). To prove the claim in (3.4), we use the same general strategy which was used to prove (3.). Indeed, if 1 q p then because of (3.1) wehave w l (p)e c p q / e c p q / w l (q)e c q /4 e c p q / C l e c p q /, which is better than (3.4). Alternatively if p q, then with (3.) wehave w l (p)e c p q / e c p q / w l (p)e c p /4 e c p q / C l e c p q /. The only remaining case is 1 q p q for which the estimate (3.4) is obvious. The last term to estimate is K 1 χ = K 1 χ K 1 χ 1. Notice that K 1 χ 1 (h) = dq k 1 χ R 3 1 (p, q) h(q), where from (.4) and (3.1), k 1 χ 1 (p, q) = (1 χ(g)) J(q)J(p) S dωv ø σ(g,θ). To estimate K 1 χ 1 we apply Cauchy-Schwartz to obtain wl K 1 χ 1 (h 1 ), h dq dp w R 3 R 3 l (p) k1 χ 1 (p, q) h 1 (q)h (p) ( ) 1/ dpdq wl (p)k1 χ 1 (p, q) h 1 (q) ( ) 1/ dpdq wl (p)k1 χ 1 (p, q) h (p). We will estimate the kernel of each term above. We further split k 1 χ 1 (p, q) = k 1 χ 1 (p, q)φ R (p, q) + k 1 χ 1 (1 φ R ) = k 1 χ 1S + k 1 χ 1L. The value of R 1 used here is independent of the case considered previously. The R here will be independent of ɛ.from(.7) and (1.9), in general we have S dωv ø σ(g,θ) s p q g 1 b π dθ sin 1+γ θ s p q g 1 b. (3.5)

22 R. M. Strain For g ɛ as in (3.1),with (3.8) we conclude that p q ɛ p q. (3.6) Furthermore, on the support of φ R we notice that additionally p q 4ɛR. Then if b (1, 4), with the formula for k 1 χ 1 (p, q), (3.5) and then (3.8), we obtain s dp wl (p) k1 χ 1S (p, q) C R dp g 1 b J(q)J(p) p q 4ɛ R p q C R dp g 1 b J 1/4 (q)j 1/4 (p) p q 4ɛ R ( ) p q b 1 C R dp J 1/4 (q)j 1/4 (p) p q 4ɛ R p q CR 4 b ɛ 4 b J 1/8 (q). The last inequality above follows easily from ( ) b 1 p q J 1/8 (q)j 1/8 (p) C and also J 1/8 (p) dp p q 4ɛ R p q b 1 CR4 b ɛ 4 b. Thus when b (1, 4) we have wl K 1 χ 1S (h 1 ), h C R ɛ 4 b J 1/16 h 1 J 1/16 h. (3.7) This is much stronger than the desired estimate for ɛ = ɛ(r) > chosen sufficiently small. Alternatively if b [, 1] then with (3.5) wehave dp w R 3 l (p) k1 χ 1S (p, q) C dp J 1/4 (q)j 1/4 (p) CR 3 ɛ 3 J 1/4 (q). p q 4ɛ R Thus when b [, 1] we have wl K 1 χ 1 (h 1 ), h C R ɛ 3 J 1/8 h 1 J 1/8 h. This concludes our estimate for the part containing k 1 χ 1S (p, q) for any fixed R after choosing ɛ = ɛ(r) > small enough (depending on the size of R). For the term involving k 1 χ 1L (p, q) the estimate is much easier. In this case dp wl (p) k1 χ 1L R (p, q) dp g 1 b (1 φ R (p, q)) J 1/4 (q)j 1/4 (p) 3 R 3 e R/16. (3.8) The same estimates hold for the other term in the inner product above. These estimates are independent of ɛ. We thus obtain the desired estimate for this term in the same way as for the last term; here we first choose R > sufficiently large. The last term to estimate is K 1 χ from (3.). With (3.6) we see that p p q + q ɛ p q + q ɛp + (1+ɛ) q. The first inequality in this chain can be found in [18, Ineq. A.1]. We conclude p q and similarly q p.for<ɛ<1/4 say the constant in these inequalities can be

23 Asymptotic Stability for Soft Potentials chosen to not depend upon ɛ. Furthermore from (1.11), if g ɛ and ɛ is small (say less than 1/8), then it is easy to show that p p + q 4, q p + q. (3.9) 4 These post-collisional energies are also clearly bounded from above by p and q,so that all of these variables are comparable on (3.). We thus have K 1 χ (h 1 ), h With Cauchy-Schwartz we obtain ( g ɛ ( ( + ( g ɛ g ɛ g ɛ R 3 R 3 S dωdqdp (1 χ(g)) v ø σ(g,θ)e cq cp ( h1 (p ) + h1 (q ) ) h (p). dωdqdp v ø σ(g,θ)e cq cp h1 (p ) ) 1/ dωdqdp v ø σ(g,θ)e cq cp h (p) ) 1/ dωdqdp v ø σ(g,θ)e cq cp h1 (q ) ) 1/ dωdqdp v ø σ(g,θ)e cq cp h (p) ) 1/. From (3.5) and the arguments just below it, for any small η>wecan estimate dωdq v ø σ(g,θ)e cq cp ηe cp/. g ɛ Above of course we have η = η(ɛ) asɛ, and by symmetry the same estimate holds if the roles of p and q are reversed. Since the kernels of the integrals above are invariant with respect to the relativistic pre-post collisional change of variables [3], which is justified for (1.1), we may apply it as dpdq = p q p q dp dq. Putting all of this together with (1.6) wehave ( ) 1/ ( K 1 χ (h 1 ), h η(ɛ) dp e cp h 1 (p) dp e cp h (p) ) 1/. For ɛ>small enough, this is stronger than the estimate which we wanted to prove. We have now completed the proof of this lemma. We will at this time use the prior lemma to prove the next lemma.

24 R. M. Strain Proof of Lemma 3.4. Most of this lemma is standard, see e.g. [1]. We only prove the coercive lower bound for the linear operator. Assuming the converse grants a sequence of functions h n (p) satisfying Ph n =, h n ν = νhn, h n =1 and Lh n, h n = h n ν Khn, h n 1 n. Thus {h n } is weakly compact in ν with limit point h. By weak lower-semi continuity h ν 1. Furthermore, We claim that Lh n, h n =1 Kh n, h n. lim n Khn, h n = Kh, h. The claim will follow from the prior Lemma 3.3. Thisclaim implies Or equivalently = 1 Kh, h. Lh, h = h ν 1. Since L, we have h ν = 1 which implies h = Ph. On the other hand since h n ={I P}h n the weak convergence implies h ={I P}h. This is a contradiction to h ν = 1. We now establish the claim. For any small η>, we split K = K c + K s as in Lemma 3.3. Then K s h n, h n η. Also K c is a compact operator in L ν so that lim K ch n K c h ν =. n We conclude by first choosing η small and then sending n. We are now ready to prove Lemma 3.5. We point out that similar estimates, but with strong Sobolev norms, have been established in recent years [8,9,31,51] via the macroscopic equations for the coefficients a, b and c. We will use the approach from [33], which exploits the hyperbolic nature of the transport operator, to prove our Lemma 3.5 in the low regularity L setting. Proof of Lemma 3.5. We use the method of contradiction, if Lemma 3.5 is not valid then for any k 1 we can find a sequence of normalized solutions to (.11) which we denote by f k that satisfy 1 ds {I P} f k ν (s) 1 k Equivalently the normalized function Z k (t, x, p) = 1 ds P f k ν (s). f k (t, x, p), 1 ds P f k ν (s)

25 Asymptotic Stability for Soft Potentials satisfies and 1 1 ds PZ k ν (s) = 1, ds {I P}Z k ν (s) 1 k. (3.3) Moreover, from (.6) the following integrated conservation laws hold: 1 ds dx dp 1 p J(p) Z k (s, x, p) =. (3.31) T 3 R 3 Furthermore, since f k satisfies (.11), so does Z k. Clearly 1 sup k 1 Hence there exists Z(t, x, p) such that p ds Z k ν (s) 1. (3.3) Z k (t, x, p) Z(t, x, p), as k, weakly with respect to the inner product 1 ds (, ) ν of the norm 1 ds ν. Furthermore, from (3.3) we know that 1 ds {I P}Z k ν (s). (3.33) We conclude that {I P}Z k {I P}Z and {I P}Z = from (3.33). It is then straightforward to verify that Hence PZ k PZ weakly in 1 ds ν. Z(t, x, p) = PZ ={a(t, x) + p b(t, x) + p c(t, x)} J. (3.34) At the same time notice that LZ k = L{I P}Z k and we have (3.33). Send k in (.11) forz k to obtain, in the sense of distributions, that t Z + ˆp x Z =. (3.35) At this point our main strategy is to show, on the one hand, Z has to be zero from (3.33), the periodic boundary conditions, and the hyperbolic transport equation (3.35), and (3.31). On the other hand, Z k will be shown to converge strongly to Z in 1 ds ν with the help of the averaging lemma [1] in the relativistic formulation [47] and 1 ds Z ν (s) >. This would be a contradiction.

26 R. M. Strain Strong convergence. We begin by proving the strong convergence, and then later we will prove that the limit is zero. Split Z k (t, x, p) as Z k (t, x, p) = PZ k + {I P}Z k = 5 Z k (t, x, ), e j e j (p) + {I P}Z k, j=1 where e j (p) are an orthonormal basis for (3.3)in ν. To prove the strong convergence in 1 ds ν, recalling (3.33), we will show 1 ds Z k, e j e j Z, e j e j ν (s). 1 j 5 Since e j (p) are smooth with exponential decay when p, it suffices to prove 1 ds dx Z k, e j Z, e j. (3.36) T 3 We will now establish (3.36) using the averaging lemma. Choose any small η>and a smooth cut off{ function χ} 1 (t, x, p) in (, 1) T 3 R 3 such that χ 1 (t, x, p) 1in[η, 1 η] T 3 p η 1 and χ 1 (t, x, p) outside { } [η/, 1 η/] T 3 p η. Split Z k (t, x, ), e j = (1 χ 1 ) Z k (t, x, ), e j + χ 1 Z k (t, x, ), e j. (3.37) For the first term above, notice that 1 ds dx (1 χ 1 ) Z k Z, e j T (1 χ 1 ) Z k e j + T 3 R 3 (1 χ 1 ) Z e j T 3 R s η 1 η s 1 p 1/η Since e j = e j (p) has exponential decay in p we have the crude bound e j (p) C η, for p 1/η. Thus all three integrals above can be bounded by ) Cη sup ( Z k (s) + Z (s) C s 1 ( Z k () + Z() ) η Cη, (3.38) which will hold for Z k uniformly in k. These bounds follow from (.11) and (3.35). The second term in (3.37), χ 1 Z k (t, x, ), e j, is actually uniformly bounded in H 1/4 ([, 1] T 3 ). To prove this, notice that (.11) implies that χ 1 Z k satisfies [ t + ˆp x ] (χ 1 Z k ) = χ 1 L[Z k ] + Z k [ t + ˆp x ]χ 1. (3.39)

27 Asymptotic Stability for Soft Potentials The goal is to show that each term on the right-hand side of (3.39) is uniformly bounded in L ([, 1] T 3 R 3 ). This would imply the H 1/4 bound by the averaging lemma. It clearly follows from (3.3) that Z k [ t + ˆp x ]χ 1 L ([, 1] T 3 R 3 ). Furthermore, it follows from Lemma 3.1 and Lemma 3.3 with l = that dp χ 1 L[Z k ] dp χ 1 ν(p)z k + dp χ 1 K (Z k ) Z k (t, x) R 3 R 3 R 3 ν. Thus the right-hand side of (3.39) is uniformly bounded in L ([, 1] T 3 R 3 ). By the averaging lemma [1,47] it follows that χ 1 Z k (t, x, ), e j = dp χ 1 (t, x, p)z k (t, x, p)e j (p) H 1/4 ([, 1] T 3 ). R 3 This holds uniformly in k, which implies up to a subsequence that χ 1 Z k (t, x, ), e j χ 1 Z(t, x, ), e j in L ([, 1] T 3 ). Combining this last convergence with (3.38) concludes the proof of (3.36). As a consequence of this strong convergence we have which implies that 1 ds Z k Z ν (s), 1 ds PZ ν (s) = 1. Now if we can show that at the same time PZ =, then we have a contradiction. The limit function Z(t, x, p) =. By analysing the equations satisfied by Z, we will show that Z must be trivial. We will now derive the macroscopic equations for PZ s coefficients a, b and c. Since {I P}Z =, we see that PZ solves (3.35). We plug the expression for PZ in (3.34) into Eq. (3.35), and expand in the basis (3.3) to obtain { a + p { } j j a + p j p { } } { i i b j + p j { b j + j c + p c} } J 1/ (p) =, p p where = t and j = x j. By a comparison of coefficients, we obtain the important relativistic macroscopic equations for a(t, x), b i (t, x) and c(t, x): which hold in the sense of distributions. c =, (3.4) i c + b i =, (3.41) (1 δ ij ) i b j + j b i =, (3.4) i a =, (3.43) a =, (3.44)

28 R. M. Strain We will show that these Eqs. (3.4) (3.44), combined with the periodic boundary conditions imply that any solution to (3.4) (3.44) is a constant. Then the conservation laws (3.31) will imply that the constant can only be zero. We deduce from (3.44) and (3.43) that a(t, x) = a(, x), a.e. x, t, a(s, x 1 ) = a(s, x ), a.e. s, x 1, x. Thus a is a constant for almost every (t, x). From(3.4), we have c(t, x) = c(x) for a.e. t. Then from (3.41) for some spatially dependent function b i (x) we have From (3.4) and the above which implies Similarly if i = j we have so that = j b i (t, x) + i b j (t, x) = b i (t, x) = i c(x)t + b i (x). = i b i (t, x) = i i c(x)t + i b i (x), i i c(x) =, i b i (x) =. ( ) j i c(x) + i j c(x) t + j b i (x) + i b j (x), j i c(x) = i j c(x), j b i (x) = i b j (x), which implies i c(x) = c i, and c(x) is a polynomial. By the periodic boundary conditions c(x) = c R. We further observe that b i (t, x) = b i (x) is a constant in time a.e. from (3.41) and the above. From (3.4)again i b i = so that trivially i i b i =. Moreover (3.4) further implies that j j b i =. Thus for each i, b i (x) is a periodic polynomial, which must be a constant: b i (x) = b i R. We compute from (3.34) that dp p i J 1/ (p) Z(t, x, p) = b i dp p R 3 R 3 i J(p), i = 1,, 3, dp J 1/ (p) Z(t, x, p) = a dp J(p) + c dp p J(p), R 3 R 3 R 3 dp p J 1/ (p) Z(t, x, p) = a dp p J(p) + c dp p J(p). R 3 As in [51] we ine ρ 1 = J(p)dp = 1, ρ = p J(p)dp, ρ = p R 3 R 3 R 3 J(p)dp. ( ) ρ1 ρ Now the matrix given by is invertible because ρ ρ ρ <ρ 1ρ. It then follows from the conservation law (3.31) which is satisfied by the limit function Z(t, x, p) that the constants a, b i, c must indeed be zero. R 3 R 3