Nonlinear diffusions as limit of kinetic equations with relaxation collision kernels

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1 Nonlinear diffusions as limit of kinetic equations with relaxation collision kernels Jean Dolbeault, Peter Markowich, Dietmar Oelz, Christian Schmeiser Abstract Kinetic transport equations with a given confining potential and non-linear relaxation type collision operators are considered. General monotone energy dependent equilibrium distributions are allowed with a chemical potential ensuring mass conservation. Existence and uniqueness of solutions is proven for initial data bounded by equilibrium distributions. The diffusive macroscopic limit is carried out using compensated compactness theory. The result are drift-diffusion equations with nonlinear diffusion. The most notable examples are of the form t ρ = ρ m + ρ V, ranging from porous medium equations to fast diffusion, with the exponent satisfying < m < 5/ in. Key words Kinetic equation, Macroscopic limit, Diffusion limit, Boltzmann equation, Equilibrium distribution function, Gibbs state, Porous medium equation, Fast diffusion equation, Relaxation time approximation, Compensated compactness MSC classification 2: Primary: 82C4, 5B4; Secondary: 5A5, 5K55, 45K5, 82D5, 85A5 We consider the scaled kinetic equation. Introduction ε 2 t f + ε [v x f x V x v f] = Qf,. Qf := G f f, G f := γ 2 v 2 µρ f x, t,.2 where the distribution function f = fx, v, t depends on position x, velocity v, and time t >. The collision model is a simple relaxation kernel towards a generalized local Gibbs state G f. The chemical potential µρ f is determined implicitly by the condition R G f dv = ρ f := R f dv, or equivalently ρ = 2 v 2 µρ dv.. γ We are interested in the diffusion limit ε which corresponds to a large time scale and a high collision frequency limit. We prove that in the limit ε, the distribution function f is a local Gibbs state: f = G f, whose density is subject to a nonlinear diffusion equation t ρ = x x νρ + ρ x V x. The main modelling ingredient is the energy dependent equilibrium profile γe, which is assumed to be nonincreasing. The given external potential V x will be assumed to be confining. An appropriate definition of this property depends on the profile γ and will be given below. Functions of the total energy v 2 /2 + V x constitute the kernel of the transport operator on the left hand side of.. Therefore, the definition of the quasi Fermi potential µ ρ x, t := µρx, t + V x will be convenient. In particular, equilibrium distributions with constant quasi Fermi potential are steady state solutions of.. The equation. is considered subject to initial conditions fx, v, = f I x, v.4

2 2 Jean Dolbeault, Peter Markowich, Dietmar Oelz, Christian Schmeiser with f I L +. The total mass M := R 6 f I x, vdxdv is formally preserved by the evolution, i.e., R 6 fx, v, tdxdv = M for all t >. For notational convenience, we restrict our attention to the three-dimensional problem. Generalizations of our results to other dimensions are straightforward... Modelling and references Reducing kinetic equations to macroscopic equations is a standard procedure, at least from a formal point of view. At the kinetic level, it is easy to relate the parameters with simple physical quantities since characteristics in the phase space can be interpreted in terms of particle dynamics. The price to pay is the high dimensionality of the phase space. On the other hand, hydrodynamical equations or parabolic models are in principle simpler to compute, but their direct derivation is far less intuitive. This motivates the study of hydrodynamics or diffusion limits, with the idea that the models are easier to build at the kinetic level, but that one is mostly interested only in the macroscopic observables. Nonlinear diffusion equations have attracted a lot of attention over the last few years, because of their deep mathematical properties and the physical interpretation of their naturally associated Lyapunov functionals. A very striking point of view is to consider these equations as the gradient flow of their Lyapunov functionals with respect to some appropriate notion of distance. However the derivation of such nonlinear diffusion equations in a physical context up to now looks rather unclear, although some properties like finite diffusion speed in the porous media case, make them very appealing from a modelling point of view. It is the purpose of this paper to provide a justification of nonlinear diffusion equations as limits of appropriate simple kinetic models. Notice that it is also possible to proceed the other way around and to reconstruct the Gibbs state at the kinetic level knowing the explicit form of the diffusion coefficient at the macroscopic level. The appropriate way to design general collision kernels and on which, experimental or theoretical, physical grounds they should be established is completely out of the scope of this paper. What we intend to do here relies on a much more pragmatic approach which can be decomposed as follows: If the kinetic equilibrium or Gibbs states, are known, or equivalently the energy profiles of such equilibria, then local equilibria, or local Gibbs states, are easily derived. From a physical point of view, such local equilibria make sense, whenever local relaxation phenomena occur on a faster time scale than the global evolution of the solution, thus giving rise to solutions at local equilibrium in the velocity space. This is particularly the case in models with collisions. Transfer of momentum during a collision is generally assumed to occur on a faster time scale than transport related effects like the ones induced by mean field forces, and are usually considered to be instantaneous. Thus local or global Gibbs states will be considered in our approach as basic input for the modeling. This is a very standard assumption for instance in semiconductor theory when one speaks of Fermi-Dirac distributions, or when one considers polytropic distribution functions in stellar dynamics. 2 Non monotone energy profiles result in various pathologies like linear and nonlinear instabilities. On the opposite, a monotonically decreasing energy profile provides a consistent way of finding a convex Lyapunov functional which, under appropriate constraints like mass conservation, allows to characterize the global Gibbs state as its unique minimizer. From a mathematical point of view, the Lyapunov functional is the sum of the total energy and of a convex nonlinear entropy based on the Legendre transform of a primitive up to a sign of the energy profile. In our approach, we say nothing about the physical phenomena responsible for the relaxation towards the local Gibbs state and, on the long time range, towards the global Gibbs state. We only derive the nonlinear diffusion limit in a way which is consistent with the Gibbs state. This is why we introduce at the kinetic level a caricature of a collision kernel, which is simply a projection onto the local Gibbs state with the same spatial density, thus introducing a local Lagrange multiplier which will be referred to as the chemical potential. In the mathematical literature, such a collision kernel is known as a relaxationtime kernel. At least in the long time asymptotics, such a kernel is generally believed to be a reasonable approximation of more realistic physical kernels. Diffusion limits appear when collision effects become dominant, but also when one is interested in long time effects so that phenomena due to convection, which would be essential in the hydrodynamical regime, are also dominated by diffusion. This can be derived in physical situations by an adimensionalization of the equation and a proper scaling. To stick to our purpose and since this is by many aspects standard, we refer for instance to [8] for the equations with correct physical parameters in the semiconductor context. Here ε will simply be a small positive parameter and we are interested in the singular limit ε. To come back to the two fields of applications quoted above, let us mention that in astrophysics, power law Gibbs states are well known see, e.g., [5], and [6] for some mathematical properties of such

3 Nonlinear diffusion limits equilibrium states, thus justifying on reasonable grounds how diffusion models can be introduced starting from more fundamental models see for instance [7] for recent results in astrophysics and two dimensional turbulence, and references therein for earlier papers. Note by the way that nonlinear diffusions with vanishing diffusion coefficients, like for the porous media model, cure one of the major problem of linear diffusion, namely the non-existence of global Gibbs states with finite mass coupled with gravitational interaction in a Euclidean space. Lyapunov functionals corresponding to porous media in an astrophysical context are usually called Tsallis entropies, referring to [22]. In the mathematical study of diffusion limits for semiconductor physics, more results are known, starting with [, 2]. The reference paper has been written by Goudon and Poupaud [5], but many other works deal with some more specific cases [2,,,4,8,9,4,]. Some of the results of this paper were written at a formal level in [2]..2. Formal macroscopic limit Consider formal asymptotic expansions f = f +εf +Oε 2, G f = G +εg +Oε 2, µ ρf = µ +Oε, and ρ f = ρ + Oε. Then, by going to the limit ε in., we obtain, at lowest order in ε, f x, v, t = G x, v, t = γ v 2 /2 + V x µ x, t and ρ = f dv. Notice that µ = µρ + V. The Oε-terms in. give v x f x V v f = G f, which can be rewritten as f = v γ v 2 /2 + V µ x µ + G..5 Now we pass to the limit in the continuity equation and obtain t ρ f + ε x vf dv =, t ρ + x vf dv =. For the evaluation of the flux, we use.5. Since G is an even function of v, it does not contribute, and we end up with Equivalently, since µ = µρ + V, we may write with t ρ = x ρ x µ. t ρ = x Dρ x ρ + ρ x V = x x νρ + ρ x V x,.6 Dρ = ρ µ ρ and νρ := This equation has to be supplemented with the initial condition ρ D ρ d ρ..7 ρ x, = ρ I x := f I x, v dv..8 Formally, this can be derived considering an initial layer governed by the equation τ f = Qf, τ = t/ε 2, and using mass conservation. The main result of this paper is a rigorous justification of this formal asymptotic expansion.

4 4 Jean Dolbeault, Peter Markowich, Dietmar Oelz, Christian Schmeiser.. Reconstruction of energy profiles Consider the inverse problem of finding an equilibrium profile γ producing a given macroscopic equation of the form.6. The macroscopic model is determined by the chemical potential function µρ, which is itself determined by.. The inverse of µ is explicitly given by µ θ = 4π 2 Differentiation with respect to θ leads to the Abelian equation µ θ = 2π 2 θ γq θ + q dq. θ γq θ + q dq..9 This identity can be inverted see [2], pp. 9, and gives an explicit expression for γ in terms of µ, which can be written as d 2 γe = 2π 2 2 de 2 µ θ E dθ. θ We refer to [] for a generalization to other space dimensions. Note that convexity of µ is sufficient for obtaining a nonnegative equilibrium profile. A precise classification of all µ corresponding to nonnegative γ is not known, however see [] for a discussion..4. Assumptions Our rigorous justification of the macroscopic limit covers all the examples of equilibrium profiles described in the last section. This involves quite a range of different qualitative behaviours including equilibrium profiles with compact support. This generality comes at the expense of a number of technical assumptions. a γ in the case E 2 = satisfying.. b γ with < E < E 2 <. Fig.. Exemplary graphs of the energy profile γ. Assumption The energy profile, or Gibbs state, γ CE,, R + with E is nonincreasing and nonnegative. It is continuously differentiable on its support E, E 2 with γ E < for E < E < E 2. There exists a Ê < E 2, such that γ is either convex or concave on the interval Ê, E 2. The following two assumptions refer to the cases of bounded and unbounded supports separately. Assumption 2 If E 2 =, there are constants E and τ > such that either or γ E γe γ E γe τ τ Moreover, in case. cf. figure a, there exists a δ > such that for E > E,. for E > E.. γe = OE 5/2 δ as E..2

5 Nonlinear diffusion limits 5 By.,., the decay is bounded by an exponential either from above or from below, and by.2 it is fast enough to ensure the existence of second order velocity moments. As a map from E, E 2 to, γe, γ is invertible. The inverse will be used below. Assumptions and 2 also guarantee that the function µρ defined by. is invertible as a map from, ρ to E 2, E, where the maximal density ρ := lim γ v 2 /2 µ dv. µ E + can be finite for E >. Assumption If E 2 < cf. figure b, there are constants k > and C > such that γe CE 2 E k for Ê < E < E 2..4 For the initial data we assume boundedness by a stationary solution of.. Assumption 4 There is a constant quasi Fermi energy µ < E such that f I x, v f x, v := γ 2 v 2 + V x µ, x, v R 6..5 It will be shown in the following section that this bound is propagated by.. Finally, we collect our assumptions on the confining potential V. Assumption 5 The potential V is bounded from below by zero, and satisfies V x V C,..6 It is confining in the sense that the upper bound f for the initial data has finite mass and energy: + 2 v 2 + V x f x, v dv dx <..7 R 6 Remark If γ has compact support, then f x, v has compact support as a function of v. If additionally V x E 2 + µ outside of a compact set in x, then f has compact support in R 6. Obviously,.7 is satisfied in this situation. Our main result is the following Theorem Under Assumptions 5, for any ε >, the problem..4 has a unique weak solution f ε C, ; L L p R 6 for all p <. As ε, f ε weakly converges to a local Gibbs state f given by f x, v, t = γ 2 v 2 + V x µρx, t x, v, t R +, where ρ is a solution of the nonlinear diffusion equation with initial data ρx, = ρ I x := f I x, vdv. Here ν is given by t ρ = x x νρ + ρ x V x.8 νρ = ρ s µ s ds. Moreover, f ε dv strongly converges to ρ in L p loc as ε. Remark 2 Notice that as a consequence of Theorem, there exists a global weak solution of.8 with initial data ρ I. We are however not aware of a result of uniqueness under such general assumptions. Since we are using compactness arguments in the proof, the diffusion limit has to be understood up to the extraction of a sequence ε i i N converging to zero, whenever uniqueness is not guaranteed. For simplicity, we will however abusively speak of the convergence as ε. The rest of this article is organized as follows. In Section 2, existence and uniqueness of solutions of the initial value problem for. is proven with some additional details. The main part is Section, where the macroscopic limit is rigorously justified. Finally, Section 4 contains a number of examples satisfying the above assumptions. Unless we are explicitly considering the limit ε, we will simply write f instead of f ε.

6 6 Jean Dolbeault, Peter Markowich, Dietmar Oelz, Christian Schmeiser 2. Existence and uniqueness Proposition 2 Let Assumptions, 4, and 5 by satisfied. Then the problem..4 has a unique weak solution in f C, ; L L p R 6 for all p [, satisfying f L R 6,, ρ f L,, with f,, t f and ρ, t ρ := f dv for all t >. Proof. The result is an adaption of a result stated in [2], where, due to a different choice of the scattering operator, the inhomogenities of the linear problems involved in the proof take a different form and therefore have to be treated with different arguments. For a given f V := {f C, T; L R 6 : f f }, let g = Γf be the solution of the linear transport problem ε 2 t g + ε v x g x V v g = G f g, with G f = γ 2 v 2 µρ f, gt = = f I, constructed by the method of characteristics. Obviously, fixed points of Γ correspond to solutions of.,.4. First we show that Γ maps V into itself, and then that it is a contraction for sufficiently small time intervals. As a consequence of the nonnegativity of f I and G f, g is nonnegative by the maximum principle. The function r := f g solves the linear transport problem ε 2 t r + ε v x r x V v r + r = f G f =: S, rt = = f f I, where S = G f G f is nonnegative because γ is decreasing, µ is increasing and ρ = f dv f dv = ρ f by assumption. Therefore also r is nonnegative and V is stable under the action of Γ. In order to prove the contraction property, consider two functions f, f 2 in V and let w := Γf 2 Γf. Then w is a solution of the problem ε 2 t w + ε v x w ε x V v w + w = G f2 G f =: U, 2. wt = =. 2.2 A multiplication by signw transforms 2. into an equation for w with the inhomogeneity replaced by signwu. The integration of this equation with respect to x and v implies ε 2 d dt w.,., t L R 6 ε 2 d dt w.,., t L R 6 + w.,., t L R 6 U.,., t L R 6. As γ and µ are respectively monotonically decreasing and increasing functions, the sign of G f2 G f equals the sign of ρ f2 ρ f and does not depend on v. Hence for any x, t R +, Gf2 G f dv = ρf2 ρ f, and therefore This finally leads to the estimate U.,., t L R 6 = ρ f2 ρ f dx f 2 f L R 6. ε 2 Γf 2.,., t Γf.,., t L R 6 t f 2.,., s f.,., s L R 6 ds, implying that Γ is a contraction on C, T; L R 6 for T < ε 2. The same is true for the spaces C, T; L p R 6, p < by interpolation using the L -bound. The global L L bound implies that the solution globally exists.

7 Nonlinear diffusion limits 7 Let the free energy be defined by [ ] Ff := 2 v 2 + V f + β γ f R 6. The Drift-Diffusion limit f dv dx, where β γ f := γ s ds. As γ is monotonically increasing, β γ is a convex function. The definition of β γ implies that β γ =, that it achieves its minimum on R + at f := γmax{, E } and that it is negative on, f. Here we recall and extend some of the results stated on a formal level in [2]. Let the microscopic energy associated to a distribution function f be denoted by E f x, v, t := 2 v 2 + V x µ ρf x, t = 2 v 2 µ ρ f x, t. If f is a solution of., an elementary computation shows that ε 2 d dt F f.,., t = γef fe f γ f dv dx := Df.. R 6 This holds provided f has a sufficient decay at infinity to justify the integrations by parts. Moreover we used the fact that G f fµ f dv =, which is a straightforward consequence of mass conservation. Since γ is monotonically increasing, Df is nonnegative. However, in the case when γ has compact support, Df does not provide a good control of the distance between f and the local equilibrium G f. This will be a major difficulty in our analysis below. We first establish some a priori estimates on the free energy. By integration of the entropy production. on the time interval, T we obtain [ ε 2 F f,.,., t F ] T f I = Dft dt,.2 which proves that F f.,., T Ff I for any T >. If f =, β γ is increasing on R + and we conclude Ff I Ff. As we shall see below, Ff <. In the case f > we use meas R 6{f > f} f R 6 f dv dx and obtain Ff I R 6 2 v 2 + V x f I dv dx + R 6 f I > f β γ f I dv dx 2 v 2 + V x f dv dx + β γ f L f f dv dx < R 6 by.7. It remains to prove that F f is bounded from below. For fixed x, t R +, the functional F loc : [ ] fx,., t E f x, v, tfx, v, t + β γ fx, v, t dv =: F loc fx,., t is convex and achieves its unique critical point if and only if = E f x, v, t + β γ f = E fx, v, t γ f, i.e., if f = G f is a local equilibrium distribution function. As a consequence: F loc fx,., t F loc G f x,., t x, t R +, which, after an integration with respect to x, proves that using f dv = G f dv again. Ff FG f,.

8 8 Jean Dolbeault, Peter Markowich, Dietmar Oelz, Christian Schmeiser In the case of a local equibrium gx, v, t = γ v 2 /2+V x µx, t, we may integrate by parts twice with respect to v i, i {, 2, }. Using β γ γ E = E γ E, we get β γ g dv = v i R 2 v 2 + V µ vi γ 2 v 2 + V µ dv = 2 v 2 + vi 2 + V µ g dv. This yields Fg = R 6 γ 2 v 2 + V µµ v 2 dv dx = µρg νρ g dx, with ρ g = µ µ V, and νρ := v 2 γ R 2 v 2 µρ dv..4 Notice that ν ρ = ρ µ ρ as follows from an integration by parts. The above definition is therefore compatible with the one in.7. We will come back on this question in Lemma 4. As a consequence, by. we obtain that Ff Fρ f where F is again a convex functional defined on L + R by ρ µρ ρ νρ dx = ρ µρ + ρv x νρ dx. Under the constraint ρ dx = M, F achieves its global minimum at ρ such that µρ x + V x = µ, where µ is the Lagrange multiplier implicitly defined by the constraint µ µ V x dx = f dv dx = M R 6 and f := γ 2 v 2 + V x µ. Observe that Fρ > by.4 and.7. We summarize all these observations in the following result. Lemma Let Assumptions -5 hold, and let f be the solution of.,.4, constructed in Proposition 2. Then i the mass is preserved along the evolution fx, v, t dv dx = R 6 f I x, v dv dx R 6 t R +, ii The following estimates hold < Ff FG f.,., t Ff.,., t Ff I < t R +. Proof. All these estimates can be justified by standard regularizations of the initial data f I and of the potential V. Passing to the limit in the regularization parameter, the estimates on the free energy hold by semi-continuity, due to the convexity of the functional µ has to be considered as a Lagrange multiplier for fixed x, t and strongly converges by standard averaging lemmata. The conservation of mass is then a consequence of the Dunford-Pettis lemma. Notice that as a consequence of the Dunford-Pettis lemma, any solution f is contained in a relatively weakly compact set of L R 6 R +,loc. From now on, weak convergence at least means convergence in this sense. We continue by using the entropy production.2 and a technical lemma based on the assumptions on the energy profile γ to prove uniform estimates on the first and second moments of the solution f. Let f f. Consider a partition of the support of f given by Ω + f := {x, v, t suppf, T : E f = 2 } v 2 µρ f x, t < E 2, Ω f := {x, v, t suppf, T : E f = 2 }.5 v 2 µρ f x, t E 2,

9 Nonlinear diffusion limits 9 and define + f := {v R : x, v, t Ω + f} and f := {v R : x, v, t Ω f}..6 Lemma 4 Let Assumptions hold. Then, for any nonnegative function fx, v, t f x, v there exists a constant M, which does not depend on x and t, such that v 2m γe f f γ dv M, f E f for m =, 2, and + := Ωx,t + f. + Proof. The proof relies on the sign of γ in Ê, E 2 see Assumption. Observe that f x, v > γê implies v 2 < 2Ê + µ. Assume first that γ is convex on Ê, E 2. We first use the mean value theorem: v 2m γ Ẽ dv sup γ E v 2m dv + {f >γê} µ E Ê + {f >γê} γ E 4π 2 Ê + µ 2 +m = C. sup µ E Ê On Ê, E 2, γ is decreasing, and Ẽ γ f = v 2 /2 + V x µ implies v 2m i γ Ẽ dv + {f γê} vi 2m γ v 2 /2 + V x µ dv = 2m vi 2m 2 f dv C 2. Consider now the case where γ is concave on Ê, E 2. For fixed x and t, let Λ x,t := {v : max f, G f x, v, t γ Ê}. The function v γ γ fx, v, t is bounded on + \ Λx,t, because γ is of class C on E, E 2. We infer v 2m γ Ẽ dv 4π 2 Ê + µ 2 +m sup γ Ẽx,., t = C. + \Λx,t + \Λx,t Observe that if γ is concave on Ê, E 2, then E 2 is finite. The function s χs := E 2 γ s s is an increasing and positive function on, γê R. This allows us to write γ f E f G f f = E 2 E f f E2 E f + G f G f f G f We conclude by making use of Assumptions, 4, and 5: v 2m i + Λx,t G f f γ dv f E f C v 2m i + Λx,t E 2 E f k E 2 E f v 2m i + Λx,t where we used C to represent various constants. E 2 γ f f G f dv E 2 E f dv = C = χg f + f χg f χf G f f χg f >. v 2m [ i E2 E f k] dv k v + Λx,t i C v 2m 2 i E 2 E f k + dv CE 2 + µ k+ 2 +m C 2,

10 Jean Dolbeault, Peter Markowich, Dietmar Oelz, Christian Schmeiser Starting to investigate the limit ε, we denote the solution of. by f ε from now on. With the definition.5, the equation.2 can be rewritten as Oε 2 = Ω +f ε γef ε f ε γ f ε E f ε dx dv dt }{{} + Ω f ε f ε E f ε E 2 }{{} + E 2 γ f ε dx dv dt..7 }{{} Let us define the scaled flux and nonequilibrium part of the stress tensor, j ε := vf ε dv = v fε G fε dv and κ ε := v v fε G fε dv..8 ε R ε R ε Lemma 5 Let Assumptions -5 hold and let U [, T be open and bounded. Then there are two constants M U and M2 U, which do not depend on ε, such that Proof. We have to verify that, for m {, 2}, j ε L 2 x,t U M U and κ ε L 2 x,t U M 2 U. U v m fε G fε R ε dv 2 dx dt.9 is bounded uniformly in ε. Depending on the function f ε we split the domains of integration by making use of the respective sets defined in.6: U 2 v i m fε G fε dv dx dt R ε v m fε G f ε dv + v m fε G f ε dv dx dt.. U ε + ε We then use Lemma 4 and.7 to estimate the first term: U 2 v m fε G f ε dv dx dt ε + v 2m G f ε fε γ f ε E f ε U + dv ε 2 G f ε f ε γ f ε E f ε dv dxdt C.. + On Ω the local equilibrium satisfies G f ε = by definition, and Ω is nonempty only if E 2 <, implying that suppf and, thus, Ω is bounded in the velocity direction: Hence we need to prove that U v 2 2E 2 + µ for x, v, t Ω. v m fε G f ε ε is bounded uniformly in ε. Let us define gx, t := 2 2 f dv ε dx dt C U ε dv dx dt.2 E f E 2 fx, v, t dv, such that and < gx, t and U g dxdt = Oε2 by.7 and, for some A > that we will finally choose large enough, let U A := { x, t U : gx, t < A }.

11 Nonlinear diffusion limits We split again the domain of integration in.2 according to U 2 f dv ε dx dt U A 2 2 f dv ε dx dt + f dv ε dx dt. U\U A Let x, t U A and let R > be some radius, for which will choose a specific value later on. We estimate f dv f L dv + E f E 2 f dv.. <E f E 2<R R The volume of the set {v : < E f E 2 < R} is given by { φ ρf R := meas } v : < E f E 2 < R = meas { v : 2 E 2 + µρ f < v < 2 R + E 2 + µρ f }. Observe that for fixed R > ρ φ ρ R is a monotone function and conclude φ ρf R φ µ µ R = 4π /2 2E 2 + µ R /2 + E 2 + µ =: ψr, where that function ψr is given by ψr = C + C 2 /2 with constants C > and C 2 > since E 2 + µ >. Otherwise the support of f would indeed be empty. From. we obtain for a positive constant C where the function h is given by f dv C ψr R R + gx, t = hr,.4 R hr := C ψr R r + gx, t. r This convex function assumes its minium at r = gx,t R C ψr. The minimal value is given h r = 2 gx, tψrc R..5 Now we choose R to be given by a solution of the equation gx, tr R = r = C ψr,.6 gx,t R which exists, since lim gx,t R R C ψr R > and lim R C ψr R = for fixed gx, t >. Observe that ψr/r C C /2 2 R for any R >. Using this estimate in.6 yields R 5/4 C gx, t for a constant C >. From.4 and.5 and by using the assumption gx, t < A we infer gx, tψr f dv C gx, tr 4 C gx, tgx, t C gx, t, R where C represents various positive constants. In the case x, t U \ U A we estimate U\U A on ρ given by f dv dxdt by choosing A larger than the bound 2 f dv ρx, t 2 ρ 2 L A gx, t.

12 2 Jean Dolbeault, Peter Markowich, Dietmar Oelz, Christian Schmeiser Finally by.7 we obtain U 2 f dv ε dx dt Cg dx dt + g dx dt U A U\U A C g dx dt = C U U E f E 2 f ε x, v, t dx dt = Oε 2. Hence.2 is bounded uniformly in ε and by combining this with. and. we conclude that this also applies to.9. The result follows now directly or, for the mixed second moments, by application of the Cauchy-Schwarz inequality. Before proving rigorously the macroscopic limit in Proposition, we will make preparatory statements on second moments of local equilibria, the derivative of which will turn out to represent the diffusivity, and on the strong convergence of the macroscopic density ρ f. Lemma 6 Let ρ < ρ max = µ µ and denote by Id the identity matrix in. Then the second order moments of the local equilibrium g ρ v := γ v 2 /2 µρ are given by v v g ρ dv = νρid where νρ := ρ σ µ σ dσ. Moreover, on the closed interval [, ρ max ], the derivative ν has either an upper bound or a strictly positive lower bound. Proof. Observe that and define v i v j g ρ dv = for any i, j =, 2, i j νρ := v 2 γ dv = v2 2 γ dv = v 2 γ dv. Then by an integration by parts with respect to v i, we get ν ρ = vi 2 γ R 2 v 2 µρ µ ρ dv = µ ρ vi 2 γ R 2 v 2 µρ dv = ρ µ ρ. Now consider an interval [δ, ρ max ] for some δ >. If µ does not behave badly in this interval, the same must be true for µ θ on [ µδ, µρ max ]. As.9 can be written as µ θ = 2π γq θ 2 dq q we infer for µδ θ µρ max < mδ := µ µδ µ θ µ µρ max =: M <. Hence M µ ρ < mδ < where ρ [δ, ρ max ]. Consequently also the product ρ µ ρ is well behaved in this sense and the values and may only be approached as ρ. As far as the behaviour at ρ = is concerned we have to distinguish the case where E 2 < and the two cases. and. corresponding to energy profiles which are, compared to an exponential decay, converging to zero not faster, or respectively, not slower. Let E 2 <, then lim ρ ν ρ = lim θ E 2 µ θ µ θ = lim θ E 2 = lim θ E 2 2 E2+θ 2 E2+θ γp θ p dp E2+θ γp θ p dp γp θ p p dp E2+θ γp θ p dp E2+θ lim E 2 + θ γp θ θ E 2 E2+θ 2 γp θ p dp p dp =.

13 Nonlinear diffusion limits In the case. we obtain lim inf ρ ν ρ = liminf θ µ θ µ θ = liminf θ In the case. we obtain by an analogous computation which concludes the proof for all possible cases. γp θ p dp γ p θ p dp liminf θ lim sup ν ρ τ, ρ γp θ p dp τ γp θ p dp = τ. Proposition 7 Under Assumptions.4,.5 and.6 ρ ε ρ in L p loc strongly for all p,. Proof. The proof relies on the Div-Curl Lemma in a similar way as in [5]. By integrating the kinetic equation. with respect to dv and v dv respectively, we obtain the following system t ρ ε + x j ε =, ε 2 t j ε + x v v f ε dv = j ε ρ ε.7 x V. Now we split the second moments of f ε into an equilibrium part and a perturbation, v v f ε dv = v v G f ε dv + v v f ε G f ε dv = νρ ε I + εκ ε, where we used the result of Lemma 6. We use this decomposition to rewrite the system.7 as a system of four scalar equations. { t ρ ε + x j ε =, x νρ ε = j ε ρ ε x V ε x κ ε ε 2 t j ε.8. We apply the Div-Curl Lemma to U ε := ρ ε, j ε, V ε := νρ ε,,,. With these definitions and the convention curlw ij = w i x j w j x i.8 becomes { divt,x U ε =, curl t,x V ε,2...4 = j ε ρ ε x V ε x κ ε ε 2 t j ε. By assumption.6, Proposition 2 and Lemma 5, ρ ε x V and j ε are bounded in L 2,loc x,t precompact in H,loc x,t. Also by Lemma 5, ε κ ε and ε 2 j ε are compact in L 2,loc x,t therefore compact in H,loc x,t. and therefore and their derivatives are Due to Proposition 2, the family ρ ε is weakly * compact in L,loc x,t and hence weakly compact in L 2,loc x,t. The same applies to νρ ε and ρνρ ε as ν is a continuous function. Let ε i i N be a sequence converging to such that ρ εi and νρ εi weakly converge in L 2,loc x,t consider the corresponding limits for U εi and V εi : U εi ρ, j and ρ εi νρ εi converges weak * in L,loc x,t and V εi ν,,, in L 2,loc x,t as i. The Div-Curl Lemma see [9] states the following: Let U R m be an open, bounded and smooth set and let v k k N and w k k N be two bounded sequences in L 2 U, R n, such that the sequences div v k k N and curlw k k N lie in compact subsets of H U and H U, R n n respectively. Then the sequence of scalar products v k w k k N converges to v w in the sense of distributions, where v and w are the weak limits in L 2 U, R n of v k k N and w k k N. Due to the considerations above we obtain from the application of the Div-Curl Lemma that U εi V εi = ρ εi νρ εi ρ ν in the sense of distributions. By uniqueness of the limit in D x,t and the weak * limit in L,loc x,t we get ρ εi νρ εi ρ ν and in L,loc x,t as i..9 From this identity we shall conclude, in a similar way as in [7], that the convergence of ρ εi is strong. In [7] the proof of strong convergence relies on strict convexity to proof the strong convergence. We

14 4 Jean Dolbeault, Peter Markowich, Dietmar Oelz, Christian Schmeiser will instead make use of the strict monotonicity of the function ν, which replaces, in this respect, strict convexity. We associate the Young measure family η x,t to the weak * convergence of ρ εi in L,loc x,t and obtain ρ max νρ εi ν = νρ dη x,t ρ, ρ max ρ = ρ dη x,t ρ,.2 ρ εi ρ εi νρ εi ρ max where the final identity is due to.9. Now observe that ρ νρ dη x,t ρ = ρ ν, νρ = νρ + ν ρρ ρ for some ρ, ρ max. From this and.2 we conclude = ρ max ρ max νρρ ρ dη x,t ρ = νρρ ρ dη x,t ρ }{{} + = ρ max ν ρρ ρ 2 dη x,t ρ C ρ max ρ ρ 2 dη x,t ρ. Here we relied on the second alternative in Lemma 6, which states that ν ρ assumes its minimum C > on [, ρ max ]. The measure η x,t is therefore the Dirac point mass δρ ρ, which means that the sequence ρ εi strongly converges. In the alternative case, in which ν ρ is bounded from above on [, ρ max ] ρ, we have to go back to.2. Observe that ν p = /ν ν p will be bounded from below on the interval [, νρ max ] and define π εi := νρ εi. By rewriting.2 we infer νρ max π εi ν = π dµ x,t π, νρ max ν π εi ρ = ν π dµ x,t π, ν π εi π εi νρ max π ν π dµ x,t π = ρ ν, where µ x,t is the Young measure associated to the weak convergence of π εi i. In the same way as above for the measure η x,t, we conclude that µ x,t is a point mass. Therefore νρ εi i strongly converges and by continuity of ν this also applies to ρ εi i. Since the limit is independent of ε i i N, the convergence holds as ε. As in Section, we denote by G f the local Gibbs state associated to f: G f x, v, t = γ v 2 /2 µρ f x, t. Lemma 8 Let f ε be the solution of. and denote by f its weak limit as ε. Under the assumptions of Proposition 7, f = G f. Proof. First we show that the differences f ε G f ε, up to the extraction of a subsequence, converge to zero a.e.: The relation.2 implies that a.e. at least one of the two factors in γe f ε f ε γ f ε E f ε converges to zero as ε. If the first factor converges to zero, then the result holds. Consider therefore the pointwise convergence at a point in phase space where the second factor converges but not the first one. Observe that γ f ε E f ε implies γ f ε mine f ε, E 2 because γ g < E 2 for all g >. As the energy profile γ is a diffeomorphism on E, E 2, we obtain γ f ε γmine f ε, E 2 f ε for some mean value f ε. Remember that γ is bounded on compact subintervals of E, E 2 and that we evaluate at a point where γe f ε f ε δ for some δ > as ε. Hence the mean value

15 Nonlinear diffusion limits 5 f ε minγe f ε, f ε, maxγe f ε, f ε stays strictly away from zero, fε > δ 2 as ε and therefore γ f ε has a strictly positive lower bound. This yields f ε γmine f ε, E 2 = f ε γe f ε = f ε G f ε. Summarizing we obtain f ε G f ε a.e. This together with Proposition 7 and the continuity of the mapping r γ v 2 /2 µr implies the result. Lemma 9 Let j ε be the perturbation of the first moment as defined in.8. Then j ε converges to j in D x,t, where the limit j is given by j = x νρ ρ x V. Proof. We define r ε := ε f ε γ 2 v 2 µρ ε. After multiplication by ε, the left hand side of. weakly converges to v x f x V v f = v x G f x V v G f, where we used Lemma 8. As the right hand side of. can be written as ε r ε, we infer that r ε weakly converges to r = x V v G f v x G f. Now observe that v r dv = ρ x V + x νρ = j. To prove the convergence j ε j in D x,t, let φ D x,t. For R >, we choose ψ R Cc v with ψ, ψ on B R and supp ψ R B R+ and obtain lim φj ε j dx dt = lim φv r ε r dv dx dt ε R 4 ε R 7 lim lim R ε φψ R vvr ε r dv dx dt R 7 + ψ R v φ v 2 r ε dv dx dt + R φ ψ R vv r dv dx dt R 7 R 7 = lim + meassupp φ M 2 R R supp φ + φ ψ R vv r dv dx dt =, R 7 where we used the convergence r ε r in D x,v,t, the uniform boundedness of κ ε by Lemma 5 together with an interpolation and the convergence of the last integral. Proposition Let f ε be the solution of. and denote by f its weak limit as ε. Under the assumptions of Proposition 7, ρ := f dv satisfies.6,.8 in the weak sense. Proof. By integration of. in the velocity space and by observing v γ v 2 /2 + V µ ρf dv = we obtain the macroscopic continuity equation t ρ ε + x j ε =, with the perturbation of the flux j ε as defined in.8. We multiply this equation by a test function φx, t DR 4, integrate and obtain after integration by parts ρ ε t φ + x φ j ε dx dt = φ, xρ I x dx. By Proposition 7 and Lemma 9 we can find a sequence ε j j N and pass to the limit as ε j in the above identity. We obtain [ ] ρ t φ ρ x V x φ + νρ x φ dx dt = φ, xρ I x dx, which is a weak formulation of.6,.8.

16 6 Jean Dolbeault, Peter Markowich, Dietmar Oelz, Christian Schmeiser 4. Examples Example Consider the case of a power law with negative exponent, γe := DE k, with D > and k > 5/2. Then Assumptions and 2 are obviously satisfied with E = and E 2 =. We easily compute µρ = ρ Dβk Substituting this expression in.6 yields t ρ = /2 k, where βk := 4π 2 Θ ρ k 5/2 k /2 + ρ V, where Θ := s s + k ds. k 5/2 Dβk k /2. Since < k 5/2 k /2 < this is a fast diffusion equation. The confinement condition.7 in Assumption 4 is satisfied if, outside of a finite ball, the potential grows faster than a certain power: V x C x q, for x > R, with q > k 5/2. The same threshold was obtained in [6] HV 6, remark 6, e in order to guarantee there that the entropy is finite. Example 2 For the Maxwellian distribution, γe = exp E, we compute and, thus, the linear drift-diffusion equation µρ = log ρ 2 log2π. t ρ = ρ + ρ V. By making use of.4 we conclude that the growth assumption on the potential in.7 is satisfied if V x q log x, for x > R, with q >. Example Let γ be a cut-off power with positive exponent: with D, k >. We obtain µρ = γe = E 2 E k +, ρ k+/2 E2, where αk = 4π 2 D αk u u k du. and, thus, the porous medium type macroscopic equations t ρ = Θ ρ k+5/2 k+/2 + ρ V, where Θ := D αk k+/2, k + 5/2 with the exponent satisfying The confinement assumption in.7 is satisfied if < k + 5/2 k + /2 < 5. E2 + µ V x + = O x q as x, with q > k + /2, where µ is the upper bound for the Fermi energy in assumption.5. This assumption is satisfied if V E 2 + µ outside of a compact set. In this situation, our results guarantee that the supports of both the kinetic as well as of the macroscopic solutions remain in fixed compact sets for all times. At the macroscopic level, this behaviour of the porous medium equation is well known.

17 Nonlinear diffusion limits 7 In the following two examples, we will use the polylogarithmic function which satisfies, for σ = ±, Li n z := k= z k k n, k s dk expk ν σ = σγs + Li +sσ expν and d dz Li nz = z Li n z. Example 4 Let γ be the energy profile of the Fermi-Dirac distribution, γe = expe+α. We obtain and therefore µ θ = 4π 2 α As macroscopic equation we obtain p dp 2π/2 expp θ log α + = Li /2 αe θ, α µρ = log Li where, by Lemma 6, the diffusivity Dρ is given by Dρ = ν ρ = ρ µ ρ = αρ /2 2π /2 α t ρ = Dρ ρ + ρ V, α 2π /2. ρ Li /2 Li αρ /2. 2π /2 Moreover the expansion of Dρ at ρ = gives 2 α ρ Dρ = + 4 2π + /2 8 2 α 2 ρ 2 9 2π + Oρ. Example 5 Let γ be the energy profile of the Bose-Einstein distribution, γe = expe α. It satisfies Assumptions - with E = logα, E 2 =. We obtain and therefore µ θ = 4π 2 α As macroscopic equation we obtain where the diffusivity Dρ is given by p dp expp θ log α = 2π/2 Li /2 αe θ, α µρ = log Dρ = ν ρ = ρ µ ρ = Li /2 αρ 2π /2 α t ρ = Dρ ρ + ρ V, α 2π /2. ρ Li /2 Li /2 αρ 2π /2. The maximal density ρ as defined in. is finite in the case of the Bose-Einstein distribution and given by ρ = 2π/2 ζ/ α α with the Riemann Zeta function given by ζs := Li s = k= k. Observe that lim s ρ ρ ν ρ = and lim ρ ν ρ =. Acknowledgment: This work has been supported by the Austrian Science Fund FWF through the Wittgenstein 2 Award of P. Markowich and the Wissenschaftskolleg Differential Equations, by the PAI Amadeus # 258QF, and by the EU network HYKE HPRN-CT c 25 by the authors. This paper may be reproduced, in its entirety, for non-commercial purposes.

18 8 Jean Dolbeault, Peter Markowich, Dietmar Oelz, Christian Schmeiser References. N. Ben Abdallah, P. Degond, F. Deluzet, V. Latocha, R. Talaalout, and M. H. Vignal, Diffusion limits of kinetic models, in Hyperbolic problems: theory, numerics, applications, Springer, Berlin, 2, pp N. Ben Abdallah and J. Dolbeault, Relative entropies for kinetic equations in bounded domains irreversibility, stationary solutions, uniqueness, Arch. Ration. Mech. Anal., 68 2, pp N. Ben Abdallah and M. L. Tayeb, Asymptotique de diffusion pour le système de Boltzmann-Poisson unidimensionnel, C. R. Acad. Sci. Paris Sér. I Math., , pp N. Ben Abdallah and M. L. Tayeb, Diffusion approximation for the one dimensional Boltzmann-Poisson system, Discrete Contin. Dyn. Syst. Ser. B, 4 24, pp J. Binney and S. Tremaine, Galactic dynamics, Princeton university press, Princeton, J. A. Carrillo, A. Jüngel, P. A. Markowich, G. Toscani, and A. Unterreiter, Entropy dissipation methods for degenerate parabolic problems and generalized Sobolev inequalities, Monatsh. Math., 2, pp P.-H. Chavanis, Generalized thermodynamics and Fokker-Planck equations: Applications to stellar dynamics and two-dimensional turbulence, Physical review E, 2, pp. 68, P. Degond, T. Goudon, and F. Poupaud, Diffusion limit for nonhomogeneous and non-micro-reversible processes, Indiana Univ. Math. J., 49 2, pp L. C. Evans, Weak convergence methods for nonlinear partial differential equations, vol. 74 of CBMS Regional Conference Series in Mathematics, Published for the Conference Board of the Mathematical Sciences, Washington, DC, 99.. K. Fellner and C. Schmeiser, Classification of equilibrium solutions of the cometary flow equation and explicit solutions of the Euler equations for monoatomic ideal gases. Preprint, 24.. F. Golse and F. Poupaud, Fluid limit of the Vlasov-Poisson-Boltzmann equation of semiconductors, in BAIL V Shanghai, 988, vol. 2 of Boole Press Conf. Ser., Boole, Dún Laoghaire, 988, pp , Limite fluide des équations de Boltzmann des semi-conducteurs pour une statistique de Fermi-Dirac, Asymptotic Anal., 6 992, pp T. Goudon, Hydrodynamic limit for the Vlasov-Poisson-Fokker-Planck system: Analysis of the twodimensional case T. Goudon, J. Nieto, F. Poupaud, and J. Soler, Multidimensional high-field limit of the electrostatic Vlasov-Poisson-Fokker-Planck system, J. Diff. Eq., 25, to appear. 5. T. Goudon and F. Poupaud, Approximation by homogenization and diffusion of kinetic equations, Comm. Partial Differential Equations, 26 2, pp Y. Guo and G. Rein, Stable models of elliptical galaxies, Mon. Not. R. Astronom., P. Marcati and A. Milani, The one-dimensional Darcy s law as the limit of a compressible Euler flow, J. Differential Equations, 84 99, pp P. A. Markowich, C. A. Ringhofer, and C. Schmeiser, Semiconductor equations, Springer-Verlag, Vienna, J. Nieto, F. Poupaud, and J. Soler, High field limit for the Vlasov-Poisson-Fokker-Planck system, Arch. Rational Mech. Anal., 2, pp A. D. Polyanin and A. V. Manzhirov, Handbook of integral equations, CRC Press, Boca Raton, FL, F. Poupaud and C. Schmeiser, Charge transport in semiconductors with degeneracy effects, Math. Methods Appl. Sci., 4 99, pp C. Tsallis, Possible generalization of Boltzmann-Gibbs statistics, J. Statist. Phys., , pp J. Dolbeault Ceremade, Université Paris Dauphine, Place de Lattre de Tassigny, F Paris Cédex 6, France. dolbeaul@ceremade.dauphine.fr and P. Markowich Faculty of Mathematics, University of Vienna, Nordbergstr. 5, A-9 Wien, Austria. peter.markowich@univie.ac.at and D. Oelz A.S.C.: Institute for Analysis and Scientific Computing, Vienna University of Technology, Wiedner Hauptstr. 8-, 4 Wien, Austria. dietmar.oelz@univie.ac.at and C. Schmeiser A.S.C. and Johann Radon Institute for Computational and Applied Mathematics, Austrian Academy of Sciences, Altenbergerstr. 69, 44 Linz, Austria. christian.schmeiser@tuwien.ac.at