Boltzmann equation for soft potentials with integrable angular cross section The Cauchy problem

1. Boltzmann equation for soft potentials with integrable angular cross sectionThe Cauchy problem Irene M. Gamba The University of Texas at Austin Mathematics and ICES IPAM April 2009- KTWSII In collaboration with Ricardo Alonso

2. elastic collision ‘v v inelastic collision C = number of particle in the box a = diameter of the spheres N=space dimension ηthe impact direction η v* ‘v* d-1 i.e. enough intersitial space May be extended to multi-linear interactions

3. Consider the Cauchy Boltzmann problem (Maxwell, Boltzmann 1860s-80s); Grad 1950s; Cercignani 60s; Kaniel Shimbrot 80’s, Di Perna-Lions late 80’s) Find a function f (t, x, v) ≥ 0 that solves the equation (written in strong form) with Conservative interaction (elastic) • σ is the impact direction: ′u=u - 2(u· σ) σ (specular reflection condition) • Assumption on the model:The collision kernel B(u, û · σ) satisfies • B(u, û· σ) = |u|λ b(û· σ) with -n < λ ≤ 1 ; we callsoft potentials: -n < λ < 0 • Grad’s assumption: b(û · σ) ∈ L1(S n−1), that is

4. Grad’s assumption allows to split the collision operator in a gain and a loss part, Q( f, g) = Q+( f, g) − Q−( f, g) = Gain - Loss But not pointwise bounds are assumed on b(û· σ) The loss operator has the following structure Q−( f, g) = f R(g), with R(g), called the collision frequency, given by |u|λ |u|λ λ The loss bilinear form is aconvolution. We shall see also thegain is a weighted convolution

5. Recall: Q+(v) operator in weak (Maxwell) form, and then it can easily be extended to dissipative (inelastic) collisions ωis the scattering direction with respect to an elastic collision:ω= u′ /|u| were u′ and u satisfy The relation of specular reflection: u′ = u -2(u· σ) σ  cos (u· ω) = π – 2 cos(u· σ). More generally, the exchange of velocities in center of mass-relative velocity frame Energy dissipation parameter or restitution parameters with β=1 elastic interaction Same the collision kernel form With the Grad Cut-off assumption: λ Q−( f, g) = f And convolution structure in the loss term:

6. Outline • In Alonso’s lecture: • Average angular estimates (for the inelastic case as well) • weighted Young’s inequalities for 1 ≤ p , q , r ≤ ∞ (with exact constants) 0 ≤λ = 1 • Sharp constants for Maxwell type interaction for (p, q , r) = (1,2, 2) and (2,1,2) λ= 0 • Hardy Littlewood Sobolev inequalities , for 1 < p , q , r < ∞ (with exact constants) -n ≤λ < 0 • In this lecture • Existence, uniqueness and regularity estimates for the near vacuum and near (different) • Maxwellian solutions for the space inhomogeneous problem • (using Kaniel-Shimbrot iteration type solutions) elastic interactions for soft potential • and the above estimates.Lp stability estimates in the soft potential case, for 1 < p < ∞

7. Average angular estimates & weighted Young’s inequalities & Hardy Littlewood Sobolev inequalities & sharp constants R. Alonso and E. Carneiro’08, and R. Alonso and E. Carneiro, IG, 09 (ArXiv.org): by means of radial symmertrization techniques Bobylev’s variables and operator is invariant under rotations Denoting by Translation and reflection operators Bobylev’s operator on Maxwell type interactions λ=0 is the well know identity for the Fourier transform of the Q+

8. for Young’s for variable hard potentials and Maxwell type interactions 0 ≤λ = 1 Hardy-Littlewood-Sobolev type inequality for soft potentials -n <λ < 0

9. Inequalities with Maxwellian weights As an application of these ideas one can also show Young type estimates for the non-symmetric Boltzmann collision operator with Maxwellian weights. For any a > 0 define the global Maxwellianas

10. Distributional and classical solutions to the Cauchy Boltzmann problem for soft potentials with integrable angular cross section(Ricardo Alonso & I.M.G., 09 submitted) Consider the Cauchy Boltzmann problem: (1) B(u, û· σ) = |u|−λ b(û· σ) with 0 ≤ λ < n-1 with the Grad’s assumption: with Q−( f, g) = f Definition:A distributional (mild) solution in [0; T] of BTE initial value problem is a function f ϵ W1;1(0; T;L∞(R2n)) that solves (1) a.e. in (0; T] x R2n such that , satisfies

11. Kaniel & Shinbrot iteration ’78 (DP-L -11yrs) Notation and spaces: Consider the space with the norm Kaniel-Shinbrot:(also Illner & Shinbrot ’84) define the sequences {ln(t)} and {un(t)} as the mild solutions to the system which relies in choosing a initial pair of functions (l0, u0) satisfying so called the beginning condition in [0, T]: and where the pair (l1, u1) solves the system with initial state (l0, u0).

12. Theorem: Let {ln(t)}and {un(t)}the sequences defined by the mild solutions of the linear system above, such that the beginning condition is satisfied in [0, T], then (i) The sequences {ln(t)}and {un(t)}are well defined for n ≥ 1. In addition, {ln(t)}, {un(t)} are increasing and decreasing sequences respectively, and l#n (t) ≤ u#n (t) a.e. in 0 ≤ t ≤ T. (ii) If 0 ≤ ln(0) = f0 = un(0)for n ≥ 1, then lim ln(t) = lim un(t) = f(t)a.e. in [0; T]: n∞ n∞ In addition the limit f (t) ∈ C(0, T; M#α,β)is the unique distributional solution of the Boltzmann equation in [0, T] and fulfills 0 ≤ l#0(t) ≤ f #(t) ≤ u#0(t) a.e. in [0, T].

13. Hard and soft potentials case for small initial data Lemma : Assume −1 ≤ λ < n − 1. Then, for any 0 ≤ s ≤ t ≤ T and functions f#, g#that lie in L∞(0, T;M#α,β), then the following inequality holds # # with Distributional solutions for small initial data: (near vacuum) Theorem: Let B(u, û· σ) = |u|−λ b(û· σ) with -1 ≤ λ < n-1 with the Grad’s assumption Then, the Boltzmann equation has a unique global distributional solution if . Moreover for any T ≥ 0 , # As a consequence, one concludes that the distributional solution f is controlled by a traveling Maxwellian, and that It behaves like the heat equation, as mass spreads as t grows

14. Distributional solutions near local Maxwellians : RicardoAlonso, IMG’08 Previous work by Toscani ’88, Goudon’97, Mischler – Perthame ‘97 • Theorem: Let B(u, û· σ) = |u|−λ b(û· σ) with -n < λ ≤ 0with the Grad’s assumption • In addition, assume that f0is ε–close to the local Maxwellian distribution • M(x, v) = C Mα,β(x − v, v) , with 0 < α, 0 < β. • Then, for sufficiently small εthe Boltzmann equation has a unique solution satisfying • C1(t) Mα1,β1 (x − (t + 1)v, v ) ≤ f ( t, x-vt , v) ≤ C2(t) Mα2,β2 (x − (t + 1)v, v) • for some positive functions 0 < C1(t) ≤ C ≤ C2(t) < ∞, and parameters 0 < α2 ≤ α ≤ α1 and 0 < β2 ≤ β ≤ β1. • Moreover, the case α = 0 (infinite mass) is permitted as long as α 1 = α 2 = 0. • (this last part extends the result of Mishler & Perthame ’97 to soft potentials)

15. Distributional solutions near local Maxwellians : RicardoAlonso, IMG’08 Sketch of proof: Define the distance between two Maxwellian distributions Mi= CiMαi,βi for i = 1, 2 as d(M1, M2) := |C2 − C1| + |α2 − α1| + |β2 − β1|. Second, we say that f is ε–closeto the Maxwellian distribution M = C Mα,β if there exist Maxwellian distributions Mi (i = 1, 2) such that d(Mi, M) <εfor some small ε > 0, and M1 ≤ f ≤ M2. Also define and notice that for -n < λ ≤ 0

16. Following the Kaniel-Shinbrot procedure, one obtains the following non-linear system of inequations which can be solved in C1(t) and C2(t) for an initial data for t0 ≥ 1 that satisfy an admissible beginning condition. Sketch of proof: 1- So choose C1(t) and C2(t) such that (Remark: Mischler &Perthame for λ=0 and ϕ1 = ϕ2 )

17. or which clearly implies for any t0 , t ≥1 2- Therefore which has a solution of the form For

18. 3- Therefore, C2(t) will be uniformly bounded for t ≥ 1 as long as which can be obtained done by taking d (M1, M2) ≤ ϵand In particular, the ‘beginning condition’ follows since, with source and absorption coefficient fixed, a simple comparison arguments of ODE’s shows that The evolution equation for C1(t) with an initial state C1(0)=C1 , with f0 ≥ C1Mαi,βi , implies . Similarly arguments work for C2(t). Then, for sufficiently small ϵthe Boltzmann equation has a unique solution satisfying C1(t) Mα1,β1 (x − (t + 1)v, v ) ≤ f ( t, x-vt , v) ≤ C2(t) Mα2,β2 (x − (t + 1)v, v) for some positive functions 0 < C1(t) ≤ C ≤ C2(t) < ∞, and parameters 0 < α2 ≤ α ≤ α1 and 0 < β2 ≤ β ≤ β1. i.e. the distributional solution f is controlled by a traveling Maxwellian, and so it spreads its mass as t ∞,

19. Classical solutions (Different approach from Guo’03, our methods follow some of the those by Boudin & Desvilletes ‘00, plus new ones ) Definition. A classical solutionin [0, T] of problem our is a function such that , Theorem (Application of HLS inequality to Q+ for soft potentials): Let the collision kernel satisfying assumptions λ < nand the Grad cut-off, then for 1 < p < ∞ where γ= n/(n−λ)and Ci= C(n, λ, p, ||b||L1(Sn−1) ) with i = 1, 2,3. The constants can be explicitly computed and are proportional to with parameter 1 < q = q(n, λ, p) < ∞, (the singularity at s = 1 is removed by symmetrazing b(s)when f = g )

20. Theorem (space regularity, globally in time ) Fix 0 ≤ T ≤ ∞ and assume the collision kernel satisfies B(u, û· σ) = |u|−λ b(û· σ) with -1 ≤ λ < n-1 with the Grad’s assumption. Also, assume that f0satisfies the smallness assumption or is near to a local Maxwellian. In addition, assume that ∇f0 ∈ Lp(R2n) for some 1 < p < ∞. Then, there is a unique classical solution f to the problem in the interval [0, T] satisfying the estimates of these theorems, and for all t ∈ [0, T], with constant x x x Proof: set with › for a fix h > 0 and x ∈ S n−1 and the corresp. translation operator and transforming x∗ → x∗ + hx in the collision operator. › : ∫

21. Multiply by and integrate: ∫ Using HLS estimates on Q(f,g) and, since the distributional solution f(t; x; v) is controlled by a traveling Maxwellian, then with a = n/(n−λ) And estimate (by similar arguments) By Gronwall inequality x globally in time Then, as h 0 to x

22. Velocity regularity (local in time) Theorem Let f be a classical solution in [0, T] with f0satisfying the condition of the smallness assumption or is near to a local Maxwellian and ∇x f0 ∈Lp(R2n) for some 1 < p < ∞. In addition assume that ∇v f0 ∈ Lp(R2n). Then, f satisfies the estimate Proof : Take for a fix h > 0 and ˆv ∈ S n−1 and the corresp. translation operator and transforming v∗ → v∗ + hˆv in the collision operator. multiply by and : ∫ apply HLS on Q

23. Just set then (Bernoulli Eq. ) with x Which is solved by Then, by the regularity estimate with 0 < λ < n-1 x Then, as h 0 to

24. Lp and Mα,β stability Set Now, since f and g are controlled by traveling Maxwellians one has with a = n/(n−λ) and 0 < λ < n-1

25. Theorem Let f and g distributional solutions of problem associated to the initial datum f0and g0respectively. Assume that these datum satisfies the condition of theorems for small data or near Maxwellians solutions (0 < λ < n-1). Then, there exist C > 0 independent of time such that Moreover, for f0and g0sufficiently small in Mα,β Remark: The result of Ha 06 for L1 stabiltity requires b(û· σ)bounded as a function of the scattering angle. Our result is for integrable b(û · σ)…. but p >1

26. Thank you for your attention! References and preprints http://rene.ma.utexas.edu/users/gamba/publications-web.htm