Irene M. Gamba Department of Mathematics and ICES The University of Texas at Austin

1. Generation and propagation of exponential weighted estimates to solutions ofnon-linear collisional equations Irene M. Gamba Department of Mathematics and ICES The University of Texas at Austin Collaborators: A. Bobylev, Karlstad University. Ricardo Alonso, UT Austin-Rice University, Vlad Panferov, CSU, Northridge, CA, Cedric Villani, ENS Lyon, France. S. Harsha Tharkbushanam, ICES and PROS more recently J. Canizo, S. Mischler, C. Mouhot (Paris IV)

2. Statistical transport from collisional kinetic models • Rarefied ideal gases-elastic:classical conservativeBoltzmann Transport eq. • Energy dissipative phenomena: Gas of elastic or inelastic interacting systems in the presence of a thermostat with a fixed background temperature өb or Rapid granular flow dynamics: (inelastic hard sphere interactions): homogeneous cooling states, randomly heated states, shear flows, shockwaves past wedges, etc. • (Soft) condensed matter at nano scale: Bose-Einstein condensates models and charge transport in solids: current/voltage transport modeling semiconductor. • Emerging applications from stochastic dynamics for multi-linear Maxwell type interactions : Multiplicatively Interactive Stochastic Processes: • information percolation models, particle swarms in population dynamics,

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

4. A general formstatistical transport : The space-homogenous BTE with external heating sources Important examples from mathematical physics and social sciences: The term models external heating sources: • background thermostat (linear collisions), • thermal bath (diffusion) • shear flow (friction), • dynamically scaled long time limits (self-similar solutions). ‘v v inelastic collision η v* ‘v* u’= (1-β) u + β |u| σ , with σthe direction of elastic post-collisional relativevelocity Inelastic Collision

6. Exact energy identity for a Maxwell type interaction models Then f(v,t) → δ0ast → ∞ to a singular concentrated measure (unless there is ‘source’) Current issues of interest regarding energy dissipation: Can one tell the shape or classify possible stationary states and their asymptotics, such as self-similarity? Non-Gaussian (or Maxwellian) statistics!

7. Reviewing inelastic properties INELASTIC Boltzmann collision term: No classical H-Theorem if e= constant < 1 It dissipates total energy for e < 1 by Jensen's inequality: •  Inelasticity brings loss of micro reversibility • but keeps time irreversibility!!: That is, there are stationary states and, in some particular cases we can show stability to stationary and self-similar states (Multi-linear Maxwell molecule equations of collisional type and variable hard potentials for collisions with a background thermostat) •  However: Existence of NESS: Non Equilibrium Statistical States (stable stationary states are non-Gaussian pdf’s)

8. Non-Equilibrium Stationary Statistical States

9. ? Yes (ARMA’09)

10. B. Wennberg~’98 generation of moments estimates generation of exponentially weighted lower bound

11. (JPS’04) Sharp Povzner estimates Summability of moments series (I.G V.Panferov, c. Villani; ARMA’09) then (JMPA’08) VIII) Generation of exponential L1-weighted estimates (Mouhot’06) and better tails (Alonso,Canizo,IG and Mouhot in progress)

12. Key property: Summability of series of moments of BTE solutions

14. Sharp Povzner estimates: optimal control of weights in `average’ Angular Averaging Inequality: (A.Bobylev, I.G., V.Panferov, JSP’04) and of γ (rate of the intramolecular potentials)

15. ** Elastic case: β=1 d-dimensions (I.G., V.Panferov C.Villani; ARMA’08 with Our result extends the Bobylev-Povzner-type estimate (JSP'97) for d=3 and γ=1, (i.e. b(σ)=C) tod > 1and kernels with monotone angular dependence on its symmetric part satisfying **

16. Corollary 2: In order to study the behavior of mp with p = ks/2 for a good choice of s, take moments of evolution forced equations:

17. Corollary: it is possible to choose s, such that for r and R depending on the initial mass m0, energy bound m1 and some high order moments mp0 for some p∗ > 1, depending on the 'heating' force coefficient. The choice of s is done by setting: (shown in the pure diffusion case and bounded angular section γ=1 and stationary state) So, in order to control zp+1/2 we need to divide by Γ(a(p+1/2) + b) and find a suitable value of a such that we can get control of a corresponding recursion inequality relation that produces a geometric growth control forzP

18. Then Add the time derivative to compute the Corresponding evolution estimate ∂tzp+ a careful choice of a = 4/3 and b < 1 cancels the coefficients for the two terms proportional to zp-1 , and the right hand side term (from the gain term) is controlled by a constant!! Similarly for the other cases: diffusion with friction: s = 2. Self-similar (homogeneous cooling) s = 1 Shear flow: at least s = 1 but anisotropy is admissible, so other direction might decay faster.

19. In the elastic case with no sources, for 0 < γ≤ 1 and b(θ) integrable: Exponential moments propagation (I.G. Panferov and Villani, arXiv’06, ARMA’09) moments of equation collision operator Loss op. Gain op. Bernoulli type eq. can also “create” moments (Desvillates 93 B. Wennberg~’98) (i.e. a=1 and s=2) or “generate”

20. In the elastic case, for 0 < γ≤ 1 and b(θ)integrable: Propagation and generation estimates (i.e. a=1 and s=2) • for r and R depending on the initial mass m0, energy bound m1 and some high order moments mp0 for some p∗ > 1, but uniform in t !  • Summability of the series of moments is uniform intime  • Propagation of exponential moments • Same argument holds for controlling moments of the derivatives of f(t,v) by iterative methods (R. Alonso &I.G. JMPA’08) • SS solutions to Elastic collisions with a cold thermostat for the choice ofa= γ/2 and s= γ and existence (Alonso, Canizo, IG, Mischler, Mouhot, in preparation ) • Generation of moments for a= γ/4 and s= γ/2 (Mouhot JSP’06) for initial data with only 2+ moments • Improvement in moments generation by taking a=a(t, γ) to s=γ(Alonso, Canizo, IG, and Mouhot, in preparation )

21. Upper point-wise uniform bounds for large energy tails for elastic hard spheres or γ-variable potentials in d-dimensions (IG, Vlad Panferov, Cedric Villani, arXiv.org’06 - ARMA’08, and R. Alonso and I.G- JMPA’08) STRATEGY : Find a comparison theorem & construct a suitable barrier function ⇒ Compare to obtain point-wise bounds: Comparison principle: Q is multi-linear, symmetric, conservative, and L1-contractive for its linear restriction (Crandall & Tartar ’80, also Vandenjapin & Bobylev 75, Kaniel & Shimbrot ’84, Lions ’94 ) Remark: it also works in the space inhomogeneous case. and

22. Crucial point:

23. In order to find the barrier probability distribution: we need That can be obtain by the following key estimate:

24. Tool: Carleman integral representation of

27. Remark:these propagation properties in L1 and L∞ Maxwellians weighted norms also hold for all the derivatives if initial data have all derivatives under such control (Ricardo Alonso and I.G.; ): we use iterative arguments.

28. Estimates for Existence theory: Average angular estimates & weighted Young’s inequalities R. Alonso and E. Carneiro’08, and R. Alonso and E. Carneiro, IG, 08 with Angular average inequality

29. Young’s inequality for variable hard potentials : 1 ≥λ≥ 0 Hardy-Littlewood-Sobolev type inequality for soft potentials : 0 > λ≥-n These two constants C depends linearly of the expression given above for the constant of the angular averaging lemma

30. by spectral-Lagrangian based methods for non Conservative energy (IG,H.Tharkabhushanam, JCP’09

31. Testing: BTE with (Gaussian) hot and (singular) cold Thermostat explicit solution problem of asymptotics of same mass particles mixture Maxwell Molecules model Rescaling of spectral modes exponentially by the continuous spectrum with λ(1)=-2/3

32. Testing: BTE with cold Thermostat Moments calculations: Explicit solutions problem IG & Bobylev and I.G., JSP06 Rescaling time by the Kinetic energy And velocity with the corresponding thermal Speed  Moments of order q ≥ 1.5 will become unbounded in time

33. Recent work related to the problem: Cercignani'95(inelastic BTE derivation); Bobylev, JSP 97 (elastic,hard spheres in 3 d: propagation of L1-exponential estimates ); Bobylev, Carrillo and I.M.G., JSP'00 (inelastic Maxwell type interactions- self similarity- mean field); Bobylev, Cercignani , and with Toscani, JSP '02 &'03 (inelastic Maxwell type interactions); Bobylev, I.M.G, V.Panferov, C.Villani, JSP'04, CMP’04 (inelastic + heat sources); Mischler and Mouhout, Rodriguez Ricart JSP '06 (inelastic + self-similar hard spheres); Bobylev and I.M.G. JSP'06 (Maxwell type interactions-inelastic/elastic + thermostat), Bobylev, Cercignani and I.M.G arXiv.org,06 (CMP’09); (generalized multi-linear Maxwell type interactions-inelastic/elastic: global energy dissipation) I.M.G, V.Panferov, C.Villani, arXiv.org’07, ARMA’09 (elastic n-dimensional variable hard potentials Grad cut-off:: propagation of L1 and L∞-exponential estimates) C. Mouhot, CMP’06 (elastic, VHP, bounded angular cross section: creation of L1-exponential ) R. Alonso and I.M.G., JMPA’08 (Grad cut-off, propagation of regularity bounds-elastic d-dim VHP) I.M.G. and Harsha Tarskabhushanam JCP’08(spectral-lagrangian solvers-computation of singulatities) R.Alonso, E.Carneiro(ArXiv.org08)(Young’s inequality for collisional integrals with integrable (grad cut-off) angular cross section) R.Alonso, E.Carneiro, I.M.G. ArXiv.org09 (weigthed Young’s inequality and Hardy Sobolev’s inequalities for collisional integrals with integrable (grad cut-fff)angular cross section) R. Alonso and I.M.G. ArXiv.org09, (Distributional and classical solutions to the Cauchy Boltzmann problem for soft potentials with integrable angular cross section) Alonso, Canizo, I.M.G.,Mischler, Mouhot, in preparation (The homogeneous Boltzmann eqaution with a cold thermostat for variable hard potentials) Alonso, Canizo, I.M.G., Mouhot, in preparation (sharper decay for moments creation estimates for variable hard potentials)

34. Comments: • . Tails are important to understand evolution of moments (well known….!!!) They depend on the rate of collision as a function of velocity. • (Decay rates to equilibrium states depend on the angular cross section as one can get exact and best constant depending on b(θ) ) • Tails control methods to space inhomogeneous problems: may lead to local in • x-space, global in v-space contrl of the solution BTE, • …. but we do not how to do it yet… • The use of Young and Hardy Littlewood Sobolev type of inequalities allows to revisit and/or extend the existence and regularity results of the space inhomogeneous BTE with soft potentials and angular cross sections that are just integrable (Grad cut-off assumption), with data between near two different Mawellians. • Need to adjust hydrodynamic limits for non conservative phenomena: • Hydrodynamic limits with energy dissipation lack of exact/local closure formulas- • macroscopic equations may not have an accurate closed form. Thank you very much for your attention! For recent preprints and reprints see: www.ma.utexas.edu/users/gamba/research and references therein