Strict entropy production bounds and stability of the rate of convergence to equilibrium for the Boltzmann equation

We first consider the Boltzmann equation with a collision kernel such that all kinematically possible collisions are run at equal rates. This is the simplest Boltzmann equation having the compressible Euler equations as a scaling limit. For it we prove a stability result for theH-theorem which says that when the entropy production is small, the solution of the spatially homogeneous Boltzmann equation is necessarily close to equilibrium in the entropie sense, and therefore strongL ¹ sense. We use this to prove that solutions to the spatially homogeneous Boltzmann equation converge to equilibrium in the entropie sense with a rate of convergence which is uniform in the initial condition for all initial conditions belonging to certain natural regularity classes. Every initial condition with finite entropy andp ^th velocity moment for some p>2 belongs to such a class. We then extend these results by a simple monotonicity argument to the case where the collision rate is uniformly bounded below, which covers a wide class of slightly modified physical collision kernels. These results are the basis of a study of the relation between scaling limits of solutions of the Boltzmann equation and hydrodynamics which will be developed in subsequent papers; the program is described here.On leave from School of Mathematics, Georgia Institute of Technology, Atlanta, Georgia 30332.On leave from C.F.M.C. and Departamento de Matemática da Faculdade de Ciencias de Lisboa, 1700 Lisboa codex, Portugal.     

On Strong Convergence to Equilibrium for the Boltzmann Equation with Soft Potentials

The paper concerns L ¹-convergence to equilibrium for weak solutions of the spatially homogeneous Boltzmann Equation for soft potentials (−4≤γ<0), with and without angular cutoff. We prove the time-averaged L ¹-convergence to equilibrium for all weak solutions whose initial data have finite entropy and finite moments up to order greater than 2+|γ|. For the usual L ¹-convergence we prove that the convergence rate can be controlled from below by the initial energy tails, and hence, for initial data with long energy tails, the convergence can be arbitrarily slow. We also show that under the integrable angular cutoff on the collision kernel with −1≤γ<0, there are algebraic upper and lower bounds on the rate of L ¹-convergence to equilibrium. Our methods of proof are based on entropy inequalities and moment estimates. E.A. Carlen work partially supported by US National Science Foundation grant DMS 06-00037. M.C. Carvalho work partially supported by POCI/MAT/61931/2004. X. Lu work partially supported by NSF of China grant 10571101.     

On the Boltzmann equation in unbounded space far from equilibrium,and the limit of zero mean free path

This paper studies Loeb solutions of the Boltzmann equation in unbounded space under natural initial conditions of finite mass, energy, and entropy. An existence theory for large initial data is presented. Maxwellian behaviour is obtained in the limits of zero mean free path and of infinite time. In the standard, space-homogeneous, hard potential case, the infinite time limit is of strongL ¹ type.     

On boltzmann equations and fokker—planck asymptotics: Influence of grazing collisions

In this paper, we are interested in the influence of grazing collisions, with deflection angle near π/2, in the space-homogeneous Boltzmann equation. We consider collision kernels given by inverse-sth-power force laws, and we deal with general initial data with bounded mass, energy, and entropy. First, once a suitable weak formulation is defined, we prove the existence of solutions of the spatially homogeneous Boltzmann equation, without angular cutoff assumption on the collision kernel, fors ≥ 7/3. Next, the convergence of these solutions to solutions of the Landau-Fokker-Planck equation is studied when the collision kernel concentrates around the value π/2. For very soft interactions, 2<s<7/3, the existence of weak solutions is discussed concerning the Boltzmann equation perturbed by a diffusion term     

The BGK Model with External Confining Potential: Existence,Long-Time Behaviour and Time-Periodic Maxwellian Equilibria

We study global existence and long time behaviour for the inhomogeneous nonlinear BGK model for the Boltzmann equation with an external confining potential. For an initial datum f ₀≥0 with bounded mass, entropy and total energy we prove existence and strong convergence in L ¹ to a Maxwellian equilibrium state, by compactness arguments and multipliers techniques. Of particular interest is the case with an isotropic harmonic potential, in which Boltzmann himself found infinitely many time-periodic Maxwellian steady states. This behaviour is shared with the Boltzmann equation and other kinetic models. For all these systems we study the multistability of the time-periodic Maxwellians and provide necessary conditions on f ₀ to identify the equilibrium state, both in L ¹ and in Lyapunov sense. Under further assumptions on f, these conditions become also sufficient for the identification of the equilibrium in L ¹.     

Cercignani's Conjecture is Sometimes True and Always Almost True

 We establish several new functional inequalities comparing Boltzmann's entropy production functional with the relative H functional. First we prove a longstanding conjecture by Cercignani under the nonphysical assumption that the Boltzmann collision kernel is superquadratic at infinity. The proof rests on the method introduced in [39] combined with a novel use of the Blachman-Stam inequality. If the superquadraticity assumption is not satisfied, then it is known that Cercignani's conjecture is not true; however we establish a slightly weakened version of it for all physically relevant collision kernels, thus extending previous results from [39]. Finally, we consider the entropy-entropy production version of Kac's spectral gap problem and obtain estimates about the dependence of the constants with respect to the dimension. The first two results are sharp in some sense, and the third one is likely to be, too; they contain all previously known entropy estimates as particular cases. This gives a first coherent picture of the study of entropy production, according to a program started by Carlen and Carvalho [12] ten years ago. These entropy inequalities are one step in our study of the trend to equilibrium for the Boltzmann equation, in both its spatially homogeneous and spatially inhomogeneous versions. Received: 25 July 2002 / Accepted: 13 September 2002 Published online: 10 February 2003 Communicated by J. L. Lebowitz     

Some Ideas About Quantitative Convergence of Collision Models to Their Mean Field Limit

We consider a stochastic N-particle model for the spatially homogeneous Boltzmann evolution and prove its convergence to the associated Boltzmann equation when N⟶∞, with non-asymptotic estimates: for any time T>0, we bound the distance between the empirical measure of the particle system and the measure given by the Boltzmann evolution in a relevant Hilbert space. The control got is Gaussian, i.e. we prove that the distance is bigger than xN ^−1/2 with a probability of type O(e^-x2)O(e^{-x^{2}}). The two main ingredients are a control of fluctuations due to the discrete nature of collisions and a kind of Lipschitz continuity for the Boltzmann collision kernel. We study more extensively the case where our Hilbert space is the homogeneous negative Sobolev space [(H)\dot]^-s\smash {\dot {H}}^{-s}. Then we are only able to give bounds for Maxwellian models; however, numerical computations tend to show that our results are useful in practice.     

Global existence proof for relativistic Boltzmann equation

The existence and causality of solutions to the relativistic Boltzmann equation inL ¹ and inL _loc ¹ are proved. The solutions are shown to satisfy physically naturala priori bounds, time-independent inL ¹. The results rely upon new techniques developed for the nonrelativistic Boltzmann equation by DiPerna and Lions.     

Fast and Slow Convergence to Equilibrium for Maxwellian Molecules via Wild Sums

We consider the spatially homogeneous Boltzmann equation for Maxwellian molecules and general finite energy initial data: positive Borel measures with finite moments up to order 2. We show that the coefficients in the Wild sum converge strongly to the equilibrium, and quantitatively estimate the rate. We show that this depends on the initial data F essentially only through on the behavior near r=0 of the function J _F (r)=_|v|>1/r |v|² dF(v). These estimates on the terms in the Wild sum yield a quantitative estimate, in the strongest physical norm, on the rate at which the solution converges to equilibrium, as well as a global stability estimate. We show that our upper bounds are qualitatively sharp by producing examples of solutions for which the convergence is as slow as permitted by our bounds. These are the first examples of solutions of the Boltzmann equation that converge to equilibrium more slowly than exponentially.     

Global Existence of Classical Solutions to the Fokker-Planck-BGK Equation

A kinetic model of the Fokker-Planck-Boltzmann equation is introduced by replacing the original Boltzmann collision operator with the Bhatnagar-Gross-Krook collision model (BGK collision model). This model equation, which we call the Fokker-Planck-BGK equation, has many physical features that the Fokker-Planck-Boltzmann equation possesses. We first establish an L ^∞ existence result for this equation, by which we construct the approximate solutions. Then, by means of the regularizing effects of the linear Fokker-Planck operator and L ^p estimates of local Maxwellians, we obtain some uniform estimates of the approximate solutions. Finally, combining those estimates and regularizing effects, we prove by a compactness argument that the equation has a global classical solution under rather general initial conditions. Supported by the Scientific Research Foundation of Huazhong University of Science and Technology (HUST-SRF).     

Asymptotic Stability of the Relativistic Boltzmann Equation for the Soft Potentials

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{L^\infty_\ell}. 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):607–629, 1985); this resolves the open question of global existence for the soft potentials.     

A Revision on Classical Solutions to the Cauchy Boltzmann Problem for Soft Potentials

This short note complements the recent paper of the authors (Alonso, Gamba in J. Stat. Phys. 137(5–6):1147–1165, 2009). We revisit the results on propagation of regularity and stability using L ^p estimates for the gain and loss collision operators which had the exponent range misstated for the loss operator. We show here the correct range of exponents. We require a Lebesgue’s exponent α>1 in the angular part of the collision kernel in order to obtain finiteness in some constants involved in the regularity and stability estimates. As a consequence the L ^p regularity associated to the Cauchy problem of the space inhomogeneous Boltzmann equation holds for a finite range of p≥1 explicitly determined.     

Dissipative Linear Boltzmann Equation for Hard Spheres

We prove the existence and uniqueness of an equilibrium state with unit mass to the dissipative linear Boltzmann equation with hard-spheres collision kernel describing inelastic interactions of a gas particles with a fixed background. The equilibrium state is a universal Maxwellian distribution function with the same velocity as field particles and with a non-zero temperature lower than the background one. Moreover thanks to the H-Theorem we prove strong convergence of the solution to the Boltzmann equation towards the equilibrium.     

Integral Representation of the Linear Boltzmann Operator for Granular Gas Dynamics with Applications

We investigate the properties of the collision operator Q associated to the linear Boltzmann equation for dissipative hard-spheres arising in granular gas dynamics. We establish that, as in the case of non-dissipative interactions, the gain collision operator is an integral operator whose kernel is made explicit. One deduces from this result a complete picture of the spectrum of Q in an Hilbert space setting, generalizing results from T. Carleman (Publications Scientifiques de l’Institut Mittag-Leffler, vol. 2, 1957) to granular gases. In the same way, we obtain from this integral representation of Q ⁺ that the semigroup in L ¹(ℝ³×ℝ³,dx ⊗ dv) associated to the linear Boltzmann equation for dissipative hard spheres is honest generalizing known results from Arlotti (Acta Appl. Math. 23:129–144, 1991).     

Propagation of Smoothness and the Rate of Exponential Convergence to Equilibrium for a Spatially Homogeneous Maxwellian Gas

We prove an inequality for the gain term in the Boltzmann equation for Maxwellian molecules that implies a uniform bound on Sobolev norms of the solution, provided the initial data has a finite norm in the corresponding Sobolev space. We then prove a sharp bound on the rate of exponential convergence to equilibrium in a weak norm. These results are then combined, using interpolation inequalities, to obtain the optimal rate of exponential convergence in the strong L¹ norm, as well as various Sobolev norms. These results are the first showing that the spectral gap in the linearized collision operator actually does govern the rate of approach to equilibrium for the full non-linear Boltzmann equation, even for initial data that is far from equilibrium.     

A Modified Boltzmann Equation for Bose–Einstein Particles: Isotropic Solutions and Long-Time Behavior

Under some strong cutoff conditions on collision kernels, global existence, local stability, entropy identity, conservation of energy, and moment production estimates are proven for isotropic solutions of a modified (quantum effect) Boltzmann equation for spatially homogeneous gases of Bose–Einstein particles (BBE). Then applying these results with the biting-weak convergence, some results on the long-time behavior of the conservative isotropic solutions of the BBE equation are obtained, including the velocity concentration at very low temperatures and the tendency toward equilibrium states at very high temperatures.     

A maxwellian lower bound for solutions to the Boltzmann equation

We prove that the solution of the spatially homogeneous Boltzmann equation is bounded pointwise from below by a Maxwellian, i.e. a function of the formc ₁ exp(-c ₂ v ²). This holds for any initial data with bounded mass, energy and entropy, and for any positive timet≧t ₀. The constantsc ₁, andc ₂, depend on the mass, energy and entropy of the initial data, and ont ₀>0 only. A similar result is obtained for the Kac caricature of the Boltzmann equation, where the proof is easier.     

Global existence inL 1 for the modified nonlinear Enskog equation in ℝ3

A global existence theorem with large initial data inL ¹ is given for the modified Enskog equation in ³. The method, which is based on the existence of a Liapunov functional (analog of theH-Boltzmann theorem), utilizes a weak compactness argument inL ¹ in a similar way to the DiPerna-Lions proof for the Boltzmann equation. The existence theorem is obtained under certain condition on the behavior of the geometric factorY. The condition onY amounts to the fact that theL ¹ norm of the collision term grows linearly when the local density tends to infinity.     

On the linearized relativistic Boltzmann equation

The linearized relativistic Boltzmann equation inL ²(r,p) is investigated. The detailed analysis of the collision operatorL is carried out for a wide class of scattering cross sections.L is proved to have a form of the multiplication operatorv(p) plus the compact inL ²(p) perturbationK. The collisional frequencyv(p) is analysed to discriminate between relativistic soft and hard interactions. Finally, the existence and uniqueness of the solution to the linearized relativistic Boltzmann equation is proved.     

On weak and strong convergence to equilibrium for solutions to the linear Boltzmann equation

This paper considers the linear space-inhomogeneous Boltzmann equation for a distribution function in a bounded domain with general boundary conditions together with an external potential force. The paper gives results on strong convergence to equilibrium, whent, for general initial data; first in the cutoff case, and then for infinite-range collision forces. The proofs are based on the properties of translation continuity and weak convergence to equilibrium. To handle these problems generalH-theorems (concerning monotonicity in time of convex entropy functionals) are presented. Furthermore, the paper gives general results on collision invariants, i.e., on functions satisfying detailed balance relations in a binary collision.