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.     

Global existence inL 1 for the generalized Enskog equation

Various existence theorems are given for the generalized Enskog equation inR ³ and in a bounded spatial domain with periodic boundary conditions. A very general form of the geometric factorY is allowed, including an explicit space, velocity, and time dependence. The method is based on the existence of a Liapunov functional, an analog of theH-function in the Boltzmann equation, and utilizes a weak compactness argument inL ¹.     

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.     

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.     

Existence of the scattering matrix for the linearized Boltzmann equation

Following Hejtmanek, we consider neutrons in infinite space obeying a linearized Boltzmann equation describing their interaction with matter in some compact setD. We prove existence of theS-matrix and subcriticality of the dynamics in the (weak-coupling) case where the mean free path is larger than the diameter ofD uniform in the velocity. We prove existence of theS-matrix also for the case whereD is convex and filled with uniformly absorbent material. In an appendix, we present an explicit example where the dynamics is not invertible onL ₊ ¹ , the cone of positive elements inL ¹.A. Sloan fellow; research partially supported by the U.S. NSF under Grant GP 39048     

Entropy production estimates for Boltzmann equations with physically realistic collision kernels

We establish strict entropy production bounds for the Boltzmann equation with the hard-sphere collision kernel. Using these entropy production bounds, we prove results asserting that the rate at which strongL ¹ convergence to equilibrium occurs is uniform in wide classes of initial data. This extends our previous results in this direction, which applied only to a very special collision kernel. Moreover, the present results provide computable lower bounds; compactness arguments are entirely avoided. The uniformity is an important ingredient in our study of scaling limits of solutions of the non-spatially homogeneous Boltzmann equation, and is the main focus of this paper. However, the results obtained here provide the only framework known to us in which one can obtain computable estimates on the time it takes a solution of the spatially homogeneous Boltzmann equation with initial data far from equilibrium to reach any given small strongL ¹ neighborhood of equilibrium.     

On the linearized relativistic Boltzmann equation. II. Existence of hydrodynamics

Solutions are analyzed of the linearized relativistic Boltzmann equation for initial data fromL ²(r, p) in long-time and/or small-mean-free-path limits. In both limits solutions of this equation converge to approximate ones constructed with solutions of the set of differential equations called the equations of relativistic hydrodynamics.     

Lyapunov Functionals and L 1 -Stability for Discrete Velocity Boltzmann Equations

We devise Lyapunov functionals and prove uniform L¹ stability for one-dimensional semilinear hyperbolic systems with quadratic nonlinear source terms. These systems encompass a class of discrete velocity models for the Boltzmann equation. The Lyapunov functional is equivalent to the L¹ distance between two weak solutions and non-increasing in time. They result from computations of two point interactions in the phase space. For certain models with only transversal collisional terms there exist generalizations for three and multi-point interactions.     

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.     

The Broadwell model for initial values inL + 1 (ℝ)

The Cauchy problem for the Broadwell model is shown to have a global mild solution for initial data inL ₊ ¹ () with smallL ¹-norm, and a local solution for arbitrary initial data inL ₊ ¹ (). For data which are small inL ¹(), the asymptotic behaviour of the solutions ast is determined. Moreover, it is shown that a global solution exists for all initial values inL ₊ ¹ () with finite entropy if theH-Theorem holds.     

A global existence theorem for the nonlinear BGK equation

A global existence theorem with large initial data inL ¹ is given for the nonlinear BGK equation. The method, which is based on the recent averaging lemma of Golseet al., utilizes a weak compactness argument inL ¹.     

On a Taylor-Couette Type Bifurcation for the Stationary Nonlinear Boltzmann Equation

This paper studies the stationary nonlinear Boltzmann equation for hard forces, in a Taylor-Couette setting between two coaxial, rotating cylinders with given indata of Maxwellian type on the cylinders. A priori L ^q -estimates are obtained, and used to prove a Taylor type bifurcation with isolated solutions and a hydrodynamic limit control, based on asymptotic expansions together with a rest term correction. The positivity of such solutions is also considered.     

Some remarks about continuity properties of local Maxwellians and an existence theorem for the BGK model of the Boltzmann equation

Continuity of local Maxwellians in various topologies ofL ¹ is studied. The existence and convergence of approximate solutions of the nonlinear BGK model of the Boltzmann equation are proved.     

Equilibrium Solution to the Inelastic Boltzmann Equation Driven by a Particle Bath

We show the existence of smooth stationary solutions for the inelastic Boltzmann equation under the thermalization induced by a host medium with a fixed distribution. This is achieved by controlling the L ^p -norms, the moments and the regularity of the solutions to the Cauchy problem together with arguments related to a dynamical proof for the existence of stationary states.     

Lees–Edwards Boundary Conditions for Lattice Boltzmann

Lees–Edwards boundary conditions (LEbc) for Molecular Dynamics simulations⁽¹⁾ are an extension of the well known periodic boundary conditions and allow the simulation of bulk systems in a simple shear flow. We show how the idea of LEbc can be implemented in isothermal lattice Boltzmann simulations and how LEbc can be used to overcome the problem of a maximum shear rate that is limited to less then 1/L _y (with L _y the transverse system size) in traditional lattice Boltzmann implementations of shear flow. The only previous Lattice Boltzmann implementation of LEbc⁽²⁾ requires a specific fourth order equilibrium distribution. In this paper we show how LEbc can be implemented with the usual quadratic equilibrium distributions.     

On the Boltzmann Equation for Fermi–Dirac Particles with Very Soft Potentials: Averaging Compactness of Weak Solutions

The paper considers macroscopic behavior of a Fermi–Dirac particle system. We prove the L ¹-compactness of velocity averages of weak solutions of the Boltzmann equation for Fermi–Dirac particles in a periodic box with the collision kernel b(cos θ)|ρ−ρ _*|^γ, which corresponds to very soft potentials: −5 < γ ≤ −3 with a weak angular cutoff: ∫₀ ^π b(cos θ)sin ³θ dθ < ∞. Our proof for the averaging compactness is based on the entropy inequality, Hausdorff–Young inequality, the L ^∞-bounds of the solutions, and a specific property of the value-range of the exponent γ. Once such an averaging compactness is proven, the proof of the existence of weak solutions will be relatively easy.     

The Stationary Nonlinear Boltzmann Equation in a Couette Setting with Multiple, Isolated L q -solutions and Hydrodynamic Limits

This paper studies the stationary nonlinear Boltzmann equation for hard forces, in a Couette setting between two coaxial, rotating cylinders with given indata of Maxwellian type on the cylinders. A priori estimates are obtained mainly in L², leading to multiple, isolated solutions together with a hydrodynamic limit control, based on asymptotic expansions together with a rest term.     

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).     

A Quantum Boltzmann Equation for Haldane Statistics and Hard Forces; the Space-Homogeneous Initial Value Problem

The paper considers equations of Boltzmann type for Haldane exclusion statistics. Existence and some basic properties of the solutions are studied for the space homogeneous initial value problem with hard forces and angular cut-off. The approach uses strong L ¹ compactness. Some of the technical estimates are based on L ^∞ decay properties, and the control of the filling factor on range estimates for the solutions.     

Optimal Decay Estimates on the Linearized Boltzmann Equation with Time Dependent Force and their Applications

Although the decay in time estimates of the semi-group generated by the linearized Boltzmann operator without forcing have been well established, there is no corresponding result for the case with general external force. This paper is mainly concerned with the optimal decay estimates on the solution operator in some weighted Sobolev spaces for the linearized Boltzmann equation with a time dependent external force. No time decay assumption is made on the force. The proof is based on both the energy method through the macro-micro decomposition and the L ^p -L ^q estimates from the spectral analysis. The decay estimates thus obtained are applied to the study on the global existence of the Cauchy problem to the nonlinear Boltzmann equation with time dependent external force and source. Precisely, for space dimension n ≥ 3, the global existence and decay rates of solutions to the Cauchy problem are obtained under the condition that the force and source decay in time with some rates. This time decay restriction can be removed for space dimension n ≥ 5. Moreover, the existence and asymptotic stability of the time periodic solution are given for space dimension n ≥ 5 when the force and source are time periodic with the same period.