Jacek Polewczak

Classical solution of the nonlinear Boltzmann equation in allR3: Asymptotic behavior of solutions

Proof is given of the existence of a classical solution to the nonlinear Boltzmann equation in allR3. The solution, which is global in time, exists if the initial data go to zero fast enough at infinity and the mean free path is sufficiently large. The solution is smooth in the space variable if the initial value is smooth. The asymptotic behavior of solutions is also given. It is shown that ast→∞ the solution to the Boltzmann equation can be approximated by the solution to the free motion problem.

Global existence and asymptotic behavior for the nonlinear Enskog equation

The existence of a classical solution to the nonlinear Enskog equation in all

Global existence inL1 for the modified nonlinear Enskog equation in ℝ3

\mathbb{R}^3

Global existence inL1 for the generalized Enskog equation

is proved. The solution, which is global in time, exists if the initial data go to zero fast enough at infinity and the mean path is sufficiently large. The solution is smooth in space variable if the initial value is smooth. The asymptoticbehavior of solutions is also given. It is shown that as

A Global Existence Theorem for the Nonlinear BGK Equation

t \to \infty

The Kinetic Theory of Simple Reacting Spheres: I. Global Existence Result in a Dilute-Gas Case

the solution to the Enskog equation can be approximated by the solution to the free motion problem.Key words. partial differential equations, nonlinear integral equation, asymptotic behavior, kinetic theory, Boltzmann equation

Kinetic Theory of Simple Reacting Spheres I

A global existence theorem with large initial data inL1 is given for the modified Enskog equation in ℝ3. The method, which is based on the existence of a Liapunov functional (analog of theH-Boltzmann theorem), utilizes a weak compactness argument inL1 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 theL1 norm of the collision term grows linearly when the local density tends to infinity.

Transport Coefficients in Some Stochastic Models of the Revised Enskog Equation

Various existence theorems are given for the generalized Enskog equation inR3 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 inL1.

New properties of a class of generalized kinetic equations

A global existence theorem with large initial data inL1 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 inL1.

Kinetic theory of atoms and photons: An application to the Milne-Chandrasekhar problem

Existence of global-in-time, spatially inhomogeneous, and L1-renormalized solutions is proven for the model of simple reacting spheres under the assumptions that initially the system has a finite total mass, energy, and entropy.