Nonlinear Evolution Equations

The book is devoted to the questions of the long-time behavior of solutions for evolution equations, connected with kinetic models in statistical physics. There is a wide variety of problems where such models are used to obtain reasonable physical as well as numerical results (Fluid Mechanics, Gas Dynamics, Plasma Physics, Nuclear Physics, Turbulence Theory etc.). The classical examples provide the nonlinear Boltzmann equation. Investigation of the long-time behavior of the solutions for the Boltzmann equation gives an approach to the nonlinear fluid dynamic equations. From the viewpoint of dynamical systems, the fluid dynamic equations arise in the theory as a tool to describe an attractor of the kinetic equation. Contents: IntroductionKinetic Approximations of Nonlinear Evolution Equations. Formal ConstructionsBoltzmann Equation. Collision OperatorsTransport OperatorsSteady-State Solutions of the Linear Boltzmann EquationNonlinear Steady-State ProblemsInitial-Boundary Value Problems for the Boltzmann Equation Readership: Mathematicians, mathematical physicists, statistical physicists, graduates and researchers. keywords:Hyperbolic Conservation Laws;Fridman-Keller Equations;Steady Boundary Value Problems;Function Spaces;Parabolic Equations;Collision Operators;Collision Kernels;Invariants;Transport Operators;Generalized Green's Formula;Velocity Averaging Estimates;Perturbation Theory;Initial Boundary Value Problem;Existence Theorem;Incompressible Limit;Navier-Stokes Equations;Euler Equations“The book is an elegant summing-up of many results that are spread out in the literature.”Mathematical Reviews “The book is a little gem which contains quite a few results that were difficult to find before, because they were published either not at all or only in Russian.”Mathematics Abstracts