Cauchy problem for the Korteweg-de Vries equation and its generalizations

Nonlocal results on the existence, uniqueness, continuous dependence on the initial data, and intrinsic regularity of generalized solutions of the Cauchy problem for the Korteweg-de Vries equation and its generalizations are established.Translated from Trudy Seminara imeni I. G. Petrovskogo, No. 13, pp. 56–105, 1988.     

On approximate solution of the Dixon integral equation and some its generalizations

The paper is devoted to the study and numerical analytical solution of Fredholm-type integral equations of the second kind with symmetric kernels represented by homogeneous functions of degree (-1). The well-known Dixon equation and some its direct generalizations are specially considered. The equations are solved by passing to a Wiener–Hopf equation and applying the kernel averaging method. Results of numerical calculations are presented.     

Nonlinear nonequilibrium kinetic model of the boltzmann equation for monatomic gases

A model kinetic equation approximating the Boltzmann equation in a wide range of nonequilibrium gas states was constructed to describe rarefied gas flows. The kinetic model was based on a distribution function depending on the absolute velocity of the gas particles. Highly efficient in numerical computations, the model kinetic equation was used to compute a shock wave structure. The numerical results were compared with experimental data for argon.     

Analytical solution of the nonstationary boltzmann equation in the BGC model

Using the Case method, the analytical solution to the BGC nonstationary kinetic equation is constructed. As an application, the problem of a point-like source of heat or particles is considered. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 113, No. 1, pp. 139–148, October, 1997.     

On the Pompeiu problem and its generalizations

Functions are investigated whose integrals over a given collection of sets are zero. Pompeiu sets are described in terms of the approximation of their indicators by linear combinations of the indicators of balls with special radii.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 45, No. 10, pp. 1444–1448, October, 1993.     

On the convergence of solutions of the enskog equation to solutions of the boltzmann equation

For the Enskog equation with a symmetrized kernel in a box an existence theorem is proved for initial data with finite mass, energy and entropy. Then by letting the diameter of the molecules go to zero we prove the weak convergence of solutions of the Enskog equation to solutions of the Boltzmann equation.     

Local validity of the boltzmann equation

The validity of the Boltzmann equation is proved, locally in time, by showing that its solution is a limit of solutions of the BBGKY hierarchy.     

On the Kaczmarz iterative method and its generalizations

Under study is the family of iterative projection methods that stems from the work of Kaczmarz in 1937. We propose some alternating block algorithms of Kaczmarz type in Krylov subspaces with a relaxation parameter and acceleration in Krylov spaces. The results are presented and discussed of numerical experiments for some model problems.     

Incompressible navier-stokes and euler limits of the boltzmann equation

We consider solutions of the Boltzmann equation, in a d-dimensional torus, d = 2, 3, For macroscopic times τ = t/?^N, ? « 1, t ≧ 0, when the space variations are on a macroscopic scale x = ?^N?1r, N ≧ 2, x in the unit torus. Let u(x, t) be, for t ≦ t₀, a smooth solution of the incompressible Navier Stokes equations (INS) for N = 2 and of the Incompressible Euler equation (IE) for N > 2. We prove that (*) has solutions for t ≦ t₀ which are close, to O(?²) in a suitable norm, to the local Maxwellian [p/(2πT)^d/2]exp{?[v ? ?u(x,t)]²/2T } with constant density p and temperature T . This is a particular case, defined by the choice of initial values of the macroscopic variables, of a class of such solutions in which the macroscopic variables satisfy more general hydrodynamical equations. For N ≧ 3 these equations correspond to variable density IE while for N = 2 they involve higher-order derivatives of the density.