Yasushi Shizuta

On the decay of solutions to the linearized equations of electro-magneto-fluid dynamics

The linearized equations of the electrically conducting compressible viscous fluids are studied. It is shown that the decay estimate (1+t)−3/4 inL2(R3) holds for solutions of the above equations, provided that the initial data are inL2(R3)∩L1(R3). Since the systems of equations are not rotationally invariant, the perturbation theory for one parameter family of matrices is not useful enough to derive the above result. Therefore, by exploiting an energy method, we show that the decay estimate holds for the solutions of a general class of equations of symmetric hyperbolic-parabolic type, which contains the linearized equations in both electro-magneto-fluid dynamics and magnetohydrodynamics.

Initial Value Problem for Kac's Model of the Boltzmann Equation

Abstract The initial value problem for Kacs one-dimensional model of the Boltzmann equation is investigated. We perform the Fourier transformation to the linearized equation and study the associated eigenvalue problem depending on a parameter ζ. The C ∞ -dependence on ζ of eigenvalues and eigenfunctions are established. The proof makes use of the fact that the complete set of eigenfunctions of the linearized collision operator consists of Hermite functions. The global solutions for the Cauchy problem are constructed by standard arguments.

The Navier-Stokes equation associated with the discrete Boltzmann equation

Publisher Summary This chapter is a summary of the work concerning the Navier-Stokes equation derived from the discrete Boltzmann equation. It considers a model of gas whose molecular velocities are restricted to a set of m constant vectors v 1 , ,v m in IR n . The purpose of this chapter is to study the hydrodynamical equations derived from the discrete Boltzmann equation by applying the Chapman-Enskog method. This chapter shows that the Navier-Stokes equation is transformed into a symmetric system by change of the dependent variable. It is known that the Navier-Stokes equation can be transformed into a coupled system of a symmetric hyperbolic system and a symmetric strongly parabolic system, by changing the dependent variable from w t o u.

The

Il existe une infinite de modeles discrets reguliers avec le groupe de symetrie G=A 5 ×I tels que les espaces associes des invariants de sommation aient 9 dimensions