No abstract

No abstract

No abstract

No abstract

This paper is concerned with the asymptotic behavior toward the rarefaction waves of the solution of a one-dimensional model system associated with compressible viscous gas. If the initial data ate suitably close to a constant state and their asymptotic values at x = + ~ are chosen so that the Riemann problem for the corresponding hyperbolic system admits the weak rarefaction waves, then the solution is proved to tend toward the rarefaction waves as t~ + oo. The proofis given by an elementary L 2 energy method.

Travelling wave solutions with shock profile for a one-dimensional model system associated with compressible viscous gas are investigated in terms of asymptotic stability. The travelling wave solution is proved to be asymptotically stable, provided the initial disturbance is suitably small and of zero constant component. The proof is given by the elemental L 2 energy method.

The asymptotic stability of traveling wave solutions with shock profile is investigated for several systems in gas dynamics. 1) The solution of a scalar conservation law with viscosity approaches the traveling wave solution at the rate t~γ (for some γ > 0) as ί-» oo, provided that the initial disturbance is small and of integral zero, and in addition decays at an algebraic rate for |x| -> oo. 2) The traveling wave solution with Nishida and Smoller's condition of the system of a viscous heat-conductive ideal gas is asymptotically stable, provided the initial disturbance is small and of integral zero. 3) The traveling wave solution with weak shock profile of the Broadwell model system of the Boltzmann equation is asymptotically stable, provided the initial disturbance is small and its hydrodynamical moments are of integral zero. Each proof is given by applying an elementary energy method to the integrated system of the conservation form of the original one. The property of integral zero of the initial disturbance plays a crucial role in this procedure. Contents 108 2.2 Reformulation of the problem Ill 2.3 A priori estimate, I 113 2.4 A priori estimate, II 116 98 S. Kawashima and A. Matsumura Section 3. The Broadwell model system 3.1 Traveling wave solution and main theorem 118 3.2 Reformulation of the problem 121 3.3 A priori estimate 123 References 127

No abstract