\Phi_{A}(X_{jk})=AX_{jk}-X_{jk}A=A\psi_{j}\psi_{k}^{*}-\psi_{j}\psi_{k}^{*}A =(\mu_{j}-\mu_{k})\psi_{j}\psi_{k}^{*}=(\mu_{j}-\mu_{k})X_{jk} .

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

Re~;u le 9 d› 1985The Cauchy problem for the compressible Euler equation is discussed with compactly supported initials. To establish the local existence of classical solutions by the aid of the theory of quasilinear symmetric hyperbolic systems, a new symmetrization is introduced which works for initials having compact support or vanishing at infinity. It is further shown that as far as the classical solution is concerned, its support does not change, and that the life span is finite for any solution except for the trivial zero solution.

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We study the decay property of the dissipative Timoshenko system in the one-dimensional whole space. We derive the L2decay estimates of solutions in a general situation and observe that this decay structure is of the regularity-loss type. Also, we give a refinement of these decay estimates for some special initial data. Moreover, under enough regularity assumption on the initial data, we show that the solution approaches the linear diffusion wave expressed in terms of the heat kernels as time tends to infinity. The proof is based on the detailed pointwise estimates of solutions in the Fourier space.