Asymptotic behavior of solutions and stability of nonlinear waves for equations of gas motion

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We studied asymptotic behavior of solutions and stability of nonlinear waves for equations of gas motion with dissipative structure.1.We developed the energy method in the Sobolev space W^{1,p} for n-dimensional scalar viscous conservation law and derived the optimal decay estimates in W^{1,p}. The method was also applied to the stability problem for rarefaction waves and stationary waves.2.We introduced the notion of entropy for n-dimensional hyperbolic conservation laws with relaxation and developed the Chapman-Enskog theory. Moreover, we proved the global existence and optimal decay of solutions in a L^2 type Sobolev space.3.For the compressible Navier-Stokes equation in the n-dimensional half space, we proved the asymptotic stability of planar stationary waves. To develop the theory in the Sobolev space of order [n/2]+1, we need additional considerations for local existence results.4.For the dissipative Timoshenko system, we derived qualitative decay estimates of solutions by applying the energy method in Fourier space. We found that the dissipative structure is so weak in high frequency region and it causes the regularity loss in the decay estimates.5.For dissipative wave equation with a nonlinear convection term, we proved the global existence and optimal decay of solutions in L^p. Moreover, we showed that the solution approaches the nonlinear diffusion waves given in terms of the self similar solutions of the Burgers equation. Derivation of detailed pointwise estimates of the fundamental solutions is crucial in the proof.