\(L^ p\) asymptotic behavior of solutions to hyperbolic-parabolic systems of conservation laws (Q1924911)

We study the large time behavior in \(L^p\) of solutions to a general \(n\times n\) hyperbolic-parabolic system of conservation laws. Under physical assumptions on the dissipativeness of the system, we show that the solution to the Cauchy problem is approximated by the solution to a uniformly parabolic system when the initial data are smooth and small. In turn the solution of the hyperbolic-parabolic system is approximated by a superposition of linear and nonlinear diffusion waves, self-similar solutions of the heat equation and the Burgers equation. The \(L^p\) \((1\leq p\leq\infty)\) optimal convergence rates are obtained. The approach is based on the pointwise estimates in the physical space of the Green's function of the linearized system. Our result applies to the compressible Navier-Stokes equations.