Large time behavior of solutions for general quasilinear hyperbolic-parabolic systems of conservation laws.

*(English)*Zbl 0884.35073
Mem. Am. Math. Soc. 599, 120 p. (1997).

The authors discuss the time-asymptotic behavior of solutions to the following system of viscous conservation laws
\[
u_t + f(u)_x = (B(u)u_x)_x,
\]
where \(u=u(x,t)\) is the density of physical quantities, \(f(u)\) is the flux, and \(B(u)\) is the viscosity matrix. For most physical models, the viscosity matrix is not positive definite, and the system is hyperbolic-parabolic and not uniformly parabolic. This implies that the Green’s function of the linearized system may contain Dirac \(\delta\)-functions. First, by the detailed pointwise estimates for the Green’s function of the linearized system and the careful analysis of coupling of nonlinear diffusion waves, and under some smallness conditions on initial data, the authors establish a general theory on the asymptotic behavior of solutions to the above hyperbolic-parabolic system and explicit expressions of the time-asymptotic behavior of solutions. This yields optimal estimates in the integral norms. The results for perturbation decays at the heat kernel rate in all \(L^p\) (\(p\in [1, \infty]\)) norms are also obtained. When the corresponding inviscid system is non-strictly hyperbolic, the time-asymptotic state contains generalized Burgers solutions. Then, at the end of the paper they give an application of the general theory to the compressible Navier-Stokes equations and the equations of magnetohydrodynamics. One-dimensional diffusion waves have played an essential role in the study of the stability properties of viscous conservation laws, rarefactions, and contact discontinuities. T.-P. Liu probably first recongnized the importance of diffusion waves (the \(N\)-waves) for the problems in one space dimension [Nonlinear stability of shock waves for viscous conservation laws, Mem. Am. Math. Soc 328 (1985; Zbl 0617.35058)]. He proved that diffusion waves decay in \(L^p\) for all \(p > 1\), but not in \(L^1\). The present paper improves this result using an explicit and pointwise description of the one-dimensional diffusion waves, a new method to study diffusion waves. This paper will be a source of inspiration not only to those who study viscous conservation laws, but to all who like to see hard problems solved by use of basic techniques.

Reviewer: S.Jiang (Beijing)

##### MSC:

35K65 | Degenerate parabolic equations |

35L65 | Hyperbolic conservation laws |

35A08 | Fundamental solutions to PDEs |

76W05 | Magnetohydrodynamics and electrohydrodynamics |

35B40 | Asymptotic behavior of solutions to PDEs |