Initial boundary value problems for quasilinear symmetric hyperbolic systems with characteristic boundary

This paper is devoted to initial boundary value problems for quasi-linear symmetric hyperbolic systems in a domain with characteristic boundary. It extends the theory on linear symmetric hyperbolic systems established by Friedrichs to the nonlinear case. The concept on regular characteristics and dissipative boundary conditions are given for quasilinear hyperbolic systems. Under some assumptions, an existence theorem for such initial boundary value problems is obtained. The theorem can also be applied to the Euler system of compressible flow.

[1]  M. Tsuji.  Regularity of solutions of hyperbolic mixed problems with characteristic boundary , 1972 .

[2]  H. B. Veiga.  On the barotropic motion of compressible perfect fluids , 1981 .

[3]  F. Massey,et al.  Differentiability of solutions to hyperbolic initial-boundary value problems , 1974 .

[4]  Y. Tanaka.  The existence of solutions to the mixed problem for symmetric hyperbolic systems , 2004 .

[5]  P. Secchi.  Characteristic Symmetric Hyperbolic Systems with Dissipation: Global Existence and Asymptotics , 1997 .

[6]  H. B. Veiga.  Perturbation theory and well-posedness in Hadamard's sense of hyperbolic initial-boundary value problems , 1994 .

[7]  S. Kawashima,et al.  Mixed problems for quasi-linear symmetric hyperbolic systems , 1987 .

[8]  J. Rauch.  Boundary value problems with nonuniformal characteristic boundary , 1994 .

[9]  Chen Shuxing.  On reflection of multidimensional shock front , 1989 .

[10]  Kurt Friedrichs,et al.  Symmetric positive linear differential equations , 1958 .

[11]  H. B. Veiga,et al.  Perturbation theorems for linear hyperbolic mixed problems and applications to the compressible euler equations , 1993 .

[12]  S. Alinhac,et al.  Existence d'ondes de rarefaction pour des systems quasi‐lineaires hyperboliques multidimensionnels , 1989 .

[13]  R. Phillips,et al.  Local boundary conditions for dissipative symmetric linear differential operators , 1960 .

[14]  Becca Thomases,et al.  A DECAY THEOREM FOR SOME SYMMETRIC HYPERBOLIC SYSTEMS , 2006 .

[15]  Paolo Secchi,et al.  Well-posedness for a mixed problem for the equations of ideal Magneto-Hydrodynamics , 1995 .

[16]  On nonviscous compressible fluids in a time-dependent domain , 1992 .

[17]  S. Schochet.  The compressible Euler equations in a bounded domain: Existence of solutions and the incompressible limit , 1986 .

[18]  D. Ebin.  The initial boundary value problem for sub‐sonic fluid motion , 1979 .

[19]  T. Yanagisawa,et al.  The initial boundary value problem for linear symmetric hyperbolic systems with boundary characteristic of constant multiplicity , 1995 .

[20]  The initial boundary value problem for inviscid barotropic fluid motion , 1981 .

[21]  Jeffrey Rauch,et al.  Symmetric positive systems with boundary characteristic of constant multiplicity , 1985 .

[22]  Paolo Secchi,et al.  Well-posedness of characteristic symmetric hyperbolic systems , 1996 .

[23]  Toshio Ohkubo.  Well posedness for quasi-linear hyperbolic mixed problems with characteristic boundary , 1989 .

[24]  A. Matsumura,et al.  The fixed boundary value problems for the equations of ideal Magneto-Hydrodynamics with a perfectly conducting wall condition , 1991 .

[25]  Paolo Secchi,et al.  Linear symmetric hyperbolic systems with characteristic boundary , 1995 .

[26]  Shuxing Chen.  Existence of local solution to supersonic flow past a three-dimensional wing , 1992 .