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The nonlinear initial-boundary value problem and the existence of multidimensional shock wave for quasilinear hyperbolic-parabolic coupled systems. (English) Zbl 0678.35071

Consider a quasilinear hyperbolic-parabolic coupled system \[ (1)\quad u_ t+P_{ij}(u_ x,u,v)u_{x_ ix_ j}+A_ j(u_ x,u,v)v_{x_ j}+f_ 1(u_ x,u,v)=0, \]
\[ v_ t+Q_ j(u,v)u_{x_ j}+B_ j(u,v)u_{x_ j}+f_ 2(u,v)=0 \] where \(\partial /\partial t+P_{ij}\partial_{x_ ix_ j}\) is two-order Petrovsky parabolic operator and \(\partial /\partial t+Q_ j\partial_{x_ j}\) is the first order Kreiss hyperbolic operator. The Kreiss’ hyperbolic operator is defined by A. Majda [Mem. Am. Math. Soc. 275, 95 p. (1983; Zbl 0506.76075)]. The author discusses the general nonlinear initial-boundary value problem of (1) whose boundary conditions on \(\partial \Omega \times R^ 1_+\) are \[ J_ 1(u_ x,w,x,t)=0,\quad J_ 2(w,x,t)=0,\quad J_ 3(w,x,t)=0 \] and whose zero initial condition \(w(x,0)=0\) and proves the existence and the uniquess of the solution of the problem under certain assumptions. Secondly, the author considers the shock wave solution, i.e. the Cauchy problem with discontinuous initial data: to find a surface S(t) and shock wave solution \(w^{\pm}(x,t)\) satisfying (1) on two sides \(\Omega \pm (t)\) of S(t) such that on S(t) the Rankine- Hugoniot condition and p supplementary conditions hold: \[ n_ t(F_ 0(w^+)-F_ 0(w^-))+\sum^{N}_{1}n_ j(F_ j(u^+_ x,w^+)- F_ j(u^-_ x,w^-), \]
\[ n_ t(G_ 0(v^+)-G_ 0(v^- ))+\sum^{N}_{1}n_ j(G_ j(w^+)-G_ j(w))=0, \]
\[ \phi (u^+,u^-)=0;\quad \psi (u^+_ x,u^-_ x,w^+,w^-)=0, \] where \((n_ 1,n_ 2,...,n_ N)\) is the normal vector of S(t), \(S_ 0\) is a smooth surface in \(R^ n\), and \(w_ 0^{\pm}(x)\) are smooth initial values on two sides \(\Omega_{\pm}\) of \(S_ 0\) \((w^+_ 0\pm w^-_ 0)\). The existence and uniqueness theorem is also given.
Reviewer: J.Tian

MSC:

35M99 Partial differential equations of mixed type and mixed-type systems of partial differential equations
35K55 Nonlinear parabolic equations
35L60 First-order nonlinear hyperbolic equations
35K20 Initial-boundary value problems for second-order parabolic equations
35L50 Initial-boundary value problems for first-order hyperbolic systems
35L67 Shocks and singularities for hyperbolic equations
35A05 General existence and uniqueness theorems (PDE) (MSC2000)
35K15 Initial value problems for second-order parabolic equations
35L45 Initial value problems for first-order hyperbolic systems

Citations:

Zbl 0506.76075
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