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On the normal form of the symmetric hyperbolic-parabolic systems associated with the conservation laws. (English) Zbl 0699.35171

The authors study a second order nonlinear system of the form \[ (1)\quad w_ t+\sum^{n}_{j=1}f^ j(w)_{x_ j}=\sum^{n}_{i,j=1}\{G^{ij}(w)w_{x_ j}\}_{x_ i}, \] where \(f^ j\), \(j=1,...,n\), w are m-dimensional vectors and \(G^{ij}(w)\) are (m\(\times m)\) matrices. In case that \(G^{ij}\equiv 0\), (1) is a hyperbolic system of conservation laws. The case \(G^{ij}\neq 0\) is important, since the equations of viscous compressible fuids, the equations of hydrodynamics and some other problems of mathematical physics can be reduced to systems of type (1).
The main problem treated in the work is the reduction of (1) to a coupled system of a symmetric hyperbolic system and a symmetric strongly parabolic system. This reduction enables one to apply the results on the existence and uniqueness of solutions of coupled hyperbolic-parabolic systems obtained by the same authors.
A suitable reduction of (1) based on the substitution \(w=w(v)\) leads to a system of type \[ (2)\quad A^ 0(v)v_ t+\sum^{n}_{j=1}A^ j(v)v_{x_ j}=\sum^{n}_{i,j=1}B^{ij}(v)v_{x_ i}v_{x_ j}+g(v,D_ xv), \] where \(A^ 0\) is real symmetric and positive definite, \(A^ j(v)\), \(B^{ij}(v)\) are real symmetric, \(B^{ij}=B^{ji}\) and moreover the matrix \[ (3)\quad B(v,\omega)=\sum^{n}_{i,j=1}B^{ij}(v)\omega_ i\omega_ j \] is real symmetric and non-negative. For (2) the notions of symmetrizability and normal form are introduced.
The main result states that the assumptions that (1) is symmetrizable and that the matrix B(v,\(\omega)\) in (3) has a constant rank independent of v, \(\omega\) implies the existence of a transformation \(w\to v\), such that the reduced system (2) for v is of normal form.
An interesting application of the result to the system of compressible fluids and to Navier-Stokes equations are given. A relation between the symmetrizability of (1) and the existence of an entropy function is also discussed.
Reviewer: V.Georgiev

MSC:

35L65 Hyperbolic conservation laws
35M99 Partial differential equations of mixed type and mixed-type systems of partial differential equations
35G20 Nonlinear higher-order PDEs
35Q30 Navier-Stokes equations
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