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Solutions in the large for certain nonlinear parabolic systems. (English) Zbl 0578.35044
The authors prove the global existence of smooth solutions for certain parabolic systems of the form $$(1)\quad u_ t+f(u)_ x=Du_{xx},$$ with initial data $$(2)\quad u(x,0)=u_ 0(x);$$ u and f are vectors and D a constant, diagonazible matrix with positive eigenvalues. It is assumed that f is defined in a ball of radius r centered at a fixed vector $$\bar u,$$ and the existence of a local solution is obtained. These local solutions are then extended globally under the assumption that there is a suitable entropy-entropy flux pair for (1). The corresponding existence theorems are developed.
The results are shown to be applicable to the equations of (nonisentropic) gas dynamics, including a result which shows that for the Navier-Stokes equations for compressible flow, smoothing of initial discontinuities must occur for the velocity and energy, but cannot occur for the density. A brief survey of the literature is also given.
Reviewer: W.G.Engel

##### MSC:
 35K55 Nonlinear parabolic equations 35Q30 Navier-Stokes equations 35A05 General existence and uniqueness theorems (PDE) (MSC2000) 35K45 Initial value problems for second-order parabolic systems 35B65 Smoothness and regularity of solutions to PDEs 76D05 Navier-Stokes equations for incompressible viscous fluids
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