Solutions in the large for certain nonlinear parabolic systems.

*(English)*Zbl 0578.35044The 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.

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 |

##### Keywords:

global existence; smooth solutions; local solution; entropy-entropy flux; gas dynamics; Navier-Stokes equations
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\textit{D. Hoff} and \textit{J. Smoller}, Ann. Inst. Henri Poincaré, Anal. Non Linéaire 2, 213--235 (1985; Zbl 0578.35044)

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