Global Weak Solutions to the Navier-Stokes Equations for a 1D Viscous Polytropic Ideal Gas

S. Jiang and P. Zhang

Abstract: We prove the existence of global weak solutions to the Navier-Stokes equations for a one-dimensional viscous polytropic ideal gas. We require only that the initial density is in $L^\infty\cap L^2_{loc}$ with positive infimum, the initial velocity is in $L^2$ and the initial temperature is in a larger space than $L^2$ with positive infimum. The initial density and the initial velocity may have differing constant states at $x=\pm\infty$. In particular, piesewise constant data with arbitrary large jump discontinuities are included. Our results show that neither vacuum states nor concentration states can form and the temperature remains positive in finite time.

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Author(s):
S. Jiang, <jiang@mail.iapcm.ac.cn>
P. Zhang, <zp@math03.math.ac.cn>
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