Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

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Global weak solutions to the Navier-Stokes equations for a 1D viscous polytropic ideal gas


Authors: Song Jiang and Ping Zhang
Journal: Quart. Appl. Math. 61 (2003), 435-449
MSC: Primary 76N10; Secondary 35L65, 35Q30
DOI: https://doi.org/10.1090/qam/1999830
MathSciNet review: MR1999830
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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_{loc}^2$ with positive infimum, the initial velocity is in $ L_{loc}^2$, and the initial temperature is in $ L_{loc}^1$ with positive infimum. The initial density and the initial velocity may have differing constant states at $ x = \pm \infty $. In particular, piecewise 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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DOI: https://doi.org/10.1090/qam/1999830
Article copyright: © Copyright 2003 American Mathematical Society


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