Weak-strong uniqueness for the compressible Navier-Stokes equations with a hard-sphere pressure law
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We consider the Navier-Stokes equations with a pressure function satisfying a hard-sphere law. That means the pressure, as a function of the density, becomes infinite when the density approaches a finite critical value. Under some structural constraints imposed on the pressure law, we show a weak-strong uniqueness principle in periodic spatial domains. The method is based on a modified relative entropy inequality for the system. The main difficulty is that the pressure potential associated with the internal energy of the system is largely dominated by the pressure itself in the area close to the critical density. As a result, several terms appearing in the relative energy inequality cannot be controlled by the total energy.
KeywordsNavier-Stokes equations hard-sphere pressure weak-strong uniqueness
The work of Eduard Feireisl and Yong Lu leading to these results has received funding from the European Research Council under the European Union’s Seventh Framework Programme (Grant No. FP7/2007-2013) and European Research Council (ERC) Grant Agreement (Grant No. 320078). The Institute of Mathematics of the Academy of Sciences of the Czech Republic was supported by Rozvoj Výzkumné Organizace (RVO) (Grant No. 67985840).
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