Weak–Strong Uniqueness for the Isentropic Compressible Navier–Stokes System

By Pierre Germain. 2008

163 citations recorded by the source corpus.

We prove weak–strong uniqueness results for the isentropic compressible Navier–Stokes system on the torus. In other words, we give conditions on a weak solution, such as the ones built up by Lions (Compressible Models, Oxford Science, Oxford, 1998), so that it is unique. It is of fundamental importance since uniqueness of these solutions is not known in general. We present two different methods, one using relative entropy, the other one using an improved Gronwall inequality due to the author; these two approaches yield complementary results. Known weak–strong uniqueness results are improved and classical uniqueness results for this equation follow naturally.

Subfields: Applied Mathematics, Control and Systems Engineering, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/8fd17c85604b08151b34622f549a677270990349

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In Applied Mathematics

In Control and Systems Engineering

In Mathematical Physics

Citations in the corpus

Existence of global weak solutions for 3D degenerate compressible Navier–Stokes equations (2015, background): “[14] P. Germain, Weak-strong uniqueness for the isentropic compressible Navier-Stokes system.”
A Global Existence Result for the Compressible Navier–Stokes Equations in the Critical Lp Framework (2010, background): “Let us point out that the problem of uniqueness for those solutions is completely open (see [14] for recent results concerning the weak-strong uniqueness issue).”
Weak–Strong Uniqueness Property for the Full Navier–Stokes–Fourier System (2011, result): “Germain [17] proves weak-strong uniqueness for the isentropic Navier-Stokes system, unfortunately, in the class of “more regular” weak solutions which existence is not known, see also Desjardin [6] for the previous results in this direction.”
Dissipative measure-valued solutions to the compressible Navier–Stokes system (2015, background): “…(cid:12)(cid:12)(cid:12) where ̺ , u is a weak solution and r and U are arbitrary “test” functions mimicking the basic properties of ̺ , u , speciﬁcally, r is positive and U satisﬁes the relevant boundary conditions, see Feireisl et al. [10], Germain [15], Mellet and Vasseur [22], among others.”
Relative Entropies, Suitable Weak Solutions, and Weak-Strong Uniqueness for the Compressible Navier–Stokes System (2011, background): “We provide an ultimate answer to the weak-strong uniqueness problem intimately related to the fundamental questions of the well-posedness for the compressible Navier-Stokes equations addressed by several authors, Desjardin [4], Germain [10], Hoff [12], [13], among others.”