Can every measure-valued solution to the compressible Euler equations be approximated by a sequence of weak solutions? We prove that the answer is negative'/> <![CDATA[<InlineEquation ID='IEq1'> <InlineMediaObject> <ImageObject Color='BlackWhite' FileRef='10231_2016_629_Article_IEq1.gif' Format='GIF' Rendition='HTML' Type='Linedraw'/> </InlineMediaObject> <EquationSource Format='TEX'>$${mathcal {A}}$$</EquationSource> <EquationSource Format='MATHML'> <math xmlns:xlink='http://www.w3.org/1999/xlink'> <mi mathvariant='script'>A</mi> </math> </EquationSource> </InlineEquation>-free rigidity and applications to the compressible Euler system]]>
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$${mathcal {A}}$$ A -free rigidity and applications to the compressible Euler system]]>

机译： $$ { mathcal {a}} $$ a -free刚性和应用于可压缩欧拉系统]]]>

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Abstract Can every measure-valued solution to the compressible Euler equations be approximated by a sequence of weak solutions? We prove that the answer is negative: generalizing a well-known rigidity result of Ball and James to a more general situation, we construct an explicit measure-valued solution for the compressible Euler equations which cannot be generated by a sequence of distributional solutions. We also give an abstract necessary condition for measure-valued solutions to be generated by weak solutions, relying on work of Fonseca and Müller. While a priori it is not unexpected that not every measure-valued solution arises from a sequence of weak solutions, it is noteworthy that this observation in the compressible case is in contrast to the incompressible situation, where every measure-valued solution can be approximated by weak solutions, as shown by Székelyhidi and Wiedemann.
机译：<标题>抽象 ara id =“par1”>可以通过一系列弱解决方案来近似到可压缩欧拉方程的每种测量值的解决方案吗？ 我们证明答案是否定的：概括了球和詹姆斯的知名刚性结果，以更一般的情况，我们构建了一种明确的测量值，用于不能通过一系列分布解决方案产生的可压缩欧拉方程。 我们还为弱解决方案产生了衡量值的措施，依靠Fonseca和Müller的工作，给出了抽象的必要条件。 虽然并非每种测量值都是从一系列弱解决方案中出现的结果，但值得注意的是，在可压缩案例中的这种观察与不可压缩的情况相反，可以近似每种测量值 弱解决方案，如Székelyhidi和Wiedemann所示。

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