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Publication# A-free rigidity and applications to the compressible Euler system

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 Muller. 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 Szekelyhidi and Wiedemann.

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Euler equations (fluid dynamics)

In fluid dynamics, the Euler equations are a set of quasilinear partial differential equations governing adiabatic and inviscid flow. They are named after Leonhard Euler. In particular, they correspo

Weak solution

In mathematics, a weak solution (also called a generalized solution) to an ordinary or partial differential equation is a function for which the derivatives may not all exist but which is nonetheless

Distribution (mathematics)

Distributions, also known as Schwartz distributions or generalized functions, are objects that generalize the classical notion of functions in mathematical analysis. Distributions make it possible to

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 can not 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. This difference between weak and measure-valued solutions in the compressible case is in contrast with the incompressible situation, where every measure-valued solution can be approximated by weak solutions, as shown by Székelyhidi and Wiedemann.

2017Weak solutions arise naturally in the study of the Navier-Stokes and Euler equations both from an abstract regularity/blow-up perspective and from physical theories of turbulence. This thesis studies the structure and size of singular set of such weak solutions to equations of incompressible fluid dynamics from two opposite directions. First, it aims to single-out new mechanisms which allow to break the typically supercritical scaling of the equations and, in this way, prevent the formation of singularities either globally or locally in spacetime. Second, in the absence of such mechanisms, we seek to quantify how singular (in terms of dimension of the singular set, for instance) the solutions that we are actually able to construct are. This thesis collects four results pointing in the two directions outlined above which have been obtained in several collaborations during the Ph.D. studies:- a global regularity result for the fractional Navier-Stokes equation slightly blow the critical fractional order,- a global well-posedness result for the defocusing wave equation with slightly supercritical power nonlinearity,- an a.e. smoothness / partial regularity result for the supercritical surface quasigeostrophic (SQG) equation,- an estimate (and a discussion of its sharpness) on the dimension of the singular set of wild Hölder continuous solutions of the incompressible Euler equations.All results presented in the thesis have either been published or are submitted for publication.