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Concept# Weak solution

Summary

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 deemed to satisfy the equation in some precisely defined sense. There are many different definitions of weak solution, appropriate for different classes of equations. One of the most important is based on the notion of distributions.
Avoiding the language of distributions, one starts with a differential equation and rewrites it in such a way that no derivatives of the solution of the equation show up (the new form is called the weak formulation, and the solutions to it are called weak solutions). Somewhat surprisingly, a differential equation may have solutions which are not differentiable; and the weak formulation allows one to find such solutions.
Weak solutions are important because many differential equations encountered in modelling real-world phenomena do not admit of sufficient

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Weak 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.

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.

2017Can 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.