Weak and measure-valued solutions to the Euler and ideal MHD equations

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Author(s):	Leonardo Epiphanio Galvão Total Authors: 1
Document type:	Master's Dissertation
Press:	Campinas, SP.
Institution:	Universidade Estadual de Campinas (UNICAMP). Instituto de Matemática, Estatística e Computação Científica
Defense date:	2021-03-30
Examining board members:	Anne Caroline Bronzi; Helena Judith Nussenzveig Lopes; Emil Wiedemann
Advisor:	Anne Caroline Bronzi
Abstract
In this thesis we are concerned with weaker notions of solutions to some partial differential equations which describe phenomena in the field of fluid dynamics, specifically the incompressible Euler equations for ideal fluids and the ideal Magneto-hydrodynamic system of equations governing the motion of ideal incompressible fluids that are electrically conductive. The main objects of study are the method of convex integration to obtain non-uniqueness of weak solutions to both PDE systems, as well as the framework for treating parametrised measures as very weak solutions to these equations. By adapting a concept of measure-valued solution from the Euler to the ideal MHD system, we are able to obtain a global existence result for the full $3D$ system, and a weak-strong uniqueness result for solutions satisfying a planar symmetry condition (AU)

FAPESP's process:	19/05841-0 - On measure-valued solutions of hydrodynamic models
Grantee:	Leonardo Epiphanio Galvão
Support Opportunities:	Scholarships in Brazil - Master