Compensated Compactness: Unraveling the Interplay of Algebraic Constraints in PDE Theory

Abstract

Fine patterns, such as oscillations and concentrations of mass, are ubiquitous in nature, from the microstructure of materials to the behaviour of fluids. These patterns can be modelled by partial differential equations (PDEs).
“Compensated compactness” is a powerful framework for understanding fine pattern formation in PDEs by exploiting the interplay between algebraic and PDE restrictions. Algebraic restrictions in this context are constraints on the possible values of a solution to a PDE, which can strongly reduce the formation of arbitrary fine patterns. This means that, even if a PDE does not have an explicit solution, we can still learn a lot about its behaviour by studying the interplay of these restrictions.
While oscillatory behaviour is somewhat well understood, much less is known about the formation of mass concentrations and the shape of their generated singularities. In this talk, I will give a general overview of compensated compactness theory, with a focus on mass concentrations and the possible shape of their limiting singularities. I will also discuss some exciting conjectures that could lead to substantial progress in this area, particularly at the intersection of various subfields of analysis.

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