Existence and non-uniqueness of global weak solutions to inviscid primitive and Boussinesq equations

TL;DR

This paper demonstrates the existence of infinitely many global weak solutions to the inviscid primitive and Boussinesq equations in three dimensions, using convex integration methods, highlighting non-uniqueness in these fluid dynamics models.

Contribution

It applies convex integration to establish the existence of infinitely many solutions and introduces dissipative solutions for these equations, advancing understanding of their solution space.

Findings

Existence of infinitely many global weak solutions for general initial data.

Existence of infinitely many dissipative solutions for certain initial data.

Application of convex integration methods to primitive and Boussinesq equations.

Abstract

We consider the initial value problem for the inviscid Primitive and Boussinesq equations in three spatial dimensions. We recast both systems as an abstract Euler-type system and apply the methods of convex integration of De Lellis and Sz\'ekelyhidi to show the existence of infinitely many global weak solutions of the studied equations for general initial data. We also introduce an appropriate notion of dissipative solutions and show the existence of suitable initial data which generate infinitely many dissipative solutions.