Lack of uniqueness for weak solutions of the incompressible porous media equation

TL;DR

This paper demonstrates that weak solutions to the 2-D incompressible porous media equation are not unique in the space of essentially bounded functions, using a convex integration approach.

Contribution

It establishes non-uniqueness of weak solutions in $L^ abla$ space for the 2-D incompressible porous media equation, extending previous understanding.

Findings

Weak solutions in $L^ abla$ are not unique.

Convex integration method proves non-uniqueness.

Results impact the theory of porous media equations.

Abstract

In this work we consider weak solutions of the incompressible 2-D porous media equation. By using the approach of De Lellis-Sz\'ekelyhidi we prove non-uniqueness for solutions in $L^\infty$ in space and time.