Existence of Weak Solutions for the Incompressible Euler Equations

TL;DR

This paper demonstrates the existence of infinitely many global weak solutions to the incompressible Euler equations in multiple dimensions with bounded energy, using recent advances in convex integration techniques.

Contribution

It extends the theory by proving the existence of numerous weak solutions for any solenoidal initial data in dimensions two and higher.

Findings

Existence of infinitely many weak solutions in dimensions ≥ 2

Solutions have bounded energy over time

Applicable to any solenoidal initial data in L^2

Abstract

Using a recent result of C. De Lellis and L. Sz\'{e}kelyhidi Jr. we show that, in the case of periodic boundary conditions and for dimension greater or equal 2, there exist infinitely many global weak solutions to the incompressible Euler equations with initial data $v_0$, where $v_0$ may be any solenoidal $L^2$-vectorfield. In addition, the energy of these solutions is bounded in time.