A proof of Vishik’s nonuniqueness theorem for the forced 2D Euler equation

Abstract We provide an alternative proof of Vishik’s nonuniqueness theorem for the forced 2D Euler equation in the vorticity class <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msup> <m:mi>L</m:mi> <m:mn>1</m:mn> </m:msup> <m:mo>∩</m:mo> <m:msup> <m:mi>L</m:mi> <m:mi>p</m:mi> </m:msup> </m:mrow> </m:math> L^{1}\cap L^{p} with <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mn>2</m:mn> <m:mo>&lt;</m:mo> <m:mi>p</m:mi> <m:mo>&lt;</m:mo> <m:mi mathvariant="normal">∞</m:mi> </m:mrow> </m:math> 2&lt;p&lt;\infty . The main variation is the construction of a smooth and compactly supported unstable vortex, which is split into two steps: firstly, we construct a piecewise constant unstable vortex, and secondly, we find a regularization through a fixed point argument. This structure of the unstable vortex yields a simplification of the other parts of Vishik’s proof.