Unbounded Yudovich Solutions of the Euler Equations

Abstract In this article, we will study unbounded solutions of the 2D incompressible Euler equations. One of the motivating factors for this is that the usual functional framework for the Euler equations (for example based on finite energy conditions, such as $$L^2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>L</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:math> ) does not respect some of the symmetries of the problem, such as Galileo invariance. Our main result, global existence and uniqueness of solutions for initial data with square-root growth $$O(|x|^{\frac{1}{2} - \varepsilon })$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>O</mml:mi> <mml:mo>(</mml:mo> <mml:mo>|</mml:mo> <mml:mi>x</mml:mi> <mml:msup> <mml:mo>|</mml:mo> <mml:mrow> <mml:mfrac> <mml:mn>1</mml:mn> <mml:mn>2</mml:mn> </mml:mfrac> <mml:mo>-</mml:mo> <mml:mi>ε</mml:mi> </mml:mrow> </mml:msup> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> and bounded vorticity, is based on two key ingredients. First, an integral decomposition of the pressure, and secondly examining local energy balance leading to solution estimates in local Morrey type spaces. We also prove continuity of the initial data to solution map by a substantial adaptation of Yudovich’s uniqueness argument.