An elementary proof of existence and uniqueness for the Euler flow in localized Yudovich spaces
Gianluca Crippa, Giorgio Stefani
Abstract
Abstract We revisit Yudovich’s well-posedness result for the 2-dimensional Euler equations for an inviscid incompressible fluid on either a sufficiently regular (not necessarily bounded) open set $$\Omega \subset \mathbb {R}^2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>Ω</mml:mi> <mml:mo>⊂</mml:mo> <mml:msup> <mml:mrow> <mml:mi>R</mml:mi> </mml:mrow> <mml:mn>2</mml:mn> </mml:msup> </mml:mrow> </mml:math> or on the torus $$\Omega =\mathbb {T}^2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>Ω</mml:mi> <mml:mo>=</mml:mo> <mml:msup> <mml:mrow> <mml:mi>T</mml:mi> </mml:mrow> <mml:mn>2</mml:mn> </mml:msup> </mml:mrow> </mml:math> . We construct global-in-time weak solutions with vorticity in $$L^1\cap L^p_{ul}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msup> <mml:mi>L</mml:mi> <mml:mn>1</mml:mn> </mml:msup> <mml:mo>∩</mml:mo> <mml:msubsup> <mml:mi>L</mml:mi> <mml:mrow> <mml:mi>ul</mml:mi> </mml:mrow> <mml:mi>p</mml:mi> </mml:msubsup> </mml:mrow> </mml:math> and in $$L^1\cap Y^\Theta _{ul}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msup> <mml:mi>L</mml:mi> <mml:mn>1</mml:mn> </mml:msup> <mml:mo>∩</mml:mo> <mml:msubsup> <mml:mi>Y</mml:mi> <mml:mrow> <mml:mi>ul</mml:mi> </mml:mrow> <mml:mi>Θ</mml:mi> </mml:msubsup> </mml:mrow> </mml:math> , where $$L^p_{ul}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>L</mml:mi> <mml:mrow> <mml:mi>ul</mml:mi> </mml:mrow> <mml:mi>p</mml:mi> </mml:msubsup> </mml:math> and $$Y^\Theta _{ul}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>Y</mml:mi> <mml:mrow> <mml:mi>ul</mml:mi> </mml:mrow> <mml:mi>Θ</mml:mi> </mml:msubsup> </mml:math> are suitable uniformly-localized versions of the Lebesgue space $$L^p$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>L</mml:mi> <mml:mi>p</mml:mi> </mml:msup> </mml:math> and of the Yudovich space $$Y^\Theta $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>Y</mml:mi> <mml:mi>Θ</mml:mi> </mml:msup> </mml:math> respectively, with no condition at infinity for the growth function $$\Theta $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Θ</mml:mi> </mml:math> . We also provide an explicit modulus of continuity for the velocity depending on the growth function $$\Theta $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Θ</mml:mi> </mml:math> . We prove uniqueness of weak solutions in $$L^1\cap Y^\Theta _{ul}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msup> <mml:mi>L</mml:mi> <mml:mn>1</mml:mn> </mml:msup> <mml:mo>∩</mml:mo> <mml:msubsup> <mml:mi>Y</mml:mi> <mml:mrow> <mml:mi>ul</mml:mi> </mml:mrow> <mml:mi>Θ</mml:mi> </mml:msubsup> </mml:mrow> </mml:math> under the assumption that $$\Theta $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Θ</mml:mi> </mml:math> grows moderately at infinity. In contrast to Yudovich’s energy method, we employ a Lagrangian strategy to show uniqueness. Our entire argument relies on elementary real-variable techniques, with no use of either Sobolev spaces, Calderón–Zygmund theory or Littlewood–Paley decomposition, and actually applies not only to the Biot–Savart law, but also to more general operators whose kernels obey some natural structural assumptions.