Regularity in time along the coarse scale flow for the incompressible Euler equations

Abstract

One of the most remarkable features of known nonstationary solutions to the incompressible Euler equations is the phenomenon known as the Taylor hypothesis, which predicts that fine scale features of the flow are advected by the mean velocity. In this work, we develop an extensive theory of time regularity for Euler weak solutions in any dimension based on quantitative realizations of this idea. Our work provides the key estimates for showing that the particle trajectories of any Euler flow that is <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper C Superscript alpha"> <mml:semantics> <mml:msup> <mml:mi>C</mml:mi> <mml:mi>α</mml:mi> </mml:msup> <mml:annotation encoding="application/x-tex">C^\alpha</mml:annotation> </mml:semantics> </mml:math> </inline-formula> in the spatial variables uniformly in time are of class <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper C Superscript 1 slash left-parenthesis 1 minus alpha right-parenthesis"> <mml:semantics> <mml:msup> <mml:mi>C</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>1</mml:mn> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>/</mml:mo> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mn>1</mml:mn> <mml:mo>−</mml:mo> <mml:mi>α</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:msup> <mml:annotation encoding="application/x-tex">C^{1/(1-\alpha )}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> when <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="1 slash left-parenthesis 1 minus alpha right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mn>1</mml:mn> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>/</mml:mo> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mn>1</mml:mn> <mml:mo>−</mml:mo> <mml:mi>α</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">1/(1-\alpha )</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is not an integer, whether or not the trajectories or solutions are unique. In particular, we prove the smoothness of trajectories in borderline spaces such as <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="v element-of upper C Superscript 1"> <mml:semantics> <mml:mrow> <mml:mi>v</mml:mi> <mml:mo>∈</mml:mo> <mml:msup> <mml:mi>C</mml:mi> <mml:mn>1</mml:mn> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">v \in C^1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> or bounded vorticity in any dimension. An essential point is the existence and improved regularity of advective derivatives of high order.