Global regularity for solutions of the Navier–Stokes equation sufficiently close to being eigenfunctions of the Laplacian

Evan Miller

Type: Article

Publication Date: 2021-05-19

Citations: 1

DOI: https://doi.org/10.1090/bproc/62

Abstract

In this paper, we will prove a new, scale critical regularity criterion for solutions of the Navier–Stokes equation that are sufficiently close to being eigenfunctions of the Laplacian. This estimate improves previous regularity criteria requiring control on the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="ModifyingAbove upper H With dot Superscript alpha"> <mml:semantics> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mover> <mml:mi>H</mml:mi> <mml:mo>˙</mml:mo> </mml:mover> </mml:mrow> <mml:mi>α</mml:mi> </mml:msup> <mml:annotation encoding="application/x-tex">\dot {H}^\alpha</mml:annotation> </mml:semantics> </mml:math> </inline-formula> norm of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="u comma"> <mml:semantics> <mml:mrow> <mml:mi>u</mml:mi> <mml:mo>,</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">u,</mml:annotation> </mml:semantics> </mml:math> </inline-formula> with <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="2 less-than-or-equal-to alpha greater-than five halves comma"> <mml:semantics> <mml:mrow> <mml:mn>2</mml:mn> <mml:mo>≤</mml:mo> <mml:mi>α</mml:mi> <mml:mo>></mml:mo> <mml:mfrac> <mml:mn>5</mml:mn> <mml:mn>2</mml:mn> </mml:mfrac> <mml:mo>,</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">2\leq \alpha >\frac {5}{2},</mml:annotation> </mml:semantics> </mml:math> </inline-formula> to a regularity criterion requiring control on the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="ModifyingAbove upper H With dot Superscript alpha"> <mml:semantics> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mover> <mml:mi>H</mml:mi> <mml:mo>˙</mml:mo> </mml:mover> </mml:mrow> <mml:mi>α</mml:mi> </mml:msup> <mml:annotation encoding="application/x-tex">\dot {H}^\alpha</mml:annotation> </mml:semantics> </mml:math> </inline-formula> norm multiplied by the deficit in the interpolation inequality for the embedding of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="ModifyingAbove upper H With dot Superscript alpha minus 2 Baseline intersection ModifyingAbove upper H With dot Superscript alpha Baseline right-arrow with hook ModifyingAbove upper H With dot Superscript alpha minus 1 Baseline period"> <mml:semantics> <mml:mrow> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mover> <mml:mi>H</mml:mi> <mml:mo>˙</mml:mo> </mml:mover> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>α</mml:mi> <mml:mo>−</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:msup> <mml:mo>∩</mml:mo> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mover> <mml:mi>H</mml:mi> <mml:mo>˙</mml:mo> </mml:mover> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>α</mml:mi> </mml:mrow> </mml:msup> <mml:mo stretchy="false">↪</mml:mo> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mover> <mml:mi>H</mml:mi> <mml:mo>˙</mml:mo> </mml:mover> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>α</mml:mi> <mml:mo>−</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> <mml:mo>.</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\dot {H}^{\alpha -2}\cap \dot {H}^{\alpha } \hookrightarrow \dot {H}^{\alpha -1}.</mml:annotation> </mml:semantics> </mml:math> </inline-formula> This regularity criterion suggests, at least heuristically, the possibility of some relationship between potential blowup solutions of the Navier–Stokes equation and the Kolmogorov-Obhukov spectrum in the theory of turbulence.

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Proceedings of the American Mathematical Society Series B - View - PDF
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