Quantitative bounds for critically bounded solutions to the Navier-Stokes equations

Abstract

We revisit the regularity theory of Escauriaza, Seregin, and Šverák for solutions to the three-dimensional Navier-Stokes equations which are uniformly bounded in the critical <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L Subscript x Superscript 3 Baseline left-parenthesis double-struck upper R cubed right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msubsup> <mml:mi>L</mml:mi> <mml:mi>x</mml:mi> <mml:mn>3</mml:mn> </mml:msubsup> <mml:mo stretchy="false">(</mml:mo> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> <mml:mn>3</mml:mn> </mml:msup> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">L^3_x(\mathbb {R}^3)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> norm. By replacing all invocations of compactness methods in these arguments with quantitative substitutes, and similarly replacing unique continuation and backwards uniqueness estimates by their corresponding Carleman inequalities, we obtain quantitative bounds for higher regularity norms of these solutions in terms of the critical <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L Subscript x Superscript 3"> <mml:semantics> <mml:msubsup> <mml:mi>L</mml:mi> <mml:mi>x</mml:mi> <mml:mn>3</mml:mn> </mml:msubsup> <mml:annotation encoding="application/x-tex">L^3_x</mml:annotation> </mml:semantics> </mml:math> </inline-formula> bound (with a dependence that is triple exponential in nature). In particular, we show that as one approaches a finite blowup time <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper T Subscript asterisk"> <mml:semantics> <mml:msub> <mml:mi>T</mml:mi> <mml:mo>∗<!-- ∗ --></mml:mo> </mml:msub> <mml:annotation encoding="application/x-tex">T_*</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, the critical <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L Subscript x Superscript 3"> <mml:semantics> <mml:msubsup> <mml:mi>L</mml:mi> <mml:mi>x</mml:mi> <mml:mn>3</mml:mn> </mml:msubsup> <mml:annotation encoding="application/x-tex">L^3_x</mml:annotation> </mml:semantics> </mml:math> </inline-formula> norm must blow up at a rate <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis log log log StartFraction 1 Over upper T Subscript asterisk Baseline minus t EndFraction right-parenthesis Superscript c"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>log</mml:mi> <mml:mo>⁡<!-- ⁡ --></mml:mo> <mml:mi>log</mml:mi> <mml:mo>⁡<!-- ⁡ --></mml:mo> <mml:mi>log</mml:mi> <mml:mo>⁡<!-- ⁡ --></mml:mo> <mml:mfrac> <mml:mn>1</mml:mn> <mml:mrow> <mml:msub> <mml:mi>T</mml:mi> <mml:mo>∗<!-- ∗ --></mml:mo> </mml:msub> <mml:mo>−<!-- − --></mml:mo> <mml:mi>t</mml:mi> </mml:mrow> </mml:mfrac> <mml:msup> <mml:mo stretchy="false">)</mml:mo> <mml:mi>c</mml:mi> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">(\log \log \log \frac {1}{T_*-t})^c</mml:annotation> </mml:semantics> </mml:math> </inline-formula> or faster for an infinite sequence of times approaching <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper T Subscript asterisk"> <mml:semantics> <mml:msub> <mml:mi>T</mml:mi> <mml:mo>∗<!-- ∗ --></mml:mo> </mml:msub> <mml:annotation encoding="application/x-tex">T_*</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and some absolute constant <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="c greater-than 0"> <mml:semantics> <mml:mrow> <mml:mi>c</mml:mi> <mml:mo>&gt;</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">c&gt;0</mml:annotation> </mml:semantics> </mml:math> </inline-formula>.

Locations

• Proceedings of symposia in pure mathematics - View
• arXiv (Cornell University) - View - PDF
• Proceedings of symposia in pure mathematics - View
• arXiv (Cornell University) - View - PDF
• Proceedings of symposia in pure mathematics - View
• arXiv (Cornell University) - View - PDF