Global regularity for a logarithmically supercritical hyperdissipative Navier–Stokes equation

Abstract

Let [math] . We consider the global Cauchy problem for the generalized Navier–Stokes system\n¶\n<math display="block">\n<mrow>\n<msub>\n<mrow>\n<mi>∂</mi>\n</mrow>\n<mrow>\n<mi>t</mi>\n</mrow>\n</msub>\n<mi>u</mi>\n<mo class="MathClass-bin">+</mo>\n<mrow>\n<mo class="MathClass-open">(</mo>\n<mrow>\n<mi>u</mi>\n<mo class="MathClass-bin">⋅</mo>\n<mo class="MathClass-op">∇</mo>\n</mrow>\n<mo class="MathClass-close">)</mo>\n</mrow>\n<mi>u</mi>\n<mo class="MathClass-rel">=</mo>\n<mo class="MathClass-bin">−</mo>\n<msup>\n<mrow>\n<mi>D</mi>\n</mrow>\n<mrow>\n<mn>2</mn>\n</mrow>\n</msup>\n<mi>u</mi>\n<mo class="MathClass-bin">−</mo>\n<mo class="MathClass-op">∇</mo>\n<mi>p</mi>\n<mo class="MathClass-punc">,</mo>\n<mspace class="quad" width="1em"/>\n<mo class="MathClass-op">∇</mo>\n<mo class="MathClass-bin">⋅</mo>\n<mi>u</mi>\n<mo class="MathClass-rel">=</mo>\n<mn>0</mn>\n<mo class="MathClass-punc">,</mo>\n<mspace class="quad" width="1em"/>\n<mi>u</mi>\n<mrow>\n<mo class="MathClass-open">(</mo>\n<mrow>\n<mn>0</mn>\n<mo class="MathClass-punc">,</mo>\n<mi>x</mi>\n</mrow>\n<mo class="MathClass-close">)</mo>\n</mrow>\n<mo class="MathClass-rel">=</mo>\n<msub>\n<mrow>\n<mi>u</mi>\n</mrow>\n<mrow>\n<mn>0</mn>\n</mrow>\n</msub>\n<mrow>\n<mo class="MathClass-open">(</mo>\n<mrow>\n<mi>x</mi>\n</mrow>\n<mo class="MathClass-close">)</mo>\n</mrow>\n</mrow>\n</math>\n¶ for [math] and [math] , where [math] is smooth and divergence free, and [math] is a Fourier multiplier whose symbol [math] is nonnegative; the case [math] is essentially Navier–Stokes. It is folklore that one has global regularity in the critical and subcritical hyperdissipation regimes [math] for [math] . We improve this slightly by establishing global regularity under the slightly weaker condition that [math] for all sufficiently large [math] and some nondecreasing function [math] such that [math] . In particular, the results apply for the logarithmically supercritical dissipation [math] .