Global regularity for a logarithmically supercritical hyperdissipative Navier–Stokes equation

Abstract

Let d ≥ 3. We consider the global Cauchy problem for the generalized Navier-Stokes systemfor u : ‫ޒ‬ + × ‫ޒ‬ d → ‫ޒ‬ d and p : ‫ޒ‬ + × ‫ޒ‬ d → ‫,ޒ‬ where u 0 : ‫ޒ‬ d → ‫ޒ‬ d is smooth and divergence free, and D is a Fourier multiplier whose symbol m : ‫ޒ‬ d → ‫ޒ‬ + is nonnegative; the case m(ξ ) = |ξ | is essentially Navier-Stokes.It is folklore that one has global regularity in the critical and subcritical hyperdissipation regimes m(ξ ) = |ξ | α for α ≥ (d + 2)/4.We improve this slightly by establishing global regularity under the slightly weaker condition that m(ξ ) ≥ |ξ | (d+2)/4 /g(|ξ |) for all sufficiently large ξ and some nondecreasing function g : ‫ޒ‬ + → ‫ޒ‬ + such that ∞ 1 ds/(sg(s) 4 ) = +∞.In particular, the results apply for the logarithmically supercritical dissipation m(ξ ) := |ξ | (d+2)/4 / log(2 + |ξ | 2 ) 1/4 .

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• Project Euclid (Cornell University) - View - PDF
• arXiv (Cornell University) - View - PDF