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Global regularity for a logarithmically supercritical hyperdissipative Navier-Stokes equation. (English) Zbl 1190.35177
Summary: Let $$d\geq 3$$. We consider the global Cauchy problem for the generalized Navier-Stokes system
$\partial_tu+ (u\cdot \nabla)u= -D^2u- \nabla p, \quad \nabla\cdot u=0, \quad u(0,x)=u_0(x),$ for $$u:\mathbb R^+\times\mathbb R^d\rightarrow \mathbb R^d$$ and $$p:\mathbb R^+\times\mathbb R^d\rightarrow \mathbb R$$, where $$u_0: \mathbb R^d\rightarrow\mathbb R^d$$ is smooth and divergence free, and $$D$$ is a Fourier multiplier whose symbol $$m:\mathbb R^d\rightarrow\mathbb R^+$$ is nonnegative; the case $$m(\xi) =|\xi|$$ is essentially Navier-Stokes. It is folklore that one has global regularity in the critical and subcritical hyperdissipation regimes $$m(\xi)= |\xi|^\alpha$$ for $$\alpha\geq(d+2)/4$$. We improve this slightly by establishing global regularity under the slightly weaker condition that $$m(\xi)\geq|\xi|^{(d+2)/4}/g(|\xi|)$$ for all sufficiently large $$\xi$$ and some nondecreasing function $$g:\mathbb R^+\rightarrow\mathbb R^+$$ such that $$\int_1^\infty ds/(sg(s)^4)=+\infty$$. In particular, the results apply for the logarithmically supercritical dissipation $$m(\xi):= |\xi|^{(d+2)/4} \log(2+|\xi|^2)^{1/4}$$.

MSC:
 35Q30 Navier-Stokes equations 76D03 Existence, uniqueness, and regularity theory for incompressible viscous fluids 76D05 Navier-Stokes equations for incompressible viscous fluids
Keywords:
Navier-Stokes system; energy method
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