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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 \(m(\xi):= |\xi|^{(d+2)/4} \log(2+|\xi|^2)^{1/4}\).

35Q30 Navier-Stokes equations
76D03 Existence, uniqueness, and regularity theory for incompressible viscous fluids
76D05 Navier-Stokes equations for incompressible viscous fluids
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