2009/06/17 by Terence Tao, Tao, Terence · 4 citations
Mathematics · Engineering · #Navier-Stokes equation solutions #Advanced Mathematical Physics Problems #Stability and Controllability of Differential Equations
paper · pdf · doi:10.48550/arxiv.0906.3070
Let d ≥ 3. We consider the global Cauchy problem for the generalised Navier-Stokes system ∂t u + (u ⋅ ∇) u &= - D2 u - ∇ p ∇ ⋅ u &= 0 u(0,x) &= u0(x) for u: \R+ × \Rd → \Rd and p: \R+ × \Rd → \R, where u0: \Rd → \Rd is smooth and divergence free, and D is a Fourier multiplier whose symbol m: \Rd → \R+ is non-negative; the case m(ξ) = |ξ| is essentially Navier-Stokes. It is folklore (see e.g. \citekp) 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 non-decreasing function g: \R+ → \R+ 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|)1/4.