Global regularity for a logarithmically supercritical hyperdissipative Navier-Stokes equation

By Terence Tao. 2009

198 citations recorded by the source corpus.

Let $d \geq 3$. We consider the global Cauchy problem for the generalised Navier-Stokes system \partial_t u + (u \cdot \nabla) u &= - D^2 u - \nabla p \nabla \cdot u &= 0 u(0,x) &= u_0(x) for $u: \R^+ \times \R^d \to \R^d$ and $p: \R^+ \times \R^d \to \R$, where $u_0: \R^d \to \R^d$ is smooth and divergence free, and $D$ is a Fourier multiplier whose symbol $m: \R^d \to \R^+$ is non-negative; the case $m(\xi) = |\xi|$ is essentially Navier-Stokes. It is folklore (see e.g. \cite{kp}) that one has global regularity in the critical and subcritical hyperdissipation regimes $m(\xi) = |\xi|^\alpha$ for $\alpha \geq \frac{d+2}{4}$. 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 non-decreasing function $g: \R^+ \to \R^+$ such that $\int_1^\infty \frac{ds}{sg(s)^4} = +\infty$. In particular, the results apply for the logarithmically supercritical dissipation $m(\xi) := |\xi|^{\frac{d+2}{4}} / \log(2 + |\xi|)^{1/4}$.

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/c2aa23e72aca0f2958cb83e4e386d66cf7993560

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Citations in the corpus

Global regularity for a slightly supercritical hyperdissipative Navier-Stokes system (2014, methodology): “, (1.2) where G : [0,∞) → [0,∞) is a non–decreasing function such that ∫ ∞ 1 ds sG(s)4 = ∞, (1.3) G(x) |x| d+24 eventually non–increasing, (1.4) then in [Tao09] it is proved1 that (1.1) has a global smooth solution for every smooth initial condition.”
Global well-posedness of slightly supercritical active scalar equations (2012, background): “The work [12] was partly inspired by the slightly supercritical Navier-Stokes regularity result of Tao [29], and partly by the recent work of [3, 4] on generalized Euler and SQG models.”
Partial Regularity of Suitable Weak Solutions to the Fractional Navier–Stokes Equations (2014, background): “1) has a unique global smooth solution for any prescribed smooth initial data (also see [6,16,18]).”
The Generalized Caffarelli‐Kohn‐Nirenberg Theorem for the Hyperdissipative Navier‐Stokes System (2017, background): “On the periodic torus a simple proof has been given in [15] when α > 5 4 , whereas the recent papers [19] and [1] improve the case α = 5 4 allowing operators with symbols −|ξ| 5 2f(ξ) for suitable logarithmic-like f .”
On a maximum principle and its application to logarithmically critical Boussinesq system (2009, background): “tial calculus combined with Theorems 1.1 and 1.3. Remarks 1.6. (1) We point out that for global well-posedness to the generalized Navier-Stokes system in dimension three, Tao proved in a recent paper [28] that we can improve the dissipation |D|52 to |D| 5 2 log 1 2(2+|D|) · (2) The space Xpis less r”
Wild solutions of the Navier–Stokes equations whose singular sets in time have Hausdorff dimension strictly less than 1 (2018, background): “In [45] it was proven that slightly below the critical threshold α = 5/4 the existence of a globally regular solution still holds when the right-hand side of the first equation in (1.”
Finite time blowup for an averaged three-dimensional Navier-Stokes equation (2014, background): “Such an approach succeeds, for instance, if the initial data u0 are sufficiently small 5 in a suitable critical norm (see [29] for an essentially optimal result in this direction), or if the dissipative operator −Δ is replaced by a hyperdissipative operator (−Δ) for some α ≥ 5/4 (see [25]) or with v”