Global regularity for a slightly supercritical hyperdissipative Navier–Stokes system

Abstract

We prove global existence of smooth solutions for a slightly supercritical hyperdissipative Navier-Stokes under the optimal condition on the correction to the dissipation. This proves a conjecture formulated by Tao [Tao09]. From the generalized Fourier Navier-Stokes to the dyadic equationThis section contains one of the crucial steps in our approach. We show that the proof of Theorem 1.2 can be reduced to a proof of decay of solutions of a suitable

The D incompressible Navier–Stokes equations with partial hyperdissipation

“…Let > 0. Assume that (1) and (2) are two solutions of (1.2) satisfying, ( ) ∈ ∞ ( 0, ; 1 ( ℝ 3 )) for = 1, 2, ( Λ ) ∇ (2) 3 ∈ 2 ( ℝ 3 × (0, ) ) .…”

“…Proof. Let (1) and (2) be the pressures associated with (1) and (2) , respectively. Then the differences̃= (1) − (2) and = (1) − (2) satisfy (3.1) Dotting (3.1) with̃and invoking the divergence-free conditions ∇ ⋅ (1) = 0 and ∇ ⋅̃= 0, we obtain We estimate 1 and write its terms explicitly,…”

“…More precisely, he shows that replacing by still leads to a unique global solution. Tao's result was later improved by to allow additional power in the logarithm, namely with . Efforts have also been devoted to the incompressible magnetohydrodynamic equations with hyperdissipation and the results are not completely parallel (see, e.g., ).…”

“…We remove some components of the hyperdissipation in (1.1). More precisely, we study the following 3D Navier-Stokes equations with fractional partial dissipation, 1 where the fractional partial dissipation operators Λ with > 0 and = 1, 2, 3 are defined via the Fourier transform Λ ( ) = | |̂( ).…”

Global Well-posedness for the Logarithmically Energy-Supercritical Nonlinear Wave Equation with Partial Symmetry

“…and which is conserved in time for solutions of (NLW). When F is given by (1), one has G(u) ≈ u 4 log(3 + u 2 ), while when F is of power-type |u| p u, G(u) = 1 p+2 |u| p+2 . In recent years, beginning with work of Tao [29], several authors have studied the global well-posedness and scattering problem for various cases of the threedimensional nonlinear wave equation with slightly energy-supercritical nonlinearity, obtaining striking results which show that the global well-posedness theory can be extended from the energy-subcritical and energy-critical settings into the slightly energy-supercritical regime (that is, admitting the inclusion of logarithmic factors).…”