Abstract
This paper is concerned with the regularity of Leray weak solutions to the 3D Navier‐Stokes equations in Lorentz space. It is proved that the weak solution is regular if the horizontal velocity denoted by satisfies The result is obvious and improved that of Dong and Chen (2008) on the Lebesgue space.
Highlights
Introduction and Main ResultsWe consider the regularity criterion of weak solutions of the Navier-Stokes equations in the whole space R3
Introduction and Main ResultsIn this note, we consider the regularity criterion of weak solutions of the Navier-Stokes equations in the whole space R3∂tu u · ∇ u ∇π Δu,∇ · u 0, 1.1 u x, 0 u0.Here u u1, u2, u3 and π denote the unknown velocity field and the unknown scalar pressure field. u0 is a given initial velocity
The weak solution remains regular when a part of the velocity components or vorticity is involved in a growth condition
Summary
We consider the regularity criterion of weak solutions of the Navier-Stokes equations in the whole space R3. A weak solution u of Navier-Stokes equations is regular if the growth condition on velocity field u u ∈ Lp 0, T ; Lq R3. The weak solution remains regular when a part of the velocity components or vorticity is involved in a growth condition. It should be mentioned that the weak solution remains regular if the single velocity component satisfies the higher subcritical growth conditions see Zhou 9 , Penel and Pokorny , Kukavica and Ziane , and Cao and Titi. It seems difficult to show regularity of weak solutions by imposing Serrin’s growth condition on only one component of velocity field for both Navier-Stokes equations and micropolar fluid flows. The aim of the present paper is to improve the two-component regularity criterion 1.9 from Lebesgue space to the critical Lorentz space see the definitions in Section 2 which satisfies the scaling invariance property. In order to make use of the structure of the nonlinear convection term u·∇ u, we study every component of u·∇ u, Δu and estimate them one by one with the aid of the identities ∇ · u 0
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