Abstract
We prove a regularity criterion for weak solutions to the Navier-Stokes equations in three-space dimensions. This improves the available result with .
Highlights
We study the regularity condition of weak solutions to the Navier-Stokes equations ut − Δu u·∇ u ∇π 0, 1.1 div u 0, in 0, T × R3, 1.2 u|t 0 u0 x, x ∈ R3.Here, u is the unknown velocity vector and π is the unknown scalar pressure
Correspondence should be addressed to Tohru Ozawa, txozdwa@waseda.jp Received 26 June 2008; Accepted 14 October 2008 Recommended by Michel Chipot We prove a regularity criterion ∇π ∈ L2/3 0, T ; BMO for weak solutions to the Navier-Stokes equations in three-space dimensions
While the existence of regular solutions is still an open problem, there are many interesting sufficient conditions which guarantee that a given weak solution is smooth
Summary
We study the regularity condition of weak solutions to the Navier-Stokes equations ut − Δu u·∇ u ∇π 0, 1.1 div u 0, in 0, T × R3, 1.2 u|t 0 u0 x , x ∈ R3.Here, u is the unknown velocity vector and π is the unknown scalar pressure. Correspondence should be addressed to Tohru Ozawa, txozdwa@waseda.jp Received 26 June 2008; Accepted 14 October 2008 Recommended by Michel Chipot We prove a regularity criterion ∇π ∈ L2/3 0, T ; BMO for weak solutions to the Navier-Stokes equations in three-space dimensions. 1. Introduction We study the regularity condition of weak solutions to the Navier-Stokes equations ut − Δu u·∇ u ∇π 0, 1.1 div u 0, in 0, T × R3, 1.2 u|t 0 u0 x , x ∈ R3. For u0 ∈ L2 R3 with div u0 0 in R3, Leray 1 constructed global weak solutions.
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