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Regularity criterion via the pressure on weak solutions to the 3D Navier-Stokes equations. (English) Zbl 1126.35047

The authors consider a Leray-Hopf weak solution \((u,p)\) of the Navier-Stokes equations in \(\mathbb{R}^3\times(0,T)\). They prove that this solution is regular provided that the initial velocity \(u_0\) belongs to \(L^2 (\mathbb{R}^3)\cap L^q (\mathbb{R}^3)\) for some \(q>3\) and that the pressure satisfies \[ \int^T_0\|p(t) \|_{\dot B^0_{\infty,\infty}}\,dt<\infty. \] The main tool in obtaining this result is an a-priori-estimate of the form \[ \sup_{0\leq t\leq T}\|u(t)\|_{L^s}\leq C(\| u_0\|_{L^s}+(CT)^{\frac{1} {s}}+e)^{\exp(C\int^T_0\|p(t)\|_{\dot B^0_{\infty, \infty}}\,dt)},\quad 3<s\leq 4, \] which is derived with the help of the Paley-Littlewood decomposition.

MSC:

35Q30 Navier-Stokes equations
76D03 Existence, uniqueness, and regularity theory for incompressible viscous fluids
35B65 Smoothness and regularity of solutions to PDEs
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