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On a regularity criterion in terms of the gradient of pressure for the Navier-Stokes equations in $$\mathbb R^N$$. (English) Zbl 1099.35091
The Cauchy problem for the Navier-Stokes equations is considered in $$\mathbb R^n\times(0,T)$$, $$n=3,4$$ \begin{aligned} &\frac{\partial v}{\partial t}+(v\cdot\nabla)v-\Delta v+\nabla p=0,\quad\text{div}\,v=0 \quad \text{in}\quad \mathbb R^n\times(0,T)\\ &v(x,0)=v_0(x),\quad x\in \mathbb R^n\end{aligned} Let $v_0\in L_2(\mathbb R^n)\bigcap L_q(\mathbb R^n)\quad\text{for}\;q\geq n,\quad\text{div}\,v_0=0.$ It is proved if $$v$$ is a Leray-Hopf weak solution to the problem and
$\nabla p\in L_{\alpha}\left(0,T;L_{\gamma}(\mathbb R^n)\right) \quad\text{with}\;\frac{2}{\alpha}+\frac{n}{\gamma}\leq 3,\;\frac{2}{3}<\alpha<\infty,\;\frac{n}{3}<\gamma<\infty$ then $$v$$ is regular and unique. A priori estimates for the smooth solution are the base of the proof.

##### MSC:
 35Q30 Navier-Stokes equations 76D03 Existence, uniqueness, and regularity theory for incompressible viscous fluids 76D05 Navier-Stokes equations for incompressible viscous fluids 35B45 A priori estimates in context of PDEs 35D10 Regularity of generalized solutions of PDE (MSC2000)
##### Keywords:
regularity; a priori estimates; Leray-Hopf weak solution
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