On a regularity criterion in terms of the gradient of pressure for the Navier-Stokes equations in \(\mathbb R^N\)
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.
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