An improved regularity criterion for the Navier-Stokes equations in terms of one directional derivative of the velocity field
Based on a new multiplicative Sobolev inequality and on a detailed analysis, the author investigates the regularity of solutions to the following homogeneous Navier-Stokes system in \({R}^3\) \[ \partial_t u +(u\cdot \nabla )u-\Delta u + \nabla \pi =0, \;\;\nabla u=0. \] Specifically, the author proves that for every initial velocity field \(u_0\in L^2({R}^3)\) with \(\nabla u_0=0\) and every \(T>0\), if the corresponding weak solution \(u\) satisfies \[ \frac{\partial u}{\partial x_3}\in L^p(0,T;L^q({R}^3)), \;\frac{2}{p}+\frac{3}{q}=2, \;\frac{3\sqrt{37}}{4}-3\leq q\leq 3, \] then \(u\) is smooth in \((0,T]\times {R}^3\). Thus some previous regularity results (mentioned in the paper) are improved.
scientific article; zbMATH DE number 6860128
scientific article