Regularity criteria in weak \(L^3\) for 3D incompressible Navier-Stokes equations (Q2816484)

The interior boundedness for a distributional solution of the 3D Navier-Stokes equations is studied in the paper. Let \(B_r=\{x\in\mathbb{R}^3:| x|<r\}\) the ball with the radius \(r\) and \((u,p)\) the distributional solution to the equations NEWLINE\[NEWLINE \frac{\partial u}{\partial t}-\Delta u+(u\cdot\nabla)u+\nabla p=0, \quad\operatorname{div}\,u=0,\quad x\in B_2,\quad t\in(0,1). NEWLINE\]NEWLINENEWLINENEWLINEThe authors prove that if NEWLINE\[NEWLINE p\in L^q(0,1;L^1(B_2)) \,\quad \text{and}\quad \| u\|_{L^\infty(0,1;L^{3,\infty}(B_2))}<\varepsilon NEWLINE\]NEWLINE for some \( q>2\) and small \(\varepsilon\) then \(u\in L^\infty(B_1\times(1/10,1))\). It is important that \(\varepsilon\) does not depend on \(q\) and on norm of the pressure \(p\) and there is no boundedness on \(\nabla u\) in \(B_2\). The proof is based on cut-off technique and on the investigation of the problem on the entire space \(\mathbb{R}^3\).