Morrey space techniques applied to the interior regularity problem of the Navier-Stokes equations (Q2758120)

A new criterion of the interior regularity to the weak solutions of the evolutionary Navier-Stokes equations is formulated in the framework of Morrey spaces. Namely, the following theorem is proved: Let \(\frac 72 < q \leq 5\), \((t_0,x_0) \in (0, T) \times \Omega\), \(0 < r_0 < \text{dist}(\partial \Omega, x_0)\) and \(0< t_0 - r_0^2\) with \(\partial \Omega\) the boundary of \(\Omega\). Suppose that \(u\) is a weak solution of equation NEWLINENEWLINE\[NEWLINE\partial u/ \partial t - \Delta u + \nabla \cdot (u \otimes u) + \nabla \pi = 0 \quad \text{in}\quad (0,T) \times \Omega NEWLINE\]NEWLINE NEWLINENEWLINE\[NEWLINE \text{div} u = 0 \quad \text{in}\quad (0,T) \times \Omega. NEWLINE\]NEWLINE NEWLINEThen \(u\) is regular in \((t_0-r_1^2, t_0] \times B_{r_1}(x_0)\) for \(r_1 = r_0/2\) provided that NEWLINENEWLINE\[NEWLINE \sup r^{1-5/q} \left( \int_{t-r^2}^t \int_{|x-y|< r} |u(s,y)|^q dy ds \right)^{1/q} < \epsilon NEWLINE\]NEWLINENEWLINE where the constant \(\epsilon\) is sufficiently small and the supremum is taken over all \((t-r^2, t] \times B_r(x) \subset (t_0 - r_0^2, t_0] \times B_{r_0}(x_0).\)