On the blow-up criterion for the Navier-Stokes equations with critical time order

We prove that every smooth solution u(t, x) on (0, T) of incompressible Navier-Stokes equations on Rnis extensible beyond t > T if u(t, x) is an element of L-w(r)( 0, T; L-sigma(p)) for 2/r+ n/p= 1and p > n satisfies blow-up critical time order estimate: vertical bar vertical bar u(t)vertical bar vertical bar L-p <= epsilon (T- t)(- p-n/2p) for T- delta< t< Twith sufficiently small positive constants epsilon and delta. Here L-w(r) denote the weak L-r space. It is well-known that if the solution usatisfies u is an element of L-r(0, T; L-sigma(p)) with n < p <= infinity and 2 <= r < infinity such that 2/r+ n/p= 1then uis extensible continued beyond the time T. In this paper, we consider the blow-up criterion when u is not an element of L-r(0, T; L-sigma(p)). (c) 2023 Elsevier Inc. All rights reserved.

Automated summary

This paper proves that every smooth solution u(t, x) on (0, T) of incompressible Navier-Stokes equations on Rn can be extended beyond t > T if u(t, x) is an element of L-w(r)(0, T; L-sigma(p)) and satisfies a blow-up critical time order estimate.