Finite time blow-up for the hypodissipative Navier Stokes equations with a force in $L_t^1{C}_x^{1,\epsilon }\cap L_t^{\infty }{L}_x^2$

Abstract

In this work we establish the formation of singularities of classical solutions with finite energy of the forced fractional Navier Stokes equations where the dissipative term is given by $|\nabla {|}^{\alpha }$ for any $\alpha \in [0,{\alpha }_0)$ (${\alpha }_0=\frac{22-8\sqrt{7}}{9}>0$). We construct solutions in ${\mathbb{R}}^3\times [0,T]$ with a finite $T>0$ and with an external forcing which is in $L_t^1([0,T])C_x^{1,\epsilon }\cap L_t^{\infty }{L}_x^2$, such that on the time interval $0\le t<T$, the velocity $u$ is in the space $C^{\infty }\cap L^2$ and such that as the time $t$ approaches the blow-up moment $T$, the integral ${\int }_0^t|\nabla u|ds$ tends to infinity.

33 pages. Minor corrections added