Finite time blow-up for the hypodissipative Navier Stokes equations with a force in $L^1_t C_x^{1,\epsilon}\cap L^{\infty}_{t}L_{x}^2$

TL;DR

This paper proves finite-time singularity formation in solutions to forced fractional Navier-Stokes equations with certain dissipative powers and forcing conditions, extending understanding of blow-up phenomena in fluid dynamics.

Contribution

It constructs solutions exhibiting finite-time blow-up for fractional Navier-Stokes equations with dissipative powers below a critical threshold, under specific forcing conditions.

Findings

Finite-time blow-up solutions are constructed.

Blow-up occurs with forcing in $L^1_t C_x^{1, ext{epsilon}}igcap L^{ ext{infinity}}_t L_x^2$.

The velocity becomes singular as time approaches the blow-up time.

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 $\alpha\in [0, \alpha_0)$ ($\alpha_0 = \frac{22-8\sqrt7}{9} > 0$). We construct solutions in $\mathbb{R}^3\times [0,T]$ with a finite $T>0$ and with an external forcing which is in $L^1_t([0, T]) C_x^{1,\epsilon}\cap L^{\infty}_{t}L_{x}^2$, 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.