A note on the development of singularities on solutions to the Navier-Stokes equations under super critical forcing terms

TL;DR

This paper extends recent results on finite-time blow-up solutions to the Navier-Stokes equations with super critical forcing, constructing solutions with forcing terms in various Lebesgue spaces and highlighting open cases.

Contribution

It generalizes Zhang's blow-up solutions by constructing solutions with forcing in broader Lebesgue spaces, including new cases with forcing in L^1(0,T;L^p(D)) for p<2.

Findings

Existence of blow-up solutions with forcing in L^q(0,T;L^p(D)) for suitable (q,p)

Construction of solutions with forcing in L^1(0,T;L^p(D)) for p<2

Open problem for the case p=2

Abstract

Recently Qi S. Zhang provides examples of solutions to the Navier-Stokes equations which, under suitable hypothesis, blow up in finite time. He considers axially symmetric solutions in a cylinder $D\,$ under appropriate boundary conditions and under the effect of super critical external forces $f\,.$ The loss of boundedness for the velocity field, as $t\rightarrow T\,,$ is the basic case of blow up. However a more general situation is considered below, as explained in the preamble.\par In his main result Zhang exhibits, for each $q<\infty\,,$ a blow up solution with an external force $f\in L^q(0,T;L^1(D))\,.$ Following Zhang, we construct blow up solutions with forcing terms in $L^q(0,T;L^p(D))\,,$ for suitable pairs $(q,p)\,.$ In particular our results contain Zhang's result. A significant particular case is the existence of external forces $f \in L^1(0,T;L^p(D))\,,$ for every $p<2$, for which the velocity blows up in a finite time. The significant case $p=2$ remains open.