Nonuniqueness analysis on the Navier-Stokes equation in $C_{t}L^{q}$ space

TL;DR

This paper demonstrates the existence of infinitely many weak solutions to the Navier-Stokes equations in certain function spaces, using a novel convex integration approach that avoids interpolation inequalities.

Contribution

It introduces a direct convex integration method with $L^{q}$-normalized jets to establish nonuniqueness of solutions in $C_{t}L^{q}$ spaces for $2<q extless3$, advancing the understanding of solution multiplicity.

Findings

Existence of infinitely many weak solutions starting from zero initial data.

Solutions can be regularized to $C_{t}W^{eta,q}$ spaces.

Method provides a framework for approaching critical exponent thresholds.

Abstract

In the presence of any prescribed kinetic energy, we implement the intermittent convex integration scheme with $L^{q}$-normalized intermittent jets to give a direct proof for the existence of solution to the Navier-Stokes equation in $C_{t}L^{q}$ for some uniform $2<q\ll3$ without the help of interpolation inequality. The result shows the sharp nonuniqueness that there evolve infinite nontrivial weak solutions of the Navier-Stokes equation starting from zero initial data. Furthermore, we improve the regularity of solution to be of $C_{t}W^{\alpha,q}$ in virtue of the fractional Gagliardo-Nirenberg inequalities with some $0<\alpha\ll1$. More importantly, the proof framework provides a stepping stone for future progress on the method of intermittent convex integration due to the fact that $L^{q}$-normalized building blocks carry the threshold effect of the exponent $q$ arbitrarily close to the critical value $3$.