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

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 $2<q\ll 3$ 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 $0<\alpha \ll 1$. 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 $3$.

arXiv comment27 pages, minor revision