$L^2$-Density of Wild Initial Data for the Hypodissipative Navier-Stokes Equations

Abstract

In this paper we deal with the Cauchy problem for the hypodissipative Navier-Stokes equations in the three-dimensional periodic setting. For all Laplacian exponents $\theta<\frac13$, we prove non-uniqueness of dissipative $L^2_tH^\theta_x$ weak solutions for an $L^2$-dense set of $\mathcal C^\beta$ H\"older continuous wild initial data with $\theta<\beta<\frac13$. This improves previous results of non-uniqueness for infinitely many wild initial data ([8,20]) and generalizes previous results on density of wild initial data obtained for the Euler equations ([14, 13]).