$L^2$-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 $\theta <\frac{1}{3}$, we prove non-uniqueness of dissipative $L_t^2{H}_x^{\theta }$ weak solutions for an $L^2$-dense set of ${\mathcal{C}}^{\beta }$ Hölder continuous wild initial data with $\theta <\beta <\frac{1}{3}$. 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]).

arXiv admin note: text overlap with arXiv:2004.00391 by other authors

References in corpus (3)

• Sharp nonuniqueness for the Navier-Stokes equations
• Ill-posedness of Leray solutions for the ipodissipative Navier-Stokes equations
• Statistical solutions and Onsager's conjecture

Cited by in corpus (1)

• Density of weak solutions of the fractional Navier-Stokes equations in the smooth divergence-free vector fields