Epochs of regularity for wild Hölder-continuous solutions of the hypodissipative Navier–Stokes system

Abstract

Abstract We consider the hypodissipative Navier–Stokes equations on [ 0 , T ] × T d [0,T]\times\mathbb{T}^{d} and seek to construct non-unique, Hölder-continuous solutions with epochs of regularity (smooth almost everywhere outside a small singular set in time), using convex integration techniques. In particular, we give quantitative relationships between the power of the fractional Laplacian, the dimension of the singular set, and the regularity of the solution. In addition, we also generalize the usual vector calculus arguments to higher dimensions with Lagrangian coordinates.