Epochs of regularity for wild H\"older-continuous solutions of the Hypodissipative Navier-Stokes System

TL;DR

This paper constructs non-unique, H"older-continuous solutions to hypodissipative Navier-Stokes equations that exhibit epochs of regularity, using convex integration and analyzing the relationship between dissipation, singular set size, and regularity.

Contribution

It introduces a convex integration framework for hypodissipative Navier-Stokes with epochs of regularity and extends vector calculus methods to higher dimensions.

Findings

Quantitative relationships between fractional Laplacian power, singular set dimension, and regularity.

Construction of non-unique solutions with prescribed regularity and singular set properties.

Generalization of vector calculus arguments to higher dimensions with Lagrangian coordinates.

Abstract

We consider the hypodissipative Navier-Stokes equations on $[0,T]\times\mathbb{T}^{d}$ and seek to construct non-unique, H\"older-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.