Sharp nonuniqueness for the forced 2D Navier-Stokes and dissipative SQG equations

TL;DR

This paper demonstrates sharp nonuniqueness results for the forced 2D Navier-Stokes and dissipative SQG equations, showing multiple solutions exist below certain regularity classes, challenging previous uniqueness assumptions.

Contribution

It establishes the first sharp nonuniqueness results for these equations below key regularity classes, expanding understanding of solution behavior in fluid dynamics.

Findings

Nonuniqueness of solutions below the Miura-Ju class for dissipative SQG.

Nonuniqueness below Ladyzhenskaya-Prodi-Serrin class for 2D Navier-Stokes.

Nonuniqueness below Constantin-Wu and Dong-Chen-Zhao-Liu classes for dissipative SQG.

Abstract

We prove a sharp nonuniqueness result for the forced generalized SQG equation. First, this yields nonunique $\dot{H}^s$- energy solutions below the Miura-Ju class. In particular, this shows that the solutions constructed by Resnick and Marchand for the dissipative SQG equation are not necessarily unique. Second, this establishes nonuniqueness below the Ladyzhenskaya-Prodi-Serrin class for the 2D Navier-Stokes equation, as well as below the Constantin-Wu and Dong-Chen-Zhao-Liu classes for the dissipative SQG equation.