Finite energy of generalized suitable weak solutions to the Navier-Stokes equations and Liouville-type theorems in two dimensional domains

TL;DR

This paper introduces a new class of weak solutions to the Navier-Stokes equations, proving energy inequalities and Liouville-type theorems in 2D unbounded domains, with implications for infinite energy solutions.

Contribution

It establishes the validity of energy inequalities for generalized suitable weak solutions and proves Liouville-type theorems under specific conditions in 2D domains.

Findings

Energy inequality holds for the new class of weak solutions.

Vorticity and its gradient are globally square integrable under certain conditions.

Liouville-type theorems are derived for solutions in unbounded domains.

Abstract

Introducing a new notion of generalized suitable weak solutions, we first prove validity of the energy inequality for such a class of weak solutions to the Navier-Stokes equations in the whole space $\mathbb{R}^n$. Although we need certain growth condition on the pressure, we may treat the class even with infinite energy quantity except for the initial velocity. We next handle the equation for vorticity in 2D unbounded domains. Under a certain condition on the asymptotic behavior at infinity, we prove that the vorticity and its gradient of solutions are both globally square integrable. As their applications, Loiuville-type theorems are obtained.