Weak solutions of the two-dimensional incompressible inhomogeneous Navier-Stokes equations in the presence of variable odd viscosity

TL;DR

This paper proves the existence of weak solutions for the 2D inhomogeneous Navier-Stokes equations with variable odd viscosity and explores their behavior as viscosity approaches a constant, including specific flow examples.

Contribution

It introduces a framework for weak solutions with variable odd viscosity and analyzes their limits, extending understanding of inhomogeneous fluid dynamics with odd viscosity effects.

Findings

Existence of weak solutions in both evolutionary and stationary cases.

Weak solutions converge as odd viscosity approaches a constant.

Examples of stationary solutions for specific flow configurations.

Abstract

We consider the two-dimensional incompressible inhomogeneous Navier-Stokes equations with odd viscosity, where the shear and the odd viscosity coefficients depend continuously on the unknown density function. We establish the existence of weak solutions in both the evolutionary and stationary cases. Furthermore, we investigate the limit of the weak solutions as the odd viscosity coefficient converges to a constant. Lastly, we consider examples of stationary solutions for parallel, concentric and radial flows.