Well-posedness for 2D non-homogeneous incompressible fluids with general density-dependent odd viscosity
Matthieu Pageard
TL;DR
This work addresses local well-posedness for 2D non-homogeneous incompressible fluids with density-dependent odd viscosity by reformulating the system via an Elsässer-type effective velocity. A viscous regularisation with $\varepsilon\Delta u$ is used to derive uniform-in-$\varepsilon$ bounds, and an Elsässer structure (defining $U=u-\nabla^{\perp}g(\rho)$ and $\Pi=\pi-f(\rho)\omega$) enables transport-elliptic coupling to control derivatives. The paper proves local existence and uniqueness of strong solutions in $H^s(\mathbb{R}^2)$ for $s>2$, with density bounded between $\rho_*$ and $\rho^*$, and shows energy equalities for $\sqrt{\rho}\,u$ and $\sqrt{\rho}\,U$, along with a clear pathway to recover the Euler limit when the viscosity is constant. The results extend the mathematical theory of fluids with odd viscosity to general density-dependent coefficients $f(\rho)$, including $f(\rho)=a\rho^{\alpha}+b$, and relax prior density-variation assumptions, providing a robust framework for analyzing non-dissipative transport phenomena in 2D fluids.
Abstract
We study the initial value problem for a system of equations describing the motion of two-dimensional non-homogeneous incompressible fluids exhibiting odd (non-dissipative) viscosity effects. We consider the complete odd viscous stress tensor with a general density-dependent viscosity coefficient $f(ρ)$. Under suitable assumptions, we prove the local existence and uniqueness of strong solutions in $H^s(\mathbb{R}^2)$ $(s>2)$, for a class of viscosity coefficients covering the particular case $f(ρ)=aρ^α+b$ for any $(a,b,α)\in\mathbb{R}^3$, generalising the result of Fanelli, Granero-Belinchón and Scrobogna, devoted to the case $f(ρ)=ρ$. Additionally, we are able to do so without requiring the initial density variation to belong to $L^2(\mathbb{R}^2)$. As a major step of the proof, we exhibit an effective velocity for this sytem, generalising the so-called "Elsässer formulation" recently derived by Fanelli and Vasseur.