Well-posedness for 2D Non-homogeneous Incompressible Fluids with General Density-Dependent Odd Viscosity

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 \(f(\rho )\). 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 \(f(\rho )=a\rho ^\alpha +b\) for any \((a,b,\alpha )\in \mathbb {R}^3\), generalising the result of [26] devoted to the case \(f(\rho )=\rho \). 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 in [27].