Well-posedness for 2D non-homogeneous incompressible fluids with general density-dependent odd viscosity

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 Fanelli, Granero-Belinch\'on and Scrobogna, 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 "Els\"asser formulation" recently derived by Fanelli and Vasseur.