In many physical fluid systems, the constituent particles present a parity-breaking intrinsic angular momentum: this is the case, for instance, of quantum fluids and super-fluids, polyatomic gases, chiral active matter and vortex dynamics. In such situations, only the skew-symmetric component of the total viscous stress tensor, often dubbed odd viscosity, is non-zero, implying that the viscosity becomes non-dissipative.
At the level of the mathematical model, the odd viscosity term is responsible for a loss of regularity, as it involves higher order space derivatives of the velocity field and, in the case of non-homogeneous fluids, of the density.
In this talk we consider the dynamics of non-homogeneous incompressible fluids having odd viscosity and we set up a well-posedness theory in Sobolev spaces for the related system of equations. The proof is based on the introduction of a set of suitable 'good unknowns' for the system, which allow to put in evidence an underlying hyperbolic structure and to circumvent, in this way, the loss of derivatives created by the odd viscosity term.
The talk is based on a joint work with Rafael Granero-Belinchón (Universidad de Cantabria) and Stefano Scrobogna (Università degli Studi di Trieste).[-]

The patch solutions of the 2D Euler and (modified) SQG equations have form $\omega(x, t)=\chi_{\Omega(t)}(x)$ of a characteristic function of a domain $\Omega(t)$ evolving in time according to the Biot-Savart law $u=\nabla^{\perp}(-\Delta)^{-1+\alpha} \omega$, here $\alpha=0$ corresponds to the Euler case and $0<\alpha<1$ to the modified SQG family. For the Euler case, the first proof of global regularity for pathes was given by Chemin in Hölder spaces $C^{k, \beta}, 0<\beta<1$. For the modified SQG family, the problem remains largely open - with the only finite time singularity formation result available in the presence of boundary and for small $\alpha[5,2]$. I will discuss some recent conditional results on the possible scenarios for finite time blow up. Also, for the Euler patch case, I will describe a construction of patches that are $C^{2}$ at the initial and all integer times, but lack this regularity for all other times - without being time periodic. This result is based on the analysis of the curvature evolution equation, which may also be useful for other applications.[-]

We consider the physically relevant fully compressible setting of the Rayleigh-Bénard problem of a fluid confined between two parallel plates, heated from the bottom, and subjected to the gravitational force. Under suitable restrictions imposed on the constitutive relations we show that this open system is dissipative in the sense of Levinson, meaning there exists a bounded absorbing set for any global-in-time weak solution. In the second part of the talk we discuss also the motion of a compressible viscous fluid in a container with impermeable boundary subject to time periodic heating and under the action of a time periodic potential force. We show the existence of a time periodic weak solution for arbitrarily large physically admissible data. The talk is based on two papers.[-]