Stability studies of first-order spin-hydrodynamic frameworks

We study the stability of first-order dissipative spin-hydrodynamic frameworks. considered two different The first one considers spin chemical potential ($\omega^{\alpha\beta}$) to be order ($\mathcal{O}(\partial)$) in hydrodynamic gradient expansion. ordering is a debatable issue within frameworks hydrodynamics. Therefore as second choice, we also consider equations with $\omega^{\alpha\beta}\sim\mathcal{O}(1)$. find that for both frameworks, at level linear perturbations some modes can unstable. To remove these generic instabilities Frenkel condition. argue condition helps get rid unstable solutions cases, but physical drawback case where $\omega^{\mu\nu}\sim\mathcal{O}(\partial)$.