Eulerian dynamics with a commutator forcing

Abstract

We study a general class of Euler equations driven by a forcing with a \emph{commutator structure} of the form $[\mathcal{L},\mathbf{u}](\rho)=\mathcal{L}(\rho \mathbf{u})- \mathcal{L}(\rho)\mathbf{u}$, where $\mathbf{u}$ is the velocity field and $\mathcal{L}$ is the "action" which belongs to a rather general class of translation invariant operators. Such systems arise, for example, as the hydrodynamic description of velocity alignment, where action involves convolutions with bounded, positive influence kernels, $\mathcal{L}_\phi(f)=\phi*f$. Our interest lies with a much larger class of $\mathcal{L}$'s which are neither bounded nor positive. In this paper we develop a global regularity theory in the one-dimensional setting, considering three prototypical sub-classes of actions. We prove global regularity for \emph{bounded} $\phi$'s which otherwise are allowed to change sign. Here we derive sharp critical thresholds such that sub-critical initial data $(\rho_0,u_0)$ give rise to global smooth solutions. Next, we study \emph{singular} actions associated with $\mathcal{L}=-(-\partial_{xx})^{\alpha/2}$, which embed the fractional Burgers' equation of order $\alpha$. We prove global regularity for $\alpha\in [1,2)$. Interestingly, the singularity of the fractional kernel $|x|^{-(n+\alpha)}$, avoids an initial threshold restriction. Global regularity of the critical endpoint $\alpha=1$ follows with double-exponential $W^{1,\infty}$-bounds. Finally, for the other endpoint $\alpha=2$, we prove the global regularity of the Navier-Stokes equations with density-dependent viscosity associated with the \emph{local} $\mathcal{L}=\Delta$.