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, ${\mathcal{L}}_{\varphi }(f)=\varphi \ast f$. 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} $\varphi$'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} $\mathcal{L}=\Delta$.

22 pages