Eulerian dynamics with a commutator forcing

Abstract

Abstract : We study a general class of Euler equations driven by a forcing with a commu-tator structure of the form [L; u]() = L(u)L()u, where u is the velocity eld and L isthe \action which belongs to a rather general class of translation invariant operators. Suchsystems arise, for example, as the hydrodynamic description of velocity alignment, whereaction involves convolutions with bounded, positive inuence kernels, L(f) = f. Ourinterest lies with a much larger class of L's which are neither bounded nor positive.In this paper we develop a global regularity theory in the one-dimensional setting, con-sidering three prototypical sub-classes of actions. We prove global regularity for bounded 'swhich otherwise are allowed to change sign. Here we derive sharp critical thresholds suchthat sub-critical initial data (0; u0) give rise to global smooth solutions. Next, we studysingular actions associated with L = (@xx)=2, which embed the fractional Burgers'equation of order . We prove global regularity for 2 [1; 2). Interestingly, the singularityof the fractional kernel jxj(n ), avoids an initial threshold restriction. Global regularity ofthe critical endpoint = 1 follows with double-exponential W1;1-bounds. Finally, for theother endpoint = 2, we prove the global regularity of the Navier-Stokes equations withdensity-dependent viscosity associated with the local L = .