Onsager conjectured that weak solutions of the Euler equations for incompressible fluids in R 3 conserve energy only if they have a certain minimal smoothness, (of order of 1/3 fractional derivatives) and that they dissipate energy if they are rougher. In this paper we prove that energy is conserved for velocities in the function space B 1/3 3,c(N) . We show that this space is sharp in a natural sense. We phrase the energy spectrum in terms of the Littlewood-Paley decomposition and show that the energy flux is controlled by local interactions. This locality is shown to hold also for the helicity flux; moreover, every weak solution of the Euler equations that belongs to B 2/3 3,c(N) conserves helicity. In contrast, in two dimensions, the strong locality of the enstrophy holds only in the ultraviolet range.
We study a general class of Euler equations driven by a forcing with a commutator structure of the form [L, u](ρ) = L(ρu) − L(ρ)u, where u is the velocity field and 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, L φ (f ) = φ * f . Our interest 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, considering three prototypical sub-classes of actions. We prove global regularity for bounded φ's which otherwise are allowed to change sign. Here we derive sharp critical thresholds such that sub-critical initial data (ρ 0 , u 0 ) give rise to global smooth solutions. Next, we study singular actions associated with L = −(−∂ xx ) α/2 , which embed the fractional Burgers' equation of order α. We prove global regularity for α ∈ [1, 2). Interestingly, the singularity of the fractional kernel |x| −(n+α) , avoids an initial threshold restriction. Global regularity of the critical endpoint α = 1 follows with double-exponential W 1,∞ -bounds. Finally, for the other endpoint α = 2, we prove the global regularity of the Navier-Stokes equations with density-dependent viscosity associated with the local L = ∆.
We continue our study of one-dimensional class of Euler equations, introduced in [ST2016], driven by a forcing with a commutator structure of the form [L φ , u](ρ) = φ * (ρu) − (φ * ρ)u, where u is the velocity field and φ belongs to a rather general class of influence or interaction kernels.In this paper we quantify the large-time behavior of such systems in terms of fast flocking for two prototypical sub-classes of kernels: bounded positive φ's, and singular φ(r) = r −(1+α) of order α ∈ [1, 2) associated with the action of the fractional Laplacian L φ = −(−∂ xx ) α/2 . Specifically, we prove fast velocity alignment as the velocity u(·, t) approaches a constant state, u →ū, with exponentially decaying slope and curvature bounds |u x (·, t)| ∞ +|u xx (·, t)| ∞ e −δt . The alignment is accompanied by exponentially fast flocking of the density towards a fixed traveling state ρ(·, t) − ρ ∞ (x −ūt) → 0. R φ(|x − y|)(f (y) − f (x))dy.
We show that a general class of active scalar equations, including porous media and certain magnetostrophic turbulence models, admits non-unique weak solutions in the class of bounded functions. The proof is based upon the method of convex integration recently implemented for equations of fluid dynamics.
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