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 use De Giorgi techniques to prove Hölder continuity of weak solutions to a class of drift-diffusion equations, with L 2 initial data and divergence free drift velocity that lies in L ∞ t BM O −1 x . We apply this result to prove global regularity for a family of active scalar equations which includes the advection-diffusion equation that has been proposed by Moffatt in the context of magnetostrophic turbulence in the Earth's fluid core.2000 Mathematics Subject Classification. 76D03, 35Q35, 76W05.

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We present a geometric estimate from below on the growth rate of a small perturbation of a threedimensional steady flow of an ideal fluid and thus we obtain effective criteria for local instability for Euler's equations. We use these criteria to demonstrate the instability of several simple flows and to show that any flow with a hyperbolic stagnation point is unstable.PACS numbers: 47.20.-k In a companion paper Vishik and Friedlander 1 obtain a universal geometric estimate from below on the growth rate of a small perturbation of a three-dimensional steady flow of an inviscid incompressible fluid. In this Letter we discuss certain implications concerning instability for Euler's equations that follow from the existence of this estimate and we demonstrate that the estimate gives effective criteria for local instability. In particular, the existence of a hyperbolic stagnation point implies that the steady flow is unstable. An important feature of our approach which allows us to obtain effective criteria is that, unlike many previous approaches to hydrodynamic stability, we do not study the spectrum but rather we consider the growth rate of the relevant Green's function as t-> °°.There is a very extensive literature concerning the field of hydrodynamic stability (for references see, for example, Drazin and Reid 2 ). We briefly mention some of the work whose techniques are related to those that we employ. Eckhoff and Storesletten 3 and Eckhoff 4 study the stability of azimuthal shear flows of a compressible fluid and more generally symmetric hyperbolic systems using an approach based on the generalized progressing wave expansion. 5,6 Eckhoff shows that local instability problems for hyperbolic systems can be essentially reduced to a local analysis involving ordinary differential equations (ODE) and algebraic equations only. We show that the same conclusion can be drawn for Euler's equations for an ideal fluid. These equations do not form a hyperbolic system; hence several additional technical details arise in the analysis. Bayly 7 studies the stability of quasi-twodimensional steady flows via an analysis of a Floquet system of ODE. He shows that the Floquet exponent gives the growth rate for a family of instabilities which include the Rayleigh centrifugal instability, the LeibovichStewartson columnar instability, and the elliptic vortex instability. We note that the instability criteria that we present in this Letter are equivalent to those of Bayly in the particular case of quasi-two-dimensional steady flows. Lifschitz 8 uses WKB methods to construct part of the continuous spectrum for axisymmetric steady flows. Using methods inspired by magnetohydrodynamics, he obtains a necessary stability condition for a vortex ring with respect to localized three-dimensional perturbations.Let u(x) be a steady solution of Euler's equations governing the motion in 3D of an inviscid incompressible fluid:(1)The 3D vector field u(x) denotes the velocity and the scalar field P(x) denotes the pressure in the fluid. We consider the linear...

ABSTRACT. We consider an active scalar equation that is motivated by a model for magneto-geostrophic dynamics and the geodynamo. We prove that the non-diffusive equation is ill-posed in the sense of Hadamard in Sobolev spaces. In contrast, the critically diffusive equation is well-posed (cf. [15]). In this case we give an example of a steady state that is nonlinearly unstable, and hence produces a dynamo effect in the sense of an exponentially growing magnetic field.

Properties of an infinite system of nonlinearly coupled ordinary differential equations are discussed. This system models some properties present in the equations of motion for an inviscid fluid such as the skew symmetry and the 3-dimensional scaling of the quadratic nonlinearity. It is proved that the system with forcing has a unique equilibrium and that every solution blows up in finite time in H 5/6 -norm. Onsager's conjecture is confirmed for the model system.

We present a three-dimensional vector model given in terms of an infinite system of nonlinearly coupled ordinary differential equations. This model has structural similarities with the Euler equations for incompressible, inviscid fluid flows. It mimics certain important properties of the Euler equations, namely, conservation of energy and divergence-free velocity. It is proven for certain families of initial data that the model system permits local existence in time for initial conditions in Sobolev spaces H s , s > 5 2 , and blowup occurs in the sense that the H 3/2+ norm becomes unbounded in finite time.

It is shown that for most, but not all, three-dimensional magnetohydrodynamic (MHD) equilibria the second variation of the energy is indefinite. Thus the class of such equilibria whose stability might be determined by the so-called Arnold criterion is very restricted. The converse question, namely conditions under which MHD equilibria will be unstable is considered in this paper. The following sufficient condition for linear instability in the Eulerian representation is presented: The maximal real part of the spectrum of the MHD equations linearized about an equilibrium state is bounded from below by the growth rate of an operator defined by a system of local partial differential equations (PDE). This instability criterion is applied to the case of axisymmetric toroidal equilibria. Sufficient conditions for instability, stronger than those previously known, are obtained for rotating MHD.