Susan Friedlander

Energy conservation and Onsager's conjecture for the Euler equations

Onsager conjectured that weak solutions of the Euler equations for incompressible fluids in conserve energy only if they have a certain minimal smoothness (of the 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 . 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 conserves helicity. In contrast, in two dimensions, the strong locality of the enstrophy holds only in the ultraviolet range.

Nonlinear instability in an ideal fluid

Linearized instability implies nonlinear instability under certain rather general conditions. This abstract theorem is applied to the Euler equations governing the motion of an inviscid fluid. In particular this theorem applies to all 2D space periodic flows without stagnation points as well as 2D space-periodic shear flows.

Global well-posedness for an advection–diffusion equation arising in magneto-geostrophic dynamics

Abstract We use De Giorgi techniques to prove Holder 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 ∞ BMO x − 1 . 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 Earthʼs fluid core.

Instability criteria for steady flows of a perfect fluid

An instability criterion based on the positivity of a Lyapunov-type exponent is used to study the stability of the Euler equations governing the motion of an inviscid incompressible fluid. It is proved that any flow with exponential stretching of the fluid particles is unstable. In the case of an arbitrary axisymmetric steady integrable flow, a sufficient condition for instability is exhibited in terms of the curvature and the geodesic torsion of a stream line and the helicity of the flow.

Internal waves in a rotating stratified fluid in an arbitrary gravitational field

Abstract Small amplitude oscillations of a uniformly rotating, density stratified, Boussinesq, non-dissipative fluid are examined. A mathematical model is constructed to describe timedependent motions which are small deviations from an initial state that is motionless with respect to the rotating frame of reference. The basic stable density distribution is allowed to be an arbitrary prescribed function of the gravitational potential. The problem is considered for a wide class of gravitational fields. General properties of the eigenvalues and eigenfunctions of square integrable oscillations are demonstrated, and a bound is obtained for the magnitude of the frequencies. The modal solutions are classified as to type. The eigenfunctions for the pressure field are shown to satisfy a second-order partial differential equation of mixed type, and the equation is obtained for the critical surfaces which delineate the elliptic and hyperbolic regions. The nature of the problem is examined in detail for certain speci...

On stability and instability criteria for magnetohydrodynamics

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. (c) 1995 American Institute of Physics.

Localized Instabilities in Fluids

Abstract We study the effects of localized instabilities on the behavior of inviscid fluids. We place geometric optics techniques independently developed by Friedlander–Vishik and Lifschitz–Hameiri in the framework of the general stability/instability problem of fluid dynamics. We show that, broadly put, all laminar flows are unstable with respect to localized perturbations.

The unstable spectrum of oscillating shear flows

A complete description is given for the unstable spectrum of the Euler equation linearized about shear flows with sinusoidal profiles. The spectral problem is treated analytically and numerically. The spectral asymptotics are determined using the method of averaging for general rapidly oscillating profiles. Such flows are always unstable.

NONLINEAR INSTABILITY FOR THE CRITICALLY DISSIPATIVE QUASI-GEOSTROPHIC EQUATION

We prove that linear instability implies non-linear instability in the energy norm for the critically dissipative quasi-geostrophic equation.

Internal waves in a contained rotating stratified fluid

Small-amplitude time-dependent motions of a uniformly rotating, density-stratified, Boussinesq non-dissipative fluid in a rigid container are examined for the case of the rotation axis parallel to gravity. We consider a variety of container shapes, along with arbitrary values for the (constant) Brunt-Vaisala and rotation frequencies. We demonstrate a number of properties of the eigenvalues and eigenfunctions of square-integrable oscillatory motions. Some of these properties hold generally, while others are shown for specific classes of containers (such as with symmetry about the container axis). A full solution is presented for the response of fluid in a cylindrical container to an arbitrary initial disturbance. Features of this solution which are different from the cases of no stratification or no rotation are emphasized. For the situation when Brunt-Vaisala and rotation frequencies are equal, characteristics of the oscillation frequencies and modal structures are found for containers of quite general shape. This situation illustrates, in particular, effects which are possible when rotation and stratification act together and which have been overlooked in previous investigations that assume that the vertical length scale is much smaller than the horizontal scales.