Misha Vishik

Incompressible flows of an ideal fluid with vorticity in borderline spaces of Besov type

Abstract We prove a uniqueness theorem for the Euler equations for an ideal incompressible fluid under the condition that vorticity belongs to a space of Besov type. We also prove an existence theorem in dimension two.

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.

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.

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.

Lax pair formulation for the Euler equation

Abstract A Lagrangian description is presented for the two- and three-dimensional flow of an ideal fluid. A Lax pair equation is obtained for the evolution of a quadratic form from which the Lagrangian velocity can be reconstructed. Inverse scattering in 2-D is discussed and a Hamiltonian structure is exhibited.

Robustness of instability for the two-dimensional Euler equations

We prove that in two dimensions any steady, inviscid, incompressible flow that is sufficiently close to an unstable flow is also unstable.

Unstable Eigenvalues Associated with Inviscid Fluid Flows

Abstract. An example is presented of a class of periodic, two-dimensional, inviscid fluid flows where the stability spectrum contains both discrete unstable eigenvalues and an unstable essential spectrum. The method of averaging is used to demonstrate the existence of unstable eigenvalues. For such flows spectral instability implies nonlinear instability.

Hydrodynamic instability for certain ABC flows

Abstract A sufficient condition for instability of the Euler equations [see (1.5)] is used to demonstrate the instability of certain ABC flows. In the parameter range A = 1, B 2 + C 2 > 1, instability follows from the presence of hyperbolic stagnation points. In the parameter range A = 1, B = C = «e1, instability follows from the existence of hyperbolic closed trajectories and the associated exponential stretching of the fluid particles. This result is proved analytically in Section 2 and illustrated numerically via Poincare sections in Section 3.

The spectrum of the second variation of the energy for an ideal incompressible fluid

Abstract The Arnold theory of hydrodynamic stability suggests the study of the second variation of the energy on the submanifold of isovorticial vector fields. We show that, in three or more dimensions, the spectrum of the associated Hermitian operator generally ranges from -∞ to ∞. The sole exceptional case is where the velocity field is harmonic (solenoidal and irrotational) in which case the Hessian is identically zero.

Instability Criteria in Fluid Dynamics

The following two results are presented concerning linear instability criteria in fluid dynamics. The first result states that an instability in the underlying Euler equation is necessary for the existence of an instability in the Navier-Stokes equations in the limit of high Reynolds number. The second result gives a sufficient condition for instability of the Euler equations in terms of a geometric quantity that can be considered as an analogue of a Lyapunov exponent for fluid motion. The effectiveness of this criterion is illustrated in several examples including the instability of a flow with a hyperbolic stagnation point and a generalized baroclinic instability.