P. Constantin

On the dimension of the attractors in two-dimensional turbulence

Abstract Using a new version of the Sobolev-Lieb-Thirring inequality, we derive an upper bound for the dimension of the universal attractor for two-dimensional space periodic Navier-Stokes equations. This estimate is optimal up to a logarithmic correction. The relevance of this estimate to turbulence and related results are also briefly discussed.

On the possibility of soft and hard turbulence in the complex Ginzburg-Landau equation

Abstract We present analytical methods which predict the occurrence of both soft (weak) and hard (strong) turbulence in the complex Ginzburg-Landau (CGL) equation: A t =RA+(1+iν)δA−(1+iμ)A¦A¦ 2 on a periodic domain [0,1] D in D spatial dimensions. Hard turbulence is characterised by large fluctuations away from spatial and temporal averages with a cascade of energy to small scales. This form of hard turbulence appears to occur not in 1D but only in 2D and 3D in parameter regions which are bounded by hyperbolic curves in the second and fourth quadrants of the μ-ν planes where the system is modulationally unstable (ϵ=1+μν 2n: F n =∫(¦∇ n−1 A¦ 2 +α n ¦A¦ 2n )dx , for αn > 0. For large times and large R, upper bounds exist for the infinite set of Fns, constructed from the hierarchy of differential inequalities Fn≤(2nR+cn‖A‖2∞)Fn−bnF2n/Fn−1, for cn, bn > 0 (F0≡1). Estimates for the “bottom rung” F2 give upper bounds for the whole ladder. Long time upper bounds on F2 and ‖A‖2∞ (and hence all Fn) are well controlled in the soft region but become much larger in the hard region, whereas spati al and temporal averages remain comparatively small. When the nonlinearity is A¦A¦2q, the critical case qD=2 gives parallel results.

Note on loss of regularity for solutions of the 3—D incompressible euler and related equations

One of the central problems in the mathematical theory of turbulence is that of breakdown of smooth (indefinitely differentiable) solutions to the equations of motion. In 1934 J. Leray advanced the idea that turbulence may be related to the spontaneous appearance of singularities in solutions of the 3—D incompressible Navier-Stokes equations. The problem is still open. We show in this report that breakdown of smooth solutions to the 3—D incompressible slightly viscous (i.e. corresponding to high Reynolds numbers, or “highly turbulent”) Navier-Stokes equations cannot occur without breakdown in the corresponding solution of the incompressible Euler (ideal fluid) equation. We prove then that solutions of distorted Euler equations, which are equations closely related to the Euler equations for short term intervals, do breakdown.

Global solutions for small data to the Hele-Shaw problem

The authors analyse an equation governing the motion of an interface between two fluids in a pressure field. In two dimensions, the interface is described by a conformal mapping which is analytic in the exterior of the unit disc. This mapping obeys a non-local nonlinear equation. When there is no pumping at infinity, there is conservation of area and contraction of the length of the interface. They prove global in time existence for small analytic perturbations of the circle as well as nonlinear asymptotic stability of the steady circular solution. The same method yields well-posedness of the Cauchy problem in the presence of pumping.

Spectral barriers and inertial manifolds for dissipative partial differential equations

In recent years, the theory of inertial manifolds for dissipative partial differential equations has emerged as an active area of research. An inertial manifold is an invariant manifold that is finite dimensional, Lipschitz, and attracts exponentially all trajectories. In this paper, we introduce the notion of a spectral barrier for a nonlinear dissipative partial differential equation. Using this notion, we present a proof of existence of inertial manifolds that requires easily verifiable conditions, namely, the existence of large enough spectral barriers.

On the Large Time Galerkin Approximation of the Navier–Stokes Equations

In this article we are interested in the difficult problem of relating the large time behaviour of the Navier–Stokes equations to the large time behaviour of their Galerkin approximation. We restrict ourselves to the simplest situation dealing with stationary solutions and the question is the following one; if the computed approximation of the time dependent equations “seems to converge” to some limit as

The geometry of turbulent advection: sharp estimates for the dimensions of level sets

t \to \infty

Finite-dimensional attractor for the laser equations

, is the same true for the exact problem and are the limits related? We give here some sufficient conditions (reasonably easy to verify) which guarantee a positive answer to this question.

Application: The Chaffee—Infante Reaction—Diffusion Equation

Lower bounds on the fractal dimension of level sets of advecting passive scalars in turbulent fields are derived, in the limit that the scalar diffusivity kappa goes to zero. The main result is as follows: denote the Holder exponent of the velocity field u by zeta (u), with 0 or=d-1+ zeta (T)+ zeta (u), where d is the dimension of space. The validity of this bound depends on some conditions concerning the limit kappa to 0; when these are satisfied the bound is obtained throughout the range of zeta (u), between the smooth (but random) velocity field with zeta (u)=1 to the extremely rough field with zeta (u)=0. The derivation of the lower bound calls for the introduction of a measure on the level sets and a careful treatment of the singular limit of the scalar diffusivity going to zero. Together with the upper bounds which were derived previously, i.e. D<or=d-1/2+ zeta (u)/2 we discover, when there is no multiscaling, the scaling relation 2 zeta (T)+ zeta (u)=1, which then means that the lower and the upper bounds in fact coincide.

Application: The Kuramoto—Sivashinsky Equation

The laser equations of Risken and Nummedal (1968) govern the dynamics of a ring laser cavity. They form a system of hyperbolic, semilinear, damped and driven partial differential equations with periodic boundary conditions. The Lorenz system of ordinary differential equations is an invariant subsystem of the laser equations corresponding to solutions without spatial dependence. The authors prove that the laser system admits global weak solutions for arbitrary L2 initial data. Despite the absence of parabolic diffusion they prove that the laser system enjoys a remarkable property of hyperbolic smoothing for t to infinity : the universal attractor for the L2 evolution consists of Cinfinity functions. They show, moreover, that the universal attractor is finite dimensional and estimate its dimension.