Showing papers by "Peter Constantin published in 1988"

Navier-Stokes equations

Abstract: Both an original contribution and a lucid introduction to mathematical aspects of fluid mechanics, Navier-Stokes Equations provides a compact and self-contained course on these classical, nonlinear, partial differential equations, which are used to describe and analyze fluid dynamics and the flow of gases.

Integral Manifolds and Inertial Manifolds for Dissipative Partial Differential Equations

Abstract: Contents: Introduction.- Presentation of the Approach and of the Main Results.- The Transport of Finite Dimensional Contact Elements.- Spectral Blocking Property.- Strong Squeezing Property.- Cone Invariance Properties.- Consequences Regarding the Global Attractor.- Local Exponential Decay Toward Blocked Integral Surfaces.- Exponential Decay of Volume Elements and the Dimension of the Global Attractor.- Choice of the Initial Manifold.- Construction of the Inertial Mainfold.- Lower Bound for the Exponential Rate of Convergence to the Attractor.- Asymptotic Completeness: Preparation.- Asymptotic Completeness: Proof of Theorem 12.1.- Stability with Respect to Perturbations.- Application: The Kuramoto-Sivashinsky Equation.- Application: A Nonlocal Burgers Equation.- Application: The Cahn-Hilliard Equation.- Application: A parabolic Equation in Two Space Variables.- Application: The Chaffee-Infante Reaction Diffusion Equation.- References.- Index.

Local smoothing properties of dispersive equations

Abstract: Is it possible for time evolution partial differential equations which are reversible and conservative to smooth locally the initial data? For the linear wave equation, for instance, the answer is no. However, in [10] T. Kato found a local smoothing property of the Korteweg-de Vries equation: the solution of the initial value problem is, locally, one derivative smoother than the initial datum. Kato's proof uses, in a curcial way, the algebraic properties of the symbol for the Korteweg-de Vries equation and the fact that the underlying spatial dimension is one. Actually, judging from the way several integrations by parts and cancellations conspire to reveal a smoothing effect, one would be inclined to believe this was a special property of the K-dV equation. This is not, however, the case. In this paper, we attempt to describe a general local smoothing effect for dispersive equations and systems. The smoothing effect is due to the dispersive nature of the linear part of the equation. All the physically significant dispersive equations and systems known to us have linear parts displaying this local smoothing property. To mention only a few, the K-dV, Benjamin-Ono, intermediate long wave, various Boussinesq, and Schrodinger equations are included. We study, thus, equations and systems of the form

The Beltrami spectrum for incompressible fluid flows

Abstract: Recently V. Yakhot, S. Orszag, and their co-workers have suggested that turbulent flows in various regions of space organize into a coherent hierarchy of weakly interacting superimposed approximate Beltrami flows. A mathematical framework is developed here to study organized Beltrami hierarchies in a systematic fashion. This framework is applied to several important classes of examples with universal Beltrami hierarchies. An analysis of the persistence of such Beltrami hierarchies is also presented for general solutions of the Navier-Stokes equations.

Existence of solutions to the Stommel-Charney model of the Gulf Stream

Abstract: The existence of weak solutions to the equations proposed by Stommel [Trans. Amer. Geophys. Union, 29 (1948), pp. 202–206] and Charney [Proc. Nat. Acad. Sci. U.S.A., 41 (1955), pp. 731–740] as a model of the Gulf Stream are established by means of the method of artificial viscosity.

On the evolution of nearly circular vortex patches

Abstract: Recently, the classical problem of the evolution of patches of constant vorticity was reformulated as an evolution equation for the boundary of the patch. We study this equation in the neighborhood of the circular vortex patch and introduce a hierarchy of area-preserving nonlinear approximate equations. The first of these equations is shown to have a rich rigid structure: it possesses an exhaustive increasing sequence of linear invariant manifolds of arbitrarily large finite dimensions. On each of these manifolds the equation can be written as an explicit finite system of ordinary differential equations. Solutions of these ODEs, starting from arbitrarily small neighborhoods of the circular vortex patch, are shown to blow up.