Showing papers by "Peter Constantin published in 1989"

Spectral barriers and inertial manifolds for dissipative partial differential equations

Abstract: 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.

Finite-dimensional attractor for the laser equations

Abstract: 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.

Construction of the Inertial Manifold

Abstract: Let Γ be the (n − l)-dimensional sphere {u|u = P n u,|u| = R} in P n H considered in Proposition 9.2. Let Σ be the integral manifold obtained with Γ as initial data: $$\sum { = \bigcup\limits_{t > o}^{} {s(t)\Gamma .} } $$ (10.1)

Application: The Chaffee—Infante Reaction—Diffusion Equation

Abstract: As an example of a parabolic reaction—diffusion equation with less stringent conditions than in Chapter 18, we briefly outline the construction of an inertial manifold for the Chaffee—Infante equation [H] in two dimensions: $$\frac{{\partial u}}{{\partial t}} - \Delta u + \lambda \left( {{u^3} - u} \right) = 0,\;\lambda > {\text{ }}0,\Omega = {\left[ { - \pi , + \pi } \right]^2} = {T^2},{\text{ periodic boundary conditions, }}u\left( 0 \right) = {u_0}$$ (19.1) (we do not restrict ourselves to odd periodic functions). For λ > 1, this equation admits multiple nonconstant steady states besides u = 0 and u = ± 1.