Margaret Beck : Using global invariant manifolds to understand metastability in Burgers equation with small viscosity.

Finding globally stable states can provide useful information about the behavior of solutions to PDEs: for any initial condition, the solution will eventually approach such a state. However, in some cases, the solution can exhibit long transients in its approach to the state. If the transient is long enough, it may be this behavior, rather than the limiting behavior, that is observed in numerical simulations or experiments. This is referred to as "metastability" and has been found, for example, in the 2D Navier-Stokes equations with small viscosity. A similar phenomenon has been seen in Burgers equation, which can be explained using global invariant manifolds. More precisely, it is shown that in terms of similarity, or scaling, variables there exists a one-dimensional global center manifold of stationary solutions. Through each of these fixed points there exists a one-dimensional, global, attractive, invariant manifold. Metastability corresponds to a fast transient in which solutions approach this 'metastable' manifold, followed by a slow decay along this manifold, and, finally, convergence to the globally stable state.

• Category: Presentations
• Duration: 01:34:47
• Date:  February 18, 2009 at 4:25 PM
• Views: 89
• Tags:   seminar, Department of Mathematics Seminar