Using Global Invariant Manifolds to Understand Metastability in the Burgers Equation with Small Viscosity

Abstract

The large-time behavior of solutions to Burgers equation with small viscosity is described using invariant manifolds. In particular, a geometric explanation is provided for a phenomenon known as metastability, which in the present context means that solutions spend a very long time near the family of solutions known as diffusive N-waves before finally converging to a stable self-similar diffusion wave. More precisely, it is shown that in terms of similarity, or scaling, variables in an algebraically weighted L 2 space, the selfsimilar diffusion waves correspond to a one-dimensional global center manifold of stationary solutions. Through each of these fixed points there exists a one-dimensional, global, attractive, invariant manifold corresponding to the diffusive N-waves. Thus, metastability corresponds to a fast transient in which solutions approach this "metastable" manifold of diffusive N-waves, followed by a slow decay along this manifold, and, finally, convergence to the self-similar diffusion wave.

Linear inviscid damping and enhanced dissipation for the Kolmogorov flow

“…26) and (5.27) with E j defined by (5.5) for j = 0, 1. It holds that (1) if ω e L 2 ≤ 1 and g L 2 ≤ 1, then…”

“…For general shear flows, the linear inviscid damping is also a difficult problem Date: November 7, 2017. 1 due to the presence of nonlocal part u ′′ (y)∂ x (−∆) −1 . In such case, the linear dynamics is associated with the singularities at the critical layers u = c of the solution for the Rayleigh equation…”

Dynamics of Partial Differential Equations

“…In [BW11b], Beck and I proposed a dynamical systems explanation for similar families of metastable states in Burgers equation. In [BW11b], Beck and I proposed a dynamical systems explanation for similar families of metastable states in Burgers equation.…”

“…These states, and their importance for the dynamics of the system, were first systematically investigated by Kim and Tzavaras in [KT01], where they are called "diffusive N-waves." The final step in our construction was to show that "almost every" (in a sense made precise in [BW11b]) initial condition gives rise to a solution of Burgers equation which approaches one of these metastable manifolds on a short time scale. These are analogous to the family of Oseen vortices in the 2D NSE.…”

Degenerate Hyperbolic Conservation Laws with Dissipation: Reduction to and Validity of a Class of Burgers-Type Equations

“…In fact, this global well posedness fails and there are solutions which blow up in finite time. Indeed, system (11.3) (with ν 21 = 0) is equivalent to the so-called strongly damped Boussinesq equation 4) for which the finite-time blow up is known (see Theorem 4 in Levine [20]). In summary, although we believe that the strongly damped Boussinesq equation (11.4) is a correct modulation equation for the non-semisimple case, its validity does not follow directly from the theory developed in this paper and therefore remains an open question.…”