On the exponentially slow motion of a viscous shock

Abstract

Using formal asymptotic methods, we study the internal layer behavior associated with the following viscous shock problem in the limit ε → 0: (Formula Presented.) The convex nonlinearity f(u) satisfies f(α) = f(–α). For the steady problem, we show that the method of matched asymptotic expansions fails to uniquely determine the location of the equilibrium shock layer solution. This indeterminacy, resulting from neglecting certain exponentially small effects, is eliminated by using the projection method, which exploits certain properties of the spectrum associated with the linearized operator. For the time dependent problem, we show that the viscous shock, which is formed from initial data, drifts towards the equilibrium solution on an exponentially long time interval of the order O(eC/ε), for some C > 0. This exponentially slow behavior is analyzed by deriving an equation of motion for the location of the viscous shock. For Burgers equation (f(u) = u2/2), the results give an analytical characterization of the slow shock layer motion observed numerically in Kreiss and Kreiss; see [11]. We also show that the shock layer behavior is very sensitive to small changes in the boundary operator. In addition, using a WKB‐type method, the slow viscous shock motion is studied numerically for small ε, the results comparing favorably with corresponding analytical results. Finally, we relate the slow viscous shock motion to similar slow internal layer motion for the Allen‐Cahn equation. Copyright © 1995 Wiley Periodicals, Inc., A Wiley Company