An Analytical Proof of the Linear Stability of the Viscous Shock Profile of the Burgers Equation with Fourth-Order Viscosity

Abstract

In this paper we establish the exponential decay of solutions of the equation u_t + \varphi(x) u_x = - \partial_x^4 u in an exponentially weighted norm. Here $\varphi(x)$ is the viscous shock profile corresponding to the Burgers equation with fourth-order viscosity: u_t + u u_x = - \partial_x^4 u. Because of the fact that the profile is not monotone, showing the stability is nontrivial. We extend the techniques of Koppel and Howard, (Adv. Math. 18 (1975), pp. 306-358), techniques that they employ to prove the existence of the viscous shock profile, and we use the techniques to prove the stability of the viscous shock profile. We have previously shown that the viscous shock profile is a stable solution in an exponentially weighted norm by making use of numerical results. The main advantage of our current method is that it is analytical. One sees more clearly what properties of the viscous shock profile cause it to be a stable solution of the PDE.