Nonlinear stability of an undercompressive shock for complex Burgers equation

Though there is strong numerical evidence for the stability of undercompressive shocks, their stability has not been verified analytically. In particular, the energy methods used to analyze stability of standard shocks do not apply. Here, we present the first proof of stability for a particular undercompresive shock, the real Burgers shock considered as a solution of the complex Burgers equation. Our analysis is by direct calculation of the Green's function for the linearized equations, combined with pointwise estimates of nonlinear effects. A benefit of this method is to obtain fairly detailed information about the solution, includingL1 behavior, and rates of decay in different regions of space.

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