Parabolic problems which are ill-posed in the zero dissipation limit

We consider linear second-order parabolic Cauchy problems which are ill-posed in the limit of zero dissipation. We show that for smooth coefficients the principle of frozen coefficients is applicable. That is, the growth rate predicted by this principle for a small viscosity parameter is correct. We discuss the limitation of this principle and show by examples that if the coefficients are varying on a scale that is proportional to the viscosity parameter then the problem need not be unstable. For nonlinear problems, we can conclude that smooth solutions in regions where the characteristics are complex are not stable. We show that small perturbations of smooth complex initial data for the viscous Burgersʹ equation result in a O(1) change of the solution. For two nonlinear examples, numerical calculations demonstrate that for smooth initial data the solutions rapidly form nonsmooth solutions—dissipative structures—that stabilizes the inherent instabilities.