Regularization of the backward-in-time Kuramoto–Sivashinsky equation

We are interested in solution techniques for backward-in-time evolutionary PDE problems arising in fluid mechanics. In addition to their intrinsic interest, such techniques have applications in the recently proposed retrograde data assimilation. As our model system we consider the terminal value problem for the Kuramoto–Sivashinsky equation in a 1D periodic domain. Such backward problems are typical examples of ill-posed problems, where any disturbances are amplified exponentially during the backward march. Hence, regularization is required in order to solve such a problem efficiently in practice. We consider regularization approaches in which the original ill-posed problem is approximated with a less ill-posed problem obtained by adding a regularization term to the original equation. While such techniques are relatively well understood for simple linear problems, in this work we investigate them carefully in the nonlinear setting and report on some interesting universal behavior. In addition to considering regularization terms with fixed magnitudes, we also mention briefly a novel approach in which these magnitudes are adapted dynamically using simple concepts from the Control Theory.