REGULARIZATION OF THE BACKWARDS KURAMOTO-SIVASHINSKY EQUATION

Abstract

<p>We are interested in backward-in-time solution techniques for evolutionary PDE problems arising in fluid mechanics. In addition to their intrinsic interest, such techniques have applications in recently proposed retrograde data assimilation. As our model system we consider the terminal value problem for the Kuramoto-Sivashinsky equation in a l D periodic domain. The Kuramoto-Sivashinsky equation, proposed as a model for interfacial and combustion phenomena, is often also adopted as a toy model for hydrodynamic turbulence because of its multiscale and chaotic dynamics. Such backward problems are typical examples of ill-posed problems, where any disturbances are amplified exponentially during the backward march. Hence, regularization is required to solve such problems efficiently in practice. We consider regularization approaches in which the original ill-posed problem is approximated with a less ill-posed problem, which is achieved by adding a regularization term to the original equation. While such techniques are relatively well-understood for linear problems, it is still unclear what effect these techniques may have in the nonlinear setting. In addition to considering regularization terms with fixed magnitudes, we also explore a novel approach in which these magnitudes are adapted dynamically using simple concepts from the Control Theory.</p>