Stability of Patterns

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AbstractLin and Schecter propose to use the Dafermos regularization of a system of conservation laws to approach difficult questions concerning systems of viscous conservation laws. The former is an artificial mathematical construct; the latter are ubiquitous in the sciences, where they represent conservation of mass, momentum, energy, etc. in many situations. Building on their earlier work, Lin and Schecter propose to complete their analysis of the spectrum of the linearized Dafermos operator. They propose to use this analysis to determine the stability of Riemann solutions as asymptotic states of viscous conservation laws. They also propose to investigate related issues that have arisen in the course of this work, including possible generalizations of the Exchange Lemma of geometric singular perturbation theory; extensions to third- and fourth-order regularizations of conservation laws; and a new approach to stability of Riemann solutions of systems of conservation laws without viscosity.In many areas of science and technology, various situations involving fluid flow, such as oil recovery and flow of thin liquid films used in manufacturing, can be mathematically modeled by equations called viscous conservation laws. The models become more tractable when one drops various terms, leaving only a system of conservation laws. For these equations one can often construct explicit solutions called Riemann solutions, that frequently involve jumps that move with varying speeds. An example from oil recovery using injection of water is a moving front that is mostly water on one side and mostly oil on the other; the water pushes the oil toward the well. One reason Riemann solutions are important is that it is believed that in many situations, solutions of viscous conservation laws, appropriately rescaled, tend to look more and more like Riemann solutions as time goes on. However, there are only a few, rather artificial situations is which this behavior is proved. A related fact is that we do not have good mathematical techniques to check whether Riemann solutions are stable, i.e., are really approached for a significant set of initial configurations of the viscous conservation laws. Lin and Schecter have developed a new approach to these issues using a different simplification of the viscous conservation laws, the so-called Dafermos regularization. This equation admits a smoothed-out version of the Riemann solution as a steady-state. In principle, one can check its stability by relatively familiar mathematical methods. Lin and Schecter plan to continue their work on the stability of these smoothed Riemann solutions, and to use this work to approach the physically relevant situation.