The Interplay of Evolution Partial Differential Equations and Their Model Equations

University Of Massachusetts Amherst, Amherst MA

Investigators

Abstract

In mathematical physics, equations that predict the behavior of many important physical systems (including water waves, signals in fiber optic cables, and quantum particles) have disparate scales that make the problem of understanding their solutions extremely challenging, be it through analysis or through computer simulations. However, the problem can be simplified by focusing on one salient aspect of the phenomenon and ignoring the others. The "model equations" that result from this process are generally easier to handle than the original equation. However, model equations are often derived using heuristic arguments, and attempts to confirm the accuracy of the models numerically can be inconclusive. One aim of this research project is to mathematically justify that model equations accurately describe the full equation. In some cases, analysis of solutions to such simplified model equations remains difficult, even to the point where basic facts about the behavior of such model equations (such as whether they can be solved for all times) are not known. A good theoretical understanding of these fundamental properties is important for providing confidence that the model in question is accurate. This project will explore ways in which an equation interacts with its model equations and vice versa, with the goals of a better understanding of both problems. This project deals with research involving partial differential equations (PDEs), specifically the relationship between evolution PDEs and model equations derived from them by asymptotic expansion methods. The first and most fundamental question of study from a mathematical standpoint is rigorous justification: to give a mathematical proof that a model equation derived from a formal argument provides a good approximation to the original PDE in the appropriate regime. This entails providing quantitative bounds on the errors incurred in approximation. Since such models often involve small parameters, one must have a good understanding of the well-posedness of the underlying problem for large initial data and over long time scales, both of which involve technical challenges. The other aspect of this project is to explore the idea of constructing, for a given equation known to arise from such a modeling procedure, an artificial equation constructed to force existence of the model equation for all time. This provides a new method for demonstrating global existence of solutions to equations which may be lacking structure for applying existing techniques, such as a coercive conserved energy. The first step is to demonstrate that this method applies in the context of nonlinear Schrodinger equations, which is one of the fundamental equations of mathematical physics. This project aims also to generalize the method to analyze other important model equations in which conserved quantities are either absent or otherwise unsuitable for use. A famous example of such an obstacle is the "super-criticality barrier" that prevents the establishment of global existence in a wide variety of problems.