Nonlinear Dispersive Equations

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March 6, 2002PI: Gustavo Ponce. DMS-0140023Proposal title : Nonlinear Dispersive Equations. In recent years there has been a significant progress onthe study of qualitative behavior of solution to nonlineardispersive systems. The relation between dispersive and nonlinear effectshas motivated several remarkable works. Among them one findssharp local and global wellposedness theories under minimal regularity assumptions on the data, precise blow up results as well as ill-posedness ones. As a consequence our understanding of several phenomena in nature is now more complete. The principal investigator will focus on severalrepresentative problems connected with partial differentialequations arising in wave propagation and fluid mechanics. The wave propagation in diverse mediums and the interaction ofdifferent kind of waves can be modeled by complicated systems ofpartial differential equations. These systems involve many variableswhich depends of the particular physical setting of the problem. In many cases, these systems are too complicated to be treated with reasonable accuracy, thus one has to consider approximated models. The validity of these approximations is essential to further study of the physical phenomenon. One has to give precise conditions under which the qualitative behavior of the solution of these approximated models captures the features of the physical setting. It is here where the theoretical study of these solutions become essential. Thus, one starts with special solutions and its properties, stability, long time behavior, ill-posedness, etc. In situations where these theoretical results are unavailable one needs to start the study with appropriate numerical simulations.