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Radial Strichartz, normal forms and global dynamics of nonlinear dispersive equations - 3
[2018 KAIX School in PDE]
Date: 2018-06-29
Speaker : Kenji Nakanishi (RIMS)
Abstract : Nonlinear dispersive equations are partial differential equations describing evolution of waves in various physical contexts, where the dynamics is governed by dispersion and non-linear interactions of waves. Typical examples are the nonlinear Schrodinger equation and the KdV equation. Depending on the competition between the dispersion and the inter-actions, the solutions exhibit various types of behavior in space-time, such as scattering, solitons and blow-up. Recent developments in the mathematical analysis of those PDEs have enabled us to classify all possible behavior of solutions and to predict it from the ini- tial data in some simple settings for model equations, where the global Strichartz estimates have played key roles, describing the dispersive nature of solutions by space-time integrabil- ity for linearized equations. In those model equations, however, nonlinear interactions tend to degenerate in high order for small amplitude of solutions, in order to avoid too strong competition with the dispersion. For more realistic equations with stronger interaction, the Strichartz estimate tends to be insufficient for us to control nonlinear solutions globally, and we often need much more elaborate multilinear estimates taking account of dispersion and interactions in detail. Nevertheless, in some cases, the Strichartz estimate turns out to be powerful for nonlinearity of lower order, if the solution is restricted by spherical symmetry or regularity, and if we combine it with certain quadratic transformations called normal forms. In this lecture, I will talk about the improved Strichartz estimates, the normal forms, and the global-in-time analysis of small and large solutions for the nonlinear dispersive equations, in particular the Zakharov system and the Gross-Pitaevskii equation.
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