Strongly Nonlinear Dispersive Flows

This project investigates the fundamental mathematical principles governing how waves move and interact in complex environments. These phenomena are central to a wide range of physical systems, including the motion of water waves in the oceans, the propagation of electromagnetic waves, and the dynamics of gravitation throughout the universe. By developing new mathematical methods to understand how such waves evolve over very long time scales, this research offers deeper insight into the stability, structure, and long-term behavior of complex natural systems. This work serves the national interest by advancing the progress of science and contributing to national prosperity and public welfare. In particular, the mathematical insights developed through this research enhance our ability to model and predict the evolution of ocean surface waves and related dispersive phenomena, with potential applications to maritime safety, coastal resilience, environmental forecasting, and engineering design. More broadly, the development of rigorous analytical frameworks for complex wave dynamics strengthens the mathematical foundations underlying a variety of scientific and technological disciplines. Furthermore, the project is dedicated to the education and training of the next generation of American scientists. By integrating graduate and postdoctoral researchers into cutting-edge mathematical research and developing advanced university curricula, the project ensures a robust and technically skilled workforce capable of addressing complex challenges in science and technology. The project spans an array of topics in nonlinear partial differential equations. The problems investigated are all associated to the field of nonlinear dispersive equations, focusing on models derived from fluid dynamics, electromagnetism, and general relativity, for which wave propagation and interaction are the leading evolution mechanisms. This work also has deep connections to related areas such as geometry, harmonic analysis, complex analysis and microlocal analysis. The primary goal is to establish a rigorous mathematical framework for understanding nonlinear wave interactions across multiple temporal scales, with a particular emphasis on long-time global dynamics and scattering phenomena. The research focuses on four central objectives: the analysis of long-time dynamics in strongly nonlinear dispersive flows where nonlinear effects outweigh dispersion; the study of free boundary problems in fluid dynamics, specifically the evolution of water waves; the investigation of completely integrable systems and their associated inverse scattering theory; and the analysis of geometric nonlinear wave equations. The methodology draws upon and synthesizes advanced techniques from harmonic analysis, microlocal analysis, differential geometry, and complex analysis, with the goal of developing new analytical frameworks and mathematical tools of broad applicability across nonlinear partial differential equations and related areas of mathematical physics. The anticipated contributions of this research include progress toward the resolution of long-standing questions concerning the stability and singularity formation in nonlinear wave equations, thereby advancing the state of the art in the field of nonlinear analysis and deepening the mathematical understanding of complex phenomena arising in physical models. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria. NSF Award ID: 2554866 | Program: 01002627DB NSF RESEARCH & RELATED ACTIVIT | Principal Investigator: Daniel Tataru | Institution: University of California-Berkeley, BERKELEY, CA | Award Amount: $300,000 View on NSF Award Search: https://www.nsf.gov/awardsearch/show-award/?AWD_ID=2554866 View on Research.gov: https://www.research.gov/awardapi-service/v1/awards/2554866.html