Stability of solitons and long-term dynamics of fluids

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The main goal of the project is to study solutions of several important partial differential equations. These solutions describe physical phenomena, such as the dynamics of fluids, the motion of water waves, and gravitation, which can be observed in daily life. We aim to analyze these solutions rigorously and to recover quantitative and qualitative information about their behavior as mathematical theorems. The research project involves graduate students and postdoctoral associates, and connects ideas and methods in partial differential equations, numerical analysis, geometry, and harmonic analysis.The PI will study the long term dynamics of solutions of several important evolution equations, such as the Euler equations in 2 and 3 dimensions, water wave models, and the Einstein field equations of General Relativity. Six problems of interest are explicitly identified as part of the project. These problems concern the global stability of shear flows and vortices for the 2D Euler and the generalized SQG equations, the long term dynamics of water waves in a large box in 2D and 3D, and the global stability of solutions of the Einstein equations with matter.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.