Bifurcation, Stability, and Non-uniqueness in Ideal Fluids

University Of Pittsburgh, Pittsburgh PA

Investigators

Abstract

This project seeks to promote and advance the mathematical theory of free-surface water waves and compressible fluids, which host a wide range of interesting physical phenomena and whose dynamics is governed by the Euler equations. The main goals are to develop new or extend existing analytic techniques to establish existence and stability theory for steady water waves, and to investigate uniqueness properties for weak solutions to one-dimensional system of compressible gases. Progress in this project will enhance our understanding of the mathematics of ideal fluids and develop novel mathematical tools that can provide insight into truly nonlinear phenomena in partial differential equations. This research will also involve training and collaboration with graduate students and postdoctoral researchers. This project will bring new perspectives and develop novel approaches to make progress on some fundamental problems in the study of ideal fluids. Specifically, the PI will use a novel global bifurcation theoretic machinery to construct large-amplitude solitary water waves in presence of localized disturbances coming from either submerged objects or the bottom topography. The second topic of this project concerns stability of solitary water waves. The PI will study the spectral stability of multimodal gravity-capillary internal solitary waves and prove nonlinear transverse instability of gravity-capillary internal solitary waves. The final track of the project consists of the study of the non-uniqueness of entropy solutions to the compressible isentropic Euler system. The main ingredients and techniques involved in the study include bifurcation method, complex analysis, stability analysis, convex integration machinery, and the theory of hyperbolic conservation laws. 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.