Spectral Stability and Oscillations of Dynamical Systems, Boltzmann-Like Models

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This project aims to develop the mathematical tools that can help investigate stability and instability mechanisms affecting the solutions of partial differential equations. In particular, these tools can be used to assess stability of distinguished states of a nonlinear partial differential equation, which is a key step in understanding the behavior of the physical system modeled by the equation. Stability here means the robustness of the dynamics to perturbations in initial conditions from a particular state. The distinguished state may be a nonlinear wave, pattern, or coherent structure arising in applications such as optics, fluids, neuroscience, ecology, chemical reactions, or shallow water dynamics. The stability of the given state indicates its physical realizability, while any instability suggests more complex dynamics. Understanding the nature of such instabilities can be used as a starting point for understanding the organization of the nonlinear dynamics away from the unstable state. The project includes research activities that will train undergraduate studentThe investigator's goal is to generalize the oscillation-type results for eigenvalue problems that are associated with partial differential equation models. These range from nonlinear eigenvalue problems arising in spectral stability of shock profiles of hyperbolic systems of balance laws to problems in multi-dimensional spatial domains. In particular, he exploits a new set of ideas that cast general multi-dimensional problems in a dynamical systems framework and thus offer a new characterization of the underlying issues of existence and stability of solutions. One of the key directions of the project is the introduction of the Spatial Evolutionary System, which can be viewed as a far-reaching extension of the spatial dynamics framework to general multi-dimensional domains. Undergraduate students are engaged in the research of the project.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.