Nonlinear Transport, Degenerate Diffusion, Critical Regularity and Self-Organized Dynamics

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The project is devoted to the following five aspects of nonlinear time-dependent problems. (i) Critical regularity in Eulerian dynamics: we will use spectral dynamics to investigate a new framework for vanishing viscosity solutions of the pressure-less Euler equations, for global regularity of the Euler-Poisson equations subject to sub-critical initial data, and the long-time regularity of the shallow-water driven by irrotational forcing. (ii) Entropy stability and well-balanced shallow-water schemes: we will develop, analyze and implement a new class of well-balanced schemes for the shallow-water equations. (iii) Self-organized dynamics: we will study the long-time behavior of models driven by velocity-alignment and address two interrelated issues. When does flocking occur with local interactions, depending on the connectivity of the underlying graph, and how is it realized in hydrodynamic models of flocking? We will also explore new models of self-organized dynamics in which inter-particle communication is scaled by their relative distance. (iv) Regularizing effects in quasi-linear transport-diffusion equations: we will continue our ongoing research on regularizing effects using velocity averaging in the concrete setups of nonlinear scalar conservation laws and certain systems which admit an entropic kinetic formulation. (v) Integro-differential equations for multi-scale decomposition of images: we will study the localization properties of new multi-scale integro-differential equations for image de-noising and de-blurring. The ultimate goal of this project is to construct, analyze and simulate time-dependent problems which are governed by nonlinear Partial Differential Equations (PDEs) and develop related novel computational schemes. The underlying equations involve nonlinear transport models, self-organized dynamics, and possibly different small scale decompositions into particle dynamics, kinetic distributions, or intensity of pixels; they arise in diverse applications, including fluid dynamics, collective behavioral sciences, and image processing and de-noising. We will focus on the unifying mathematical content of the equations, using a synergy of modern analytical tools and novel computational algorithms, to study the persistence of global features in these equations. The project provides a great educational experience through research for the graduate students and postdoctoral fellows involved.