Regularity and Critical Thresholds in Nonlinear Transport-Diffusion Equations

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We will use modern mathematical tools complemented by novel computational simulations to examine the phenomena of regularizing effects, critical thresholds, time decay, entropy stability, and scaling. The project will address the following questions: (i) Transport-diffusion equations: How do diffusion and entropy dissipation dictate the regularizing effect in such equations? (ii) Eulerian dynamics: How does the competition between rotation and pressure forcing determine the overall stability? (iii) Chemotaxis and related bio-related transport-diffusion models: How should we interpret the solutions beyond their critical time? (iv) Hierarchical decompositions of images: How can the inverse scale space be adapted to the data for effective image processing? The principal investigator and his collaborators plan to pursue the development of new analytical and computational tools to explore transport-diffusion models, which are expected to contribute to understanding of the dynamics of realistic models in a variety of applications. The goal of this project is to study the persistence of global features in nonlinear transport-diffusion equations, which arise in a wide variety of applications. Examples include nonlinear conservation laws with degenerate diffusion, which model sedimentation, traffic flows, and data-driven applications in image processing; the ubiquitous Eulerian dynamics governing a range of phenomena from the small scale of semi-conductors through the largest scale of star formation; and chemotaxis models found in biological applications. We focus our attention on the unifying mathematical content of the underlying transport-diffusion equations. Of primary interest are problems with critical regularity properties that hinge on a borderline balance between the nonlinear convection mechanisms, the nonlinear diffusion processes, and the possibly nonlinear forcing driving such a system.