Regularity Versus Singularity Formation in Nonlinear Partial Differential Equations

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This project is concerned with various aspects of the theory of partial differential equations related to the dichotomy between regularity and singularity formation, from both a qualitative and a quantitative point of view. The topics to be studied as part of the project are universal in the field of mathematics in the sense that the same paradigm appears, for instance, in geometry, mathematical physics, or dynamical systems. The fundamental issue the project aims to understand is the interplay, for a given system of partial differential equations, between the existence and regularity of solutions versus the appearance of singularities. This type of question is of major importance for both time-dependent and time-independent equations. The project provides research training opportunities for graduate students and postdoctoral researchers.The project lies at the interface of several areas of mathematics, such as Partial Differential Equations, Geometric Measure Theory, Geometric Analysis, and Harmonic Analysis, with many problems under consideration being motivated by relevant applications to liquid crystals, fluid dynamics, statistical physics, and various areas of differential geometry. The Principal Investigator (PI) plans to pursue research in the regularity theory and geometric properties for partial differential equations arising in the theory of minimal surfaces and harmonic maps with free boundaries, by investigating thoroughly a new model where the boundary effects are prevalent. The theory of nonlocal equations makes it possible to study problems which seem to be local at first sight but are intrinsically nonlocal, such as connected sums on the boundary of manifolds or the construction of blow-up solutions of the harmonic map flow with free boundary. The PI plans to continue this fruitful line of research. Related to the latter, the theory of a particularly important class of degenerate/singular partial differential equations connected to the fractional Laplacian has seen some recent developments. Building on recent results, the PI plans to study degenerate/singular equations in fluid dynamics involving variable coefficients. Several models in compressible fluids are a major output of this line of investigation. A possible way to understand the singularities and degeneracies in a system is via geometric microlocal analysis. A useful tool developed over the years by the PI and collaborators, called parabolic gluing, offers a very versatile method to construct new objects, possibly singular, in parabolic equations and geometric flows. As part of this project, the PI plans to continue to develop this method to investigate bubbling phenomena in several parabolic equations coming from physics (fluid dynamics, in particular) and geometry (curvature flows and complex flows, for example).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.