Generic Singularities and Fine Regularity Structure for Nonlinear Partial Differential Equations

North Carolina State University, Raleigh NC

Investigators

Abstract

Numerical solutions of nonlinear partial differential equations (PDE) play a crucial role in a variety of applications. However, if the solutions do not have sufficient regularity, the performance of numerical algorithms and their computational accuracy is a challenging issue. In this project, the PI will study some fundamental questions regarding the regularity of solutions for various classes of PDE modeling nonlinear waves, also in the presence of nonlocal source terms. A major focus of the research will be on the emergence of singularities, such as shock waves, for generic solutions. The expected results will provide an accurate asymptotic description of how new singularities are formed, and how they interact with each other, valid for almost all initial data. This will lead to a new class of numerical schemes, with high-order accuracy and a wide range of applications. In addition, the project will provide a training ground for both undergraduate and graduate students. This research project contains two main parts. The first part is concerned with the fine regularity structure of solutions to Hamilton-Jacobi equations. Here, the goals are to (i) deepen the analysis of the metric entropy of sets of solutions to Hamilton-Jacobi equations, study SBV regularity, and develop new techniques that cover a wider class of equations; and (ii) investigate the fine properties of generalized monotone functions, study the propagation of singularities, and establish new regularity estimates and controllability results for Hamilton Jacobi equations. The second part of the project will focus on generic singularities for nonlinear balance laws, also in the presence of nonlocal terms. For a wide class of such equations, it is well known that solutions with smooth initial data can lose regularity in finite time. However, they can be extended in a weak sense beyond the time when the first derivatives blowup. A major difficulty in understanding entropy weak solutions is that nonlocal source terms have a huge influence on singularity formation. For this reason, many interesting properties of entropy weak solutions, such as uniqueness, shock formation, shock interactions, and the structure of singularities, are still far from being well understood. In this part of the project the PI aims to (i) develop a quantitative analysis of the number of shocks for entropy weak solutions and study their generic regularity for various models of nonlinear waves and for nonlinear hyperbolic systems of conservation laws; and (ii) provide a detailed description of shock formation and wave breaking of entropy weak solutions for nonlocal balance laws as well as trace their impact on BV regularity and stability results. The approach that will be pursued, taking advantage of the piecewise regularity of solutions, seeks to reduce an equation defined on a space with low regularity to an equation on a more regular space coupled with an ODE on a finite dimensional manifold. In cases where the solutions of interest are known to be generically piecewise smooth, this can have an impact in the broader field of numerical analysis, suggesting new high-order computational algorithms. 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.