Euler Alignment, Nonlinear Conservation Laws and the Pressure-less System

University Of Maryland, College Park, College Park MD

Investigators

Abstract

Partial Differential Equations play a pivotal role in a wide range of applications, facilitating the study of many questions in physics, geometry, meteorology, biology, economics, and engineering sciences, to name a few. This project aims to advance the current understanding of fundamental questions that arise in the context of three canonical classes of nonlinear partial differential equations, which model (i) emergent phenomena; (ii) conservation laws; and (iii) the pressure-less early universe model. While these three classes of evolution equations are well-understood in the one-dimensional spatial setting, the questions of existence, regularity and large-time behavior of solutions for the more realistic multi-dimensional models are mostly open. The plan of this project is to develop novel paradigms to address these questions with emphasis on the multi-dimensional setting. This project also involves mentoring graduate students who will be involved in this research. This project is concerned with the following time-dependent partial differential equations in multiple spatial dimensions. (i) Euler Alignment. The system of Euler Alignment arises as the large crowd dynamics of the Cucker-Smale alignment model. The goal is to study the open question of existence of multidimensional strong solutions, subject to sub-critical initial data, and their large-time behavior with short-range communication kernels. (ii) Nonlinear Conservation Laws. Nonlinear scalar conservation laws admit a regularization effect, where the entropic time evolution of bounded data gains spatial regularity of fractional order s< 1/3. The 1/3 barrier, at least in two or more dimensions, remains an open problem. The project will address this open problem by refining a velocity averaging lemma adapted to deal with one-signed measures and quantify fractional regularity in proper Morrey spaces. (iii) The Pressure-less System. In one-dimension one encounters the inviscid Burgers’ equation. A standard embedding of the system into vanishing pressure is limited to one dimension by the formation of delta-shocks. The goal is to construct vanishing viscosity solutions of the Zeldovich pressure-less system in two or more dimensions. Note that for two or more dimensions, the equations form a non-conservative system. The development of such two-dimensional existence theory is based on spectral dynamics and priori bounds on the spectral gap coupled with compensated compactness arguments. 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.