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