CAREER: Large and Multi-Dimensional Solutions of Conservation Laws

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Understanding the properties of nonlinear Partial Differential Equations (PDE) is a fundamental challenge in modern pure and applied mathematics. The goal of the research projects is to gain new insightinto three specific areas of evolutionary PDE. (A) Large solutions of one-dimensional systems of conservation laws:The combined effects of large data and nonlinear effects pose a basic challenge as well as being of obvious importance in applications. Building on existing work this project aims at giving further examples of blowup solutions, understanding their stability properties, and identifying assumptions guaranteeing global existence of large solutions.(B) Existence and qualitative properties of solutions for particular systems of multi-dimensional systems of conservation laws. Multi-dimensional equations display an exceedingly richvariety of behaviors and there is currently no general existence result available for global solutions. Recent examples of blowup demonstrate that the small variation theory for one-dimensionalsystems cannot be generalized to several space dimensions. The research aims at a "bottom-up" approach where insight is obtained from specific cases. The systems will be chosen to provide simplified, but non-trivial, examples with strong structural constraints. The goal is to create a toolbox of methods that can be applied to more general systems. Both analytical and numerical tools will be employed to gain insight into the structure of the solutions.(C) The Navier-Stokes equations for fluid flow provide a basicmodel of importance in a wide range of applications. The goal of this project is to investigate three issues: formation of vacuums (cavitation), the effect of including temperature dependence in the transport coefficients (viscosities and heat conductivity), and multi-dimensional flows with large amplitudes. The various projects will also address the significant numerical challenges one faces in computing large or multi-dimensional solutions, and flows containing vacuums. Vice versa, exact solutionswill be used to benchmark various computational codes.The projects aims at a better understanding of how nonlinear mechanisms interact with one- or multi-dimensional effects in equations that are extensively used in physical models, ranging from properties of materials and to fluid flow and meteorology. While these issues are of independent theoretical interest they are also of obvious importance in applications to Science and Engineering. Several of the projects require development and testing of high performance computer codes with important applications in everyday simulations. A combined approach of analytical techniques, modeling, and numerical calculations will enhance basic understanding of fluid flow and its applications.