INTRODUCTION TO CONSERVATION LAWS AND THEIR STOCHASTIC COUNTERPARTS

Abstract

This master’s thesis centers on studying conservation laws and their stochastic versions. The study starts by deeply examining first-order partial differential equations (PDEs) with focus given to method of characteristics. This analytical way sets up comprehension for primary ideas about conservation laws, how they behave in deterministic systems. Moving forward, the study progresses to delve into hyperbolic conservation laws, where the definitions of conservation laws are rigorously studied alongside the concepts of weak solutions and their formulation. Notably, the Rankine-Hugoniot conditions play a crucial role in solving specific Riemann problems, with a notable example being the analysis of Burgers’ equation and the intricate understanding of shock formation phenomena within these systems. Building upon these foundational principles, the thesis incorporates insights from Sobolev spaces, as elucidated by Evans, further enhancing the analytical tools and techniques available for studying conservation laws and their dynamics. These mathematical spaces provide a rich framework for investigating the properties and behaviors of solutions to conservation laws under various conditions. The upcoming phase of the thesis will transition into the realm of stochastic conservation laws, where the focus will shift towards analyzing solutions in the presence of stochastic influences, including multiplicative noise. This analysis will encompass exploring the existence and uniqueness of solutions within stochastic frameworks, paving the way for a comprehensive understanding of how randomness and uncertainty impact the dynamics of conservation laws.