Research Projects

Hyperbolic conservation laws form a foundational mathematical framework for modeling dynamic physical systems governed by the conservation of mass, momentum, and energy. These systems, expressed as nonlinear partial differential equations (PDEs), describe phenomena with wave-like transport and nonlinear interactions. Originally developed for compressible, inviscid fluids via the Euler equations, such models now extend to areas like electromagnetism, elasticity, traffic flow, and biological transport. A key feature of hyperbolic systems is their tendency to form discontinuities—shock waves—even from smooth initial conditions. These discontinuities arise from nonlinear wave interactions and persist over time, making classical smooth solutions inadequate. Instead, weak formulations are used, where solutions satisfy the equations in an integrated sense. However, weak solutions are generally non-unique. To address this, entropy conditions—motivated by thermodynamics—are imposed to select physically relevant solutions and to derive stability estimates. Despite progress in one-dimensional and scalar cases, many challenges remain open in higher dimensions. Notably, the global well-posedness of multidimensional compressible Euler equations is unresolved, and entropy solutions may still be non-unique. Recent advances in the field are focused on three key directions: Generalized Solution Concepts: To handle non-uniqueness and fine-scale oscillations, measure-valued and statistical solution frameworks have been developed. These represent solutions as distributions or ensembles, effectively capturing subgrid phenomena and uncertainty inherent in multidimensional systems. Structure-Preserving Numerical Methods: Reliable simulation of hyperbolic systems requires numerical schemes that respect the underlying physical and mathematical structures. These include conservation, entropy dissipation, steady-state preservation, and compatibility with constraints like incompressibility or divergence-free fields. Structure-preserving schemes aim to maintain these properties at the discrete level, ensuring stability and accuracy even in complex regimes. Data-Driven Approaches: Machine learning, particularly Physics-Informed Neural Networks (PINNs), has introduced new ways to approximate PDE solutions using data and embedded physical laws. While PINNs face challenges near shocks and discontinuities, hybrid methods that integrate them with structure-preserving schemes show promise for handling complex, multiscale systems in high-dimensional or data-limited settings. Project Objective This project seeks to develop a unified framework for the analysis and computation of hyperbolic conservation laws by integrating PDE theory, numerical analysis, and machine learning. The aim is to construct tools capable of capturing the rich multiscale dynamics of these systems while ensuring physical consistency and computational robustness. By bridging theoretical insights with data-driven and structure-aware numerical techniques, the project aspires to enable accurate, stable simulations of nonlinear hyperbolic problems across a wide array of scientific and engineering domains.