Analysis of Properties of Effective Hamiltonians with Applications

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Many physical systems (e.g. composite materials and turbulent combustion) in practical applications involve small structures. It is important to “average” those small scales through suitable mathematical models (called “homogenization”) and investigate resulting macroscopic structures (“effective quantities”) that are approximations of original systems. A particularly interesting and physically relevant question is to understand how those effective quantities depend on original physical systems. Effective Hamiltonian arises from homogenizing Hamilton-Jacobi equations, which are partial differential equations (PDE) that play fundamental roles in control theory and the modeling of things like crystal growth, flame propagations, etc. For example, effective Hamiltonian serves as an important approach to model turbulent flame speed, one of the most important quantities in turbulent combustion (e.g., the spread of wild fire fanned by strong wind, the burning inside an engine), via the popular G-equation model. One goal of this project is to understand how the turbulent flame speed depends on the strength of the flow (e.g., wind velocity) and other important physical quantities like curvature effects, which could provide theoretical justifications for phenomena observed in experiments. The curvature effect is a phenomenological way to capture variance of burning temperature along the flame front. The project provides research training opportunities for graduate students.The first part of the project aims to identify optimal convergence rate in periodic homogenization of Hamilton-Jacobi equations. As the principal investigator and his collaborators have revealed in previous works, the optimal convergence rate is closely related to the shape of corresponding effective Hamiltonian. The second and third part of the project focus on studying important properties of effective Hamiltonian from perspectives of both the mathematical interest and practical applications in turbulent combustion. Although the existence of effect Hamiltonian has been established in many situations since the fundamental work of Lions-Papanicolaou-Varadhan (1987), understanding finer properties of effective Hamiltonian remains a major open problem in homogenization theory due to limitations of standard methods in PDEs. There are deep connections between properties of effective Hamiltonian and dynamical systems. In particular, the principal investigator needs to employ substantial tools/methods from ergodic theory, Aubry-Mather theory/weak KAM theory. In addition, dynamics associated with two-person zero-sum differential games is also important in dealing with non-convex Hamiltonian and the presence of curvature term (curvature effect).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.