Lagrangian, Game Theoretic and PDE Methods for Averaging G-equations in Turbulent Combustion: Existence and Beyond
arxiv(2024)
摘要
G-equations are popular level set Hamilton-Jacobi nonlinear partial
differential equations (PDEs) of first or second order arising in turbulent
combustion. Characterizing the effective burning velocity (also known as the
turbulent burning velocity) is a fundamental problem there. We review relevant
studies of the G-equation models with a focus on both the existence of
effective burning velocity (homogenization), and its dependence on physical and
geometric parameters (flow intensity and curvature effect) through
representative examples. The corresponding physical background is also
presented to provide motivations for mathematical problems of interest.
The lack of coercivity of Hamiltonian is a hallmark of G-equations. When
either the curvature of the level set or the strain effect of fluid flows is
accounted for, the Hamiltonian becomes highly non-convex and nonlinear. In the
absence of coercivity and convexity, PDE (Eulerian) approach suffers from
insufficient compactness to establish averaging (homogenization). We review and
illustrate a suite of Lagrangian tools, most notably min-max (max-min) game
representations of curvature and strain G-equations, working in tandem with
analysis of streamline structures of fluid flows and PDEs. We discuss open
problems for future development in this emerging area of dynamic game analysis
for averaging non-coercive, non-convex, and nonlinear PDEs such as geometric
(curvature-dependent) PDEs with advection.
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