Rate of convergence in periodic homogenization for convex Hamilton–Jacobi equations with multiscales

Han, Yuxi; Jang, Jiwoong

doi:10.1088/1361-6544/acf17c

Cited by 2 publications

(1 citation statement)

References 12 publications

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“…Moreover, counterexamples are constructed there when n ≥ 3. (See [21,25,32,34,49,52,53,56] and the references therein for other related works and mathematical models. )…”

Section: Curvature G-equationmentioning

confidence: 99%

Lagrangian, game theoretic, and PDE methods for averaging G-equations in turbulent combustion: existence and beyond

Xin,

Yu,

Ronney

2024

Bull. Amer. Math. Soc.

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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 nonconvex and nonlinear. In the absence of coercivity and convexity, the 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 noncoercive, nonconvex, and nonlinear PDEs such as geometric (curvature-dependent) PDEs with advection.

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“…Moreover, counterexamples are constructed there when n ≥ 3. (See [21,25,32,34,49,52,53,56] and the references therein for other related works and mathematical models. )…”

Section: Curvature G-equationmentioning

confidence: 99%

Lagrangian, game theoretic, and PDE methods for averaging G-equations in turbulent combustion: existence and beyond

Xin,

Yu,

Ronney

2024

Bull. Amer. Math. Soc.

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show abstract

A convergence rate of periodic homogenization for forced mean curvature flow of graphs in the laminar setting

Jang

2024

Nonlinear Differ. Equ. Appl.

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Rate of convergence in periodic homogenization for convex Hamilton–Jacobi equations with multiscales

Abstract: We study the rate of convergence in periodic homogenization for convex Hamilton–Jacobi equations with multiscales, where the Hamiltonian H = H ( x , y , p ) : … Show more

Cited by 2 publications

References 12 publications

Lagrangian, game theoretic, and PDE methods for averaging G-equations in turbulent combustion: existence and beyond

Lagrangian, game theoretic, and PDE methods for averaging G-equations in turbulent combustion: existence and beyond

A convergence rate of periodic homogenization for forced mean curvature flow of graphs in the laminar setting

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