Head and Tail Speeds of Mean Curvature Flow with Forcing

Gao, Hong-Wei; Kim, In-Won

doi:10.1007/s00205-019-01423-3

Cited by 6 publications

(5 citation statements)

References 20 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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“…We would like to mention that there are quite a few studies regarding the homogenization or large time limit of mean curvature motion law given by v n = a(x) − κ for a positive Z n -periodic function a(x). See ( [20,4,3,5,6,15,16], etc) and reference therein. When a(x) satisfies a special coercivity condition, homogenization has been proved in [20] for all dimensions.…”

Section: Figure 2: Curvature Effectmentioning

confidence: 99%

Existence of effective burning velocity in cellular flow for curvature G-equation

Gao¹,

Long²,

Xin³

et al. 2022

Preprint

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G-equation is a popular level set model in turbulent combustion, and becomes an advective mean curvature type evolution equation when curvature of a moving flame in a fluid flow is considered:Here d > 0 is the Markstein number and the positive part () + is imposed to avoid negative laminar flame speed that is non-physical. Assume that V : R 2 → R 2 is a typical steady two dimensional cellular flow, e.g, V (x) = A (DH(x)) ⊥ for the stream function H(x 1 , x 2 ) = sin(x 1 ) sin(x 2 ) and any flow intensity A > 0. We prove that for any unit vector p ∈ R 2 ,for a constant C depending only on d and V when G(x, 0) = p • x.The effective Hamiltonian H A (p) corresponds to the effective burning velocity (turbulent flame speed) in physics literature, which we conjecture to grow like O A log A as A increases. Our proof combines PDE methods with a dynamical analysis of the Kohn-Serfaty deterministic game characterization of the curvature G-equation while utilizing the special structure of the cellular flow.

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“…A counter-example to homogenization under the first condition can be constructed in 2-d periodic and laminar medium in the spirit of [6,7]. As explored by [6,8,16], informally speaking, non-homogenization at a given direction should imply pinning at a transversal direction (in 2-d). The example of [6] Our result Theorem 1.1 is a step towards addressing this difficult, and more general, open issue.…”

Section: Physical Motivation and Related Open Problemsmentioning

confidence: 99%

“…As mentioned before, our result is analogous to theirs, but in random media. In the direction of understanding the nature of non-homogenization, and potentially splitting non-flat traveling front solutions into traveling waves of multiple speeds, Kim and Gao [16] recently showed the existence of head and tail speeds that depend continuously on the normal direction and construct maximal and minimal speed traveling wave solutions in laminar media by a new proof.…”

Section: Introductionmentioning

confidence: 99%

Mean curvature flow with positive random forcing in 2-d

Feldman

2019

Preprint

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We consider the forced mean curvature flow in 2-d, finite range of dependence and positive random forcing. We prove flatness and existence of effective speed for initially flat propagating fronts. This is the analogue, in random media, of a result of Caffarelli and Monneau [6]. The main new tools are a large scale Lipschitz estimate for the arrival time function, and a quantitative uniqueness result which does not use uniform local regularity.

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Head and Tail Speeds of Mean Curvature Flow with Forcing

Cited by 6 publications

References 20 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

Existence of effective burning velocity in cellular flow for curvature G-equation

Mean curvature flow with positive random forcing in 2-d

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