Correctors for the homogenization of Hamilton‐Jacobi equations in the stationary ergodic setting

By Pierre-Louis Lions, Panagiotis E. Souganidis. 2003

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There has recently been considerable interest in issues related to the homogenization of fully nonlinear first-order partial differential equations (HamiltonJacobi equations). This interest stems mainly from the wide spectrum of applications requiring the homogenization (averaging) of Hamilton-Jacobi equations. These applications range from front propagation in random media [5] to turbulentreaction diffusion equations and combustion [28], ergodic properties of controlled dynamical systems [1, 2], averaging behavior of Hamiltonian systems [16, 17], and the connection with the long-time asymptotics of Hamilton-Jacobi equations [6, 7, 18, 19, 20, 32], and stochastic homogenization [31, 33]. The most important step in the analysis of the homogenization problem is, of course, the identification of the effective (averaged) Hamiltonian, which is usually accomplished by studying the cell problem, an auxiliary equation arising in the formal expansion. The cell problem can be thought as a nonlinear eigenvalue problem with the effective Hamiltonian and the solution or approximate solution playing, respectively, the roles of the eigenvalue and the eigenfunction or approximate eigenfunction. It is also expected that the fine properties of the corrector may help to obtain more precise information about the averaged (effective) Hamiltonian. The question of the existence or nonexistence of an exact or approximate corrector appears to be rather delicate if one moves away from the periodic setting. Although the existence of an approximate corrector has been established in the almost-periodic setting for general Hamiltonians by Ishii [23], and the effective Hamiltonian has been identified by one of the authors [33] in the stationary ergodic setting and for convex Hamiltonians (see also [31]), not much is known about the existence of an exact corrector. Indeed, the classical arguments, which yield the existence of correctors in the periodic setting, fail, even in the simple quasi-periodic setting, due to the lack of compactness.

Subfields: Computational Theory and Mathematics, Control and Systems Engineering, Statistical and Nonlinear Physics

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Citations in the corpus

Stochastic Homogenization of Nonconvex Hamilton‐Jacobi Equations: A Counterexample (2015, background): “This result has been extended to various frameworks, still under the assumption that the Hamiltonian is convex in p (see [19, 15, 18, 17, 22, 4, 6]).”
Asymptotic and numerical homogenization (2008, methodology): “The terminology of reconstruction operator is adopted from Godunov schemes for nonlinear conservation laws (LeVeque 1990) and gas-kinetic schemes (Xu and Prendergast 1994).”
Stochastic homogenization of Hamilton‐Jacobi‐Bellman equations (2006, background): “It turns out [12] that for Hamilton-Jacobi equations with stationary ergodic Hamiltonians, correctors may not exist in general.”
Homogenization of degenerate second-order PDE in periodic and almost periodic environments and applications (2005, methodology): “in [ LS2 ], always in the stationary ergodic setting, the homogenization of (1.7), when the”
Stochastic homogenization of Hamilton-Jacobi and degenerate Bellman equations in unbounded environments (2011, background): “We also mention the work of Caffarelli, Souganidis and Wang [4] on the stochastic homogenization of uniformly elliptic equations of second-order, Caffarelli and Souganidis [3] who obtained a rate of convergence for the latter in strongly mixing environments, and Schwab [30] on the homogenization of ”
Rates of convergence for the homogenization of fully nonlinear uniformly elliptic pde in random media (2010, background): “Lions and Souganidis investigated in [ 20 ] the issue of the existence of correctors for the random Hamilton-Jacobi case.”
Homogenization of fully nonlinear, uniformly elliptic and parabolic partial differential equations in stationary ergodic media (2005, methodology): “viscosity solutions (see [CIL]) and arguing similarly to the proof of Theorem 1.5 of [ LS2 ]”
Exact and approximate correctors for stochastic Hamiltonians: the 1-dimensional case (2009, background): “For instance in [16] Lions and Souganidis tackle this issues and exploit a Crandall’s result on Eikonal Hamiltonians to construct a 1–dimensional example showing that bounded approximate correctors in general do not exist, in contrast to what happens in the almost periodic case, see [2, 13] The poin”
Stochastic Homogenization of Level-Set Convex Hamilton–Jacobi Equations (2012, background): “The existence and nonexistence of correctors in random environments were studied by Lions and Souganidis [21], who also introduced in [23] a more direct proof of homogenization in probability.”
Stochastic homogenization of Hamilton-Jacobi equations in stationary ergodic spatio-temporal media (2009, methodology): “Moreover, counter examples to the solvability of the corrector problem with a strictly sublinear w were given in [21] and [22].”
Error estimates and convergence rates for the stochastic homogenization of Hamilton-Jacobi equations (2012, background): “[32] P. E. Souganidis.”
Homogenization of “Viscous” Hamilton–Jacobi Equations in Stationary Ergodic Media (2005, methodology): “In a subsequent work [ LS1 ], the authors showed that, in general, in”
Ergodic Type Problems and Large Time Behaviour of Unbounded Solutions of Hamilton–Jacobi Equations (2006, background): “They also arise in homogenization theory (the so-called cell problems) and a related question to ours is whether the periodicity assumption on H can be removed while keeping bounded solutions (the correctors): We refer to Ishii (2000) for the existence of bounded approximate correctors in the almost”