Homogenization of the g-equation with incompressible random drift in two dimensions

Abstract

Abstract. We study the homogenization limit of solutions to the G-equation with random drift. This Hamilton-Jacobi equation is a model for flame propagation in a turbulent fluid in the regime of thin flames. For a fluid velocity field that is statistically stationary and ergodic, we prove sufficient conditions for homogenization to hold with probability one. These conditions are expressed in terms of travel times for the associated control problem. When the spatial dimension is equal to two and the fluid velocity is divergence-free, we verify that these conditions hold under suitable assumptions about the growth of the random stream function.

A survival guide for feeble fish

“…In the two-dimensional case the estimate (7) has been obtained in [3] under slightly different assumptions on V . This estimate has been used in [3] to characterize effective behavior of solutions of the random two-dimensional G-equation, a certain Hamilton-Jacobi partial differential equation that arises in modeling propagation of flame fronts in turbulent media [4,5]. Subsequently effective behavior of solutions of the random G-equation was characterized in [1] for any dimension.…”

“…Subsequently effective behavior of solutions of the random G-equation was characterized in [1] for any dimension. The approach in [1] is different from [3]. In particular, it relies on ergodic properties of the flow V instead of its geometry.…”

Large time average of reachable sets and Applications to Homogenization of interfaces moving with oscillatory spatio-temporal velocity

“…The homogenization of time independent noncoercive level set equations in the periodic setting was established by Cardaliaguet, Lions and Souganidis [10] and recently by Ciomaga, Souganidis and Tran [14] in the random setting. The homogenization of the G-equation, which is used as model for fronts propagating with normal velocity and advection, in periodic environments was established by Cardaliaguet, Nolen and Souganidis [11] (a special case of space periodic incompressible flows was considered by Xin and Yu [30]) and by Cardaliaguet and Souganidis in [13] in random media (a special case was studied by Novikov and Nolen [23]).…”

Periodic approximations of the ergodic constants in the stochastic homogenization of nonlinear second-order (degenerate) equations

“…Cardaliaguet was partially supported by the ANR (Agence Nationale de la Recherche) project ANR-12-BS01-0008-01. homogenization of non coercive Hamilton-Jacobi equations was obtained by Cardaliaguet and Souganidis [14] for the so-called G-equation (also see Nolen and Novikov [29] who considered the same in dimension d = 2 and under additional structure conditions). The stochastic homogenization of fully nonlinear uniformly elliptic second-order equations was established by Caffarelli, Souganidis and Wang [12] and Caffarelli and Souganidis [10] obtained a rate of convergence in strongly mixing environments.…”