Stochastic homogenization of interfaces moving with changing sign velocity

Abstract

Abstract. We are interested in the averaged behavior of interfaces moving in stationary ergodic environments, with oscillatory normal velocity which changes sign. This problem can be reformulated, using level sets, as the homogenization of a Hamilton-Jacobi equation with a positively homogeneous non-coercive Hamiltonian. The periodic setting was earlier studied by Cardaliaguet, Lions and Souganidis (2009). Here we concentrate in the random media and show that the solutions of the oscillatory Hamilton-Jacobi equation converge in L ∞ -weak * to a linear combination of the initial datum and the solutions of several initial value problems with deterministic effective Hamiltonian(s), determined by the properties of the random media.

Stochastic homogenization of a nonconvex Hamilton–Jacobi equation

“…In contrast to the periodic setting, in which nonconvex Hamiltonians are not more difficult to handle than convex Hamiltonians (c.f. [9,14]), extending the results of [21,23] to the nonconvex case has remained, until now, completely open (except p h(p) = (|p| 2 − 1) 2 H(p) for the quite modest extension to level-set convex Hamiltonians [3] and the forthcoming work [6], which considers a first-order motion with a sign-changing velocity). The issue of whether convexity is necessary for homogenization in the random setting is mentioned prominently as an open problem for example in [11,16,17].…”

“…for the quite modest extension to level-set convex Hamiltonians [3] and the forthcoming work [6], which considers a first-order motion with a sign-changing velocity). The issue of whether convexity is necessary for homogenization in the random setting is mentioned prominently as an open problem for example in [11,16,17].…”

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

“…Of special interest is the study of noncoercive Hamilton-Jacobi equations associated to moving interfaces. 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]).…”

“…The problem, however, is that θ(·, (x, 0)) = +∞ in R n × [0, ∞) \ W ((x, 0)). Hence, it does not seem possible to have any estimates like (1.8) or to extend the minimal time function to the whole R n × [0, ∞) as in [14]. Because of this lack of control on the minimal time function, we found it necessary to come up with a new approach.…”

Convergence of a class of nonlinear time delays reaction-diffusion equations

“…The carrying capacity is now function of the growth rate and time dependent. For diffusion in random media and homogenization refer to [2,8,11,10,19,20], for stochastic homogenization of Hamilton Jacobi equations refer to [3,16,24] and for stochastic homogenization of moving interfaces refer to [7,9] and references therein.…”