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Register a new userOptimal homogenization rates in stochastic homogenization of nonlinear uniformly elliptic equations and systems
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Added by
Julian Fischer
Publication date
2019
fields
Informatics Engineering
and research's language is
English
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We derive optimal-order homogenization rates for random nonlinear elliptic PDEs with monotone nonlinearity in the uniformly elliptic case. More precisely, for a random monotone operator on $mathbb{R}^d$ with stationary law (i.e. spatially homogeneous statistics) and fast decay of correlations on scales larger than the microscale $varepsilon>0$, we establish homogenization error estimates of the order $varepsilon$ in case $dgeq 3$, respectively of the order $varepsilon |log varepsilon|^{1/2}$ in case $d=2$. Previous results in nonlinear stochastic homogenization have been limited to a small algebraic rate of convergence $varepsilon^delta$. We also establish error estimates for the approximation of the homogenized operator by the method of representative volumes of the order $(L/varepsilon)^{-d/2}$ for a representative volume of size $L$. Our results also hold in the case of systems for which a (small-scale) $C^{1,alpha}$ regularity theory is available.
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Corrector estimates constitute a key ingredient in the derivation of optimal convergence rates via two-scale expansion techniques in homogenization theory of random uniformly elliptic equations. The present work follows up - in terms of corrector estimates - on the recent work of Fischer and Neukamm (arXiv:1908.02273) which provides a quantitative stochastic homogenization theory of nonlinear uniformly elliptic equations under a spectral gap assumption. We establish optimal-order estimates (with respect to the scaling in the ratio between the microscopic and the macroscopic scale) for higher-order linearized correctors. A rather straightforward consequence of the corrector estimates is the higher-order regularity of the associated homogenized monotone operator.
The aim of this paper is twofold. The first is to study the asymptotics of a parabolically scaled, continuous and space-time stationary in time version of the well-known Funaki-Spohn model in Statistical Physics. After a change of unknowns requiring the existence of a space-time stationary eternal solution of a stochastically perturbed heat equation, the problem transforms to the qualitative homogenization of a uniformly elliptic, space-time stationary, divergence form, nonlinear partial differential equation, the study of which is the second aim of the paper. An important step is the construction of correctors with the appropriate behavior at infinity.
In this paper, we consider homogenization of the Landau-Lifshitz equation with a highly oscillatory material coefficient with period $varepsilon$ modeling a ferromagnetic composite. We derive equations for the homogenized solution to the problem and the corresponding correctors and obtain estimates for the difference between the exact and homogenized solution as well as corrected approximations to the solution. Convergence rates in $varepsilon$ over times $O(varepsilon^sigma)$ with $0leq sigmaleq 2$ are given in the Sobolev norm $H^q$, where $q$ is limited by the regularity of the solution to the detailed Landau-Lifshitz equation and the homogenized equation. The rates depend on $q$, $sigma$ and the the number of correctors.
We study the homogenization of elliptic systems of equations in divergence form where the coefficients are compositions of periodic functions with a random diffeomorphism with stationary gradient. This is done in the spirit of scalar stochastic homogenization by Blanc, Le Bris and P.-L. Lions. An application of the abstract result is given for Maxwells equations in random dissipative bianisotropic media.
We consider the specified stochastic homogenization of first order evolutive Hamilton-Jacobi equations on a very simple junction, i.e the real line with a junction at the origin. Far from the origin, we assume that the considered hamiltonian is closed to given stationary ergodic hamiltonians (which are different on the left and on the right). Near the origin, there is a perturbation zone which allows to pass from one hamiltonian to the other. The main result of this paper is a stochastic homogenization as the length of the transition zone goes to zero. More precisely, at the limit we get two deterministic right and left hamiltonians with a deterministic junction condition at the origin. The main difficulty and novelty of the paper come from the fact that the hamiltonian is not stationary ergodic. Up to our knowledge, this is the first specified stochastic homogenization result. This work is motivated by traffic flow applications.
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