Armstrong, Scott N. Smart, Charles K.
Published in
Archive for Rational Mechanics and Analysis

We introduce a new method for studying stochastic homogenization of elliptic equations in nondivergence form. The main application is an algebraic error estimate, asserting that deviations from the homogenized limit are at most proportional to a power of the microscopic length scale, assuming a finite range of dependence. The results are new even f...

Moakher, Maher Maddocks, John H.
Published in
Archive for Rational Mechanics and Analysis

Motivated by applications in the modeling of deformations of the DNA double helix, we construct a continuum mechanics model of two elastically interacting elastic strands. The two strands are described in terms of averaged, or macroscopic, variables plus an additional small, internal or microscopic, perturbation. We call this composite structure a ...

Julin, Vesa
Published in
Archive for Rational Mechanics and Analysis

This paper is concerned with nonlinear elliptic equations in nondivergence form F(D2u,Du,x)=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$F(D^{2}u, Du, x) = 0 $$\end...

Cagnetti, Filippo Dal Maso, Gianni Scardia, Lucia Zeppieri, Caterina Ida
Published in
Archive for Rational Mechanics and Analysis

In this paper we study the stochastic homogenisation of free-discontinuity functionals. Assuming stationarity for the random volume and surface integrands, we prove the existence of a homogenised random free-discontinuity functional, which is deterministic in the ergodic case. Moreover, by establishing a connection between the deterministic converg...

Cancès, Clément Matthes, Daniel Nabet, Flore
Published in
Archive for Rational Mechanics and Analysis

We study a non-local version of the Cahn–Hilliard dynamics for phase separation in a two-component incompressible and immiscible mixture with linear mobilities. Differently to the celebrated local model with nonlinear mobility, it is only assumed that the divergences of the two fluxes—but not necessarily the fluxes themselves—annihilate each other....

Mitake, Hiroyoshi Tran, Hung V. Yu, Yifeng
Published in
Archive for Rational Mechanics and Analysis

We study the rate of convergence of uε\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${u^\varepsilon}$$\end{document}, as ε→0+\documentclass[12pt]{minimal} \usepackage{...

Cheng, Bin Ju, Qiangchang Schochet, Steve
Published in
Archive for Rational Mechanics and Analysis

Singular limits of a class of evolutionary systems of partial differential equations having two small parameters and hence three time scales are considered. Under appropriate conditions solutions are shown to exist and remain uniformly bounded for a fixed time as the two parameters tend to zero at different rates. A simple example shows the necessi...

Abedin, Farhan Tralli, Giulio
Published in
Archive for Rational Mechanics and Analysis

We prove invariant Harnack inequalities for certain classes of non-divergence form equations of Kolmogorov type. The operators we consider exhibit invariance properties with respect to a homogeneous Lie group structure. The coefficient matrix is assumed either to satisfy a Cordes–Landis condition on the eigenvalues, or to admit a uniform modulus of...

Guardia, Marcel Kaloshin, Vadim Zhang, Jianlu
Published in
Archive for Rational Mechanics and Analysis

For the Restricted Circular Planar 3 Body Problem, we show that there exists an open set U\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal{U}}$$\end{document} ...

Jiang, Fei Jiang, Song
Published in
Archive for Rational Mechanics and Analysis

We investigate why the non-slip boundary condition for the velocity, imposed in the direction of impressed magnetic fields, can contribute to the magnetic inhibition effect based on the magnetic Rayleigh–Taylor (abbr. NMRT) problem in nonhomogeneous incompressible non-resistive magnetohydrodynamic (abbr. MHD) fluids. Exploiting an infinitesimal met...