Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 1901.09669

Local precised approximation in multiscale problems with local defects

Xavier Blanc·Marc Josien·Claude Le Bris
View original source ↗Read source document ↗Export metadata ↓

Abstract

We proceed here with our systematic study, initiated in [3], of multiscale problems with defects, within the context of homogenization theory. The case under consideration here is that of a diffusion equation with a diffusion coefficient of the form of a periodic function perturbed by an $L^r (R^d ) , 1 < r < +$\infty$$ , function modeling a localized defect. We outline the proof of the following approximation result: the corrector function, the existence of which has been established in [3,4], allows to approximate the solution of the original multiscale equation with essentially the same accuracy as in the purely periodic case. The rates of convergence may however vary, and are made precise, depending upon the $L^r$ integrability of the defect. The generalization to an abstract setting is mentioned. Our proof exactly follows, step by step, the pattern of the original proof of Avellaneda and Lin in [1] in the periodic case, extended in the works of Kenig and collaborators [13], and borrows a lot from it. The details of the results announced in this Note are given in our forthcoming publications [2,12].

Explore related subjects

math.AP

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Xavier Blanc, Marc Josien, Claude Le Bris. 2019-01-18. Local precised approximation in multiscale problems with local defects. https://doi.org/10.1016/j.crma.2018.12.005

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

A linear test approach to global controllability of third- and fifth-order nonlinear dispersive equations

We investigate third- and fifth-order nonlinear dispersive equations of KdV type on the torus and establishes approximate controllability by a fixed four-dimensional control; rather than relying solely on the saturation machinery, the analysis exploits the finite-dimensional controllability of the inviscid Burgers equation linearized around a carefully constructed return trajectory, with the trajectory itself obtained from an observable family. This ``linear test" strategy, yields more information about the structure of the control than the standard approach. In particular, the constructed control is shown to depend continuously on the initial and target states, a property that is by no means automatic in nonlinear control problems, and to decompose as a bounded linear operator applied to the data plus a fixed remainder, with the operator part interestingly independent of the order of dispersion.

math.AP↗
arXiv2026

A conditional Lagrangian clock barrier at the $C^{1,\frac{1}{3}}$ threshold for axisymmetric Euler without swirl

We consider axisymmetric no-swirl solutions to the three-dimensional incompressible Euler equations, with initial velocity in $C^{1,α}\cap L^2$, where $α\in\left[\frac{1}{3},1\right)$. In a major breakthrough, Shkoller introduced a clock-and-driver framework that he used in order to prove finite-time type I blow-up below the $C^{1,\frac{1}{3}}$ threshold in this setting. Motivated by this, we define Lagrangian classes of coherent conditional solutions for which the same mechanism yields a supercritical-critical barrier to blow-up when $α\geq\frac{1}{3}$. When $α>\frac{1}{3}$, the aforementioned barrier is genuinely depleted, whereas at the critical endpoint $α=\frac{1}{3}$, we obtain an exponential bound preventing blow-up. In the general case, we formulate a matrix-clock criterion in terms of the smallest singular value of the deformation gradient and show that, under transverse cusp-tail, longitudinal, off-clock, Dini, and suitable geometric coherence hypotheses, this singular value cannot collapse in finite time. In particular, we also show that the class of such coherent solutions includes the smooth ones locally in time. In the on-axis case, the criterion reduces to the scalar clock inequality $\displaystyle \dot{J}(t)\gtrsim -B(t)J(t)-CJ(t)^{3α}$, which rules out Shkoller-type clock collapse for $α\geq\frac{1}{3}$. These results do not enlarge the known Lorentz-space global regularity classes. Rather, they in particular identify the supercritical Lagrangian obstruction dual to Shkoller's subcritical blow-up mechanism in the case $α>\frac{1}{3}$.

math.AP↗