arXiv subjects

Thuyen Dang

Publications and source records attributed to Thuyen Dang.

9 recordsLinked to original sources

arXiv2026

A point-free theory of quantitative homogenization

We introduce a purely operator-theoretic framework for quantitative homogenization that bypasses the traditional reliance on large-scale spatial regularity and probabilistic assumptions. Inspired by Tartar's vision of a \emph{point-free} theory, we derive explicit norm resolvent estimates using only the algebraic structure of multiscale operators and the abstract geometry of Hilbert spaces. In this framework, the effective macroscopic dynamics and the abstract corrector emerge naturally from an orthogonal decomposition of the state space, governed algebraically by a Schur complement. To quantify the convergence rate, we introduce a frequency-splitting technique and solve a generalized Sylvester equation that controls the commutator between the differential structure and the highly oscillatory material properties. This abstract perspective unifies stationary, non-stationary, periodic, quasi-periodic, and stochastic homogenization. We demonstrate that the physical distinctions between these media--and their respective convergence rates--are entirely captured by the behavior of the spectral measures of the microscopic and macroscopic derivative operators near zero frequency.

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arXiv2025

Global boundedness of solutions of degenerate and non-uniform parabolic equations

Let $2 \le N\in\mathbb{N}$, $Ω$ be a bounded open in $\mathbb{R}^{N}$, $T\in (0,\infty)$, $Q=Ω\times (0,T)$, $u$ be a weak solution of parabolic equation $\displaystyle \frac{\partial u}{\partial t} -Lu= f$, where $L$ is an elliptic operator on a space of functions on $Q$. The coefficients of $L$ may not be bounded, not strictly nor uniformly elliptic, and not of Muckenhoupt type. We obtain global boundedness of $u$. Our result can be applied to $u$, which may vanish on $(A\times (0,T))\cup (Ω\times \{0\})$ of the boundary of $Q$ and is free outside this set.

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arXiv2025

Homogenization of a semilinear elliptic problem

We consider the homogenization of a semilinear elliptic equation where the coefficients of the second-order differential operator may be discontinuous. We establish the existence and uniqueness of the fine-scale solution, followed by an a priori estimate. The homogenized equation is derived using two-scale convergence, and a corrector result is also provided

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arXiv2024

Homogenization of high-contrast dielectric elastomer composites

This paper focuses on the homogenization of high-contrast dielectric elastomer composites, materials that deform in response to electrical stimulation. The considered heterogeneous material consisting of an ambient material with inserted particles is described by a weakly coupled system of an electrostatic equation with an elastic equation enriched with electrostriction. It is assumed that particles gradually become rigid as the fine-scale parameter approaches zero. This study demonstrates that the effective response of this system entails a homogeneous dielectric elastomer, described by a weakly coupled system of PDEs. The coefficients of the homogenized equations are dependent on various factors, including the composite's geometry, the original microstructure's periodicity, and the coefficients characterizing the initial heterogeneous material. Particularly, these coefficients are significantly influenced by the high-contrast nature of the fine-scale problem's coefficients. Consequently, as anticipated, the high-contrast coefficients of the original yield non-local effects in the homogenized response.

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arXiv2023

Topological Anderson Insulators by homogenization theory

A central property of (Chern) topological insulators is the presence of robust asymmetric transport along interfaces separating two-dimensional insulating materials in different topological phases. A Topological Anderson Insulator is an insulator whose topological phase is induced by spatial fluctuations. This paper proposes a mathematical model of perturbed Dirac equations and shows that for sufficiently large and highly oscillatory perturbations, the systems is in a different topological phase than the unperturbed model. In particular, a robust asymmetric transport indeed appears at an interface separating perturbed and unperturbed phases. The theoretical results are based on careful estimates of resolvent operators in the homogenization theory of Dirac equations and on the characterization of topological phases by the index of an appropriate Fredholm operator.

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arXiv2023

Explicit corrector in homogenization of monotone operators and its application to nonlinear dielectric elastomer composites

This paper concerns the rigorous periodic homogenization for a weakly coupled electroelastic system of a nonlinear electrostatic equation with an elastic equation enriched with electrostriction. Such coupling is employed to describe dielectric elastomers or deformable (elastic) dielectrics. It is shown that the effective response of the system consists of a homogeneous dielectric elastomer described by a nonlinear weakly coupled system of PDEs whose coefficients depend on the coefficients of the original heterogeneous material, the geometry of the composite and the periodicity of the original microstructure. The approach developed here for this nonlinear problem allows obtaining an explicit corrector result for the homogenization of monotone operators with minimal regularity assumptions. Two $L^p-$gradient estimates for elastic systems with discontinuous coefficients are also obtained.

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arXiv2022

Global gradient estimate for a divergence problem and its application to the homogenization of a magnetic suspension

This paper generalizes the results obtained by the authors in \cite{dangHomogenizationNondiluteSuspension2021} concerning the homogenization of a non-dilute suspension of magnetic particles in a viscous flow. More specifically, in this paper, a restrictive assumption on the coefficients of the coupled equation, made in \cite{dangHomogenizationNondiluteSuspension2021}, that significantly narrowed the applicability of the homogenization results obtained, is relaxed and a new regularity of the solution of the fine-scale problem is proven. In particular, we obtain a global $L^{\infty}$-bound for the gradient of the solution of the scalar equation $-\mathrm{div} \left[ \mathbf{a} \left( x/\varepsilon \right)\nabla φ^{\varepsilon}(x) \right] = f(x)$, uniform with respect to microstructure scale parameter $\varepsilon\ll 1$ in a small interval $(0,\varepsilon_0)$, where the coefficient $\mathbf{a}$ is only \emph{piecewise} Hölder continuous. Thenceforth, this regularity is used in the derivation of the effective response of the given suspension discussed in \cite{dangHomogenizationNondiluteSuspension2021}.

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arXiv2022

Homogenization of a non-linear strongly coupled model of magnetorheological fluids

This paper concerns the rigorous periodic homogenization for a non-linear strongly coupled system, which models a suspension of magnetizable rigid particles in a non-conducting carrier viscous Newtonian fluid. The fluid drags the particles, thus alters the magnetic field. Vice versa, the magnetic field acts on the particles, which in turn affect the fluid via the no-slip boundary condition. As the size of the particles approaches zero, it is shown that the suspension's behavior is governed by a generalized magnetohydrodynamic system, where the fluid is modeled by a stationary Navier-Stokes system, while the magnetic field is modeled by Maxwell equations. A corrector result from the theory of two-scale convergence allows us to obtain the limit of the product of several weakly convergent sequences, where the div-curl lemma, which is a typical tool in these types of problems, is not applicable.

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arXiv2021

Homogenization of Non-dilute Suspension of Viscous Fluid with Magnetic Particles

This paper seeks to carry out the rigorous homogenization of a particulate flow consisting of a non-dilute suspension of a viscous Newtonian fluid with magnetizable particles. The fluid is assumed to be described by the Stokes flow, while the particles are either paramagnetic or diamagnetic, for which the magnetization field is a linear function of the magnetic field. The coefficients of the corresponding partial differential equations are locally periodic. A one-way coupling between the fluid domain and the particles is also assumed. The homogenized or effective response of such a suspension is derived, and the mathematical justification of the obtained asymptotics is carried out. The two-scale convergence method is adopted for the latter. As a consequence, the presented result provides a justification for the formal asymptotic analysis of Lévy and Sanchez-Palencia for particulate steady-state Stokes flows.

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