arXiv subjects

Weisheng Niu

Publications and source records attributed to Weisheng Niu.

11 recordsLinked to original sources

arXiv2026

Uniform Calderón-Zygmund estimates in multiscale elliptic homogenization

This paper is concerned with the elliptic equation $-\text{div} (A_\varepsilon \nabla u_\varepsilon) = \text{div} f$ in a bounded $C^1$ domain, where $A_\varepsilon$ takes a form of $A_\varepsilon(x) = A(x/\varepsilon_1, x/\varepsilon_2,\cdots, x/\varepsilon_n)$, with $A(y_1,y_2,\cdots,y_n)$ being 1-periodic in each $y_i$. We prove the uniform Calderón-Zygmund estimate, namely, the uniform $L^p$ boundedness of the linear map $f\mapsto \nabla u_\varepsilon$ for any $p\in (1,\infty)$ with a constant independent of small parameters $(\varepsilon_1,\varepsilon_2,\cdots, \varepsilon_n) \in (0,1]^n$. Our result includes the uniform Calderón-Zygmund estimate in quasiperiodic elliptic homogenization (even without the Diophantine condition), which was previously unknown. The proof novelly combines the Dirichlet's theorem on the simultaneous Diophantine approximation from number theory, a technique of reperiodization, reiterated periodic homogenization and a large-scale real-variable argument. Using the idea of reperiodization, we also obtain some large-scale or mesoscopic-scale Lipschitz estimates.

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arXiv2025

Optimal convergence rates in multiscale elliptic homogenization

This paper is devoted to the quantitative homogenization of multiscale elliptic operator $-\nabla\cdot A_\varepsilon \nabla$, where $A_\varepsilon(x) = A(x/\varepsilon_1, x/\varepsilon_2,\cdots, x/\varepsilon_n)$, $\varepsilon = (\varepsilon_1, \varepsilon_2,\cdots, \varepsilon_n) \in (0,1]^n$ and $\varepsilon_i > \varepsilon_{i+1}$. We assume that $A(y_1,y_2,\cdots, y_n)$ is 1-periodic in each $y_i \in \mathbb{R}^d$ and real analytic. Classically, the method of reiterated homogenization has been applied to study this multiscale elliptic operator, which leads to a convergence rate limited by the ratios $\max \{ \varepsilon_{i+1}/\varepsilon_i: 1\le i\le n-1\}$. In the present paper, under the assumption of real analytic coefficients, we introduce the so-called multiscale correctors and more accurate effective operators, and improve the ratio part of the convergence rate to $\max \{ e^{-c\varepsilon_{i}/\varepsilon_{i+1}}: 1\le i\le n-1 \}$. This convergence rate is optimal in the sense that $c>0$ cannot be replaced by a larger constant. As a byproduct, the uniform Lipschitz estimate is established under a mild double-log scale-separation condition.

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arXiv2024

Well-posedness and regularity of the Darcy-Boussinesq system in layered porous media

We establish the existence of global weak solution in 2D and 3D, as well as the uniqueness of weak solution in 2D, for the Darcy-Boussinesq model for convection in layered porous media with square integrable initial data. We also derived tangential regularity in the 2D case. In addition, we obtain the existence and uniqueness of regular solution in a novel piecewise $H^2$ space in both 2D and 3D under uniform porosity assumption and $H^1$ initial data. This is the first rigorous result for this model in the physically important layered setting.

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arXiv2021

Compactness and stable regularity in multiscale homogenization

In this paper we develop some new techniques to study the multiscale elliptic equations in the form of $-\text{div} \big(A_\varepsilon \nabla u_{\varepsilon} \big) = 0$, where $A_\varepsilon(x) = A(x, x/\varepsilon_1,\cdots, x/\varepsilon_n)$ is an $n$-scale oscillating periodic coefficient matrix, and $(\varepsilon_i)_{1\le i\le n}$ are scale parameters. We show that the $C^α$-Hölder continuity with any $α\in (0,1)$ for the weak solutions is stable, namely, the constant in the estimate is uniform for arbitrary $(\varepsilon_1, \varepsilon_2, \cdots, \varepsilon_n) \in (0,1]^n$ and particularly is independent of the ratios between $\varepsilon_i$'s. The proof uses an upgraded method of compactness, involving a scale-reduction theorem by $H$-convergence. The Lipschitz estimate for arbitrary $(\varepsilon_i)_{1\le i\le n}$ still remains open. However, for special laminate structures, i.e., $A_\varepsilon(x) = A(x,x_1/\varepsilon_1, \cdots, x_d/\varepsilon_n)$, we show that the Lipschitz estimate is stable for arbitrary $(\varepsilon_1, \varepsilon_2, \cdots, \varepsilon_n) \in (0,1]^n$. This is proved by a technique of reperiodization.

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arXiv2021

Homogenization of locally periodic parabolic operators with non-self-similar scales

We investigate quantitative estimates in homogenization of the locally periodic parabolic operator with multiscales $$ \partial_t- \text{div} (A(x,t,x/\varepsilon,t/κ^2) \nabla ),\qquad \varepsilon>0,\, κ>0. $$ Under proper assumptions, we establish the full-scale interior and boundary Lipschitz estimates. These results are new even for the case $κ=\varepsilon$, and for the periodic operators $ \partial_t-\text{div}(A(x/\varepsilon, t/\varepsilon^{\ell}) \nabla ),$ $0<\varepsilon,\ell<\infty, $ of which the large-scale Lipschitz estimate down to $\varepsilon+\varepsilon^{\ell/2}$ was recently established by the first author and Shen in Arch. Ration. Mech. Anal. 236(1): 145--188 (2020). Due to the non-self-similar structure, the full-scale estimates do not follow directly from the large-scale estimates and the blow-up argument. As a byproduct, we also derive the convergence rates for the corresponding initial-Dirichlet problems, which extend the results in the aforementioned literature to more general settings.

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arXiv2020

Combined Effects of Homogenization and Singular Perturbations: Quantitative Estimates

We investigate quantitative estimates in periodic homogenization of second-order elliptic systems of elasticity with singular fourth-order perturbations. The convergence rates, which depend on the scale $κ$ that represents the strength of the singular perturbation and on the length scale $ε$ of the heterogeneities, are established. We also obtain the large-scale Lipschitz estimate, down to the scale $ε$ and independent of $κ$. This large-scale estimate, when combined with small-scale estimates, yields the classical Lipschitz estimate that is uniform in both $ε$ and $κ$.

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arXiv2020

Convergence Rates in Almost-Periodic Homogenization of Higher-order Elliptic Systems

This paper concentrates on the quantitative homogenization of higher-order elliptic systems with almost-periodic coefficients in bounded Lipschitz domains. For coefficients which are almost-periodic in the sense of H. Weyl, we establish uniform ocal $L^2$ estimates for the approximate correctors. Under an additional assumption on the frequencies of the coefficients (see (1.10)), we derive the existence of the true correctors as well as the sharp $O(\varepsilon)$ convergence rate in $H^{m-1}$. As a byproduct, the large-scale Hölder estimate and a Liouville theorem are obtained for higher-order elliptic systems with almost-periodic coefficients in the sense of Besicovish. Since (1.10) is not well-defined for the equivalence classes of almost-periodic functions in the sense of H. Weyl or Besicovish, we provide another condition that implies the sharp convergence rate in terms of perturbations on the coefficients.

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arXiv2019

Quantitative Estimates in Reiterated Homogenization

This paper investigates quantitative estimates in the homogenization of second-order elliptic systems with periodic coefficients that oscillate on multiple separated scales. We establish large-scale interior and boundary Lipschitz estimates down to the finest microscopic scale via iteration and rescaling arguments. We also obtain a convergence rate in the $L^2$ space by the reiterated homogenization method.

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arXiv2018

Convergence rates in homogenization of higher order parabolic systems

This paper is concerned with the optimal convergence rate in homogenization of higher order parabolic systems with bounded measurable, rapidly oscillating periodic coefficients. The sharp $O(\va)$ convergence rate in the space $L^2(0,T; H^{m-1}(\Om))$ is obtained for both the initial-Dirichlet problem and the initial-Neumann problem. The duality argument inspired by \cite{suslinaD2013} is used here.

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arXiv2017

Uniform Boundary Estimates in Homogenization of Higher Order Elliptic Systems

This paper focuses on the uniform boundary estimates in homogenization of a family of higher order elliptic operators $\mathcal{L}_ε$, with rapidly oscillating periodic coefficients. We derive uniform boundary $C^{m-1,λ} (0\!<\!λ\!<\!1)$, $ W^{m,p}$ estimates in $C^1$ domains, as well as uniform boundary $C^{m-1,1}$ estimate in $C^{1,θ} (0\!<\!θ\!<\!1)$ domains without the symmetry assumption on the operator. The proof, motivated by the profound work "S.N. Armstrong and C.~K. Smart, Ann. Sci. Éc. Norm. Supér. (2016), Z. Shen, Anal. PDE (2017)", is based on a suboptimal convergence rate in $H^{m-1}(Ω)$. Compared to "C.E. Kenig, F. Lin and Z. Shen, Arch. Ration. Mech. Anal. (2012), Z. Shen, Anal. PDE (2017)", the convergence rate obtained here does not require the symmetry assumption on the operator, nor additional assumptions on the regularity of $u_0$ (the solution to the homogenized problem), and thus might be of some independent interests even for second order elliptic systems.

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arXiv2017

Convergence Rates and Interior Estimates in Homogenization of Higher Order Elliptic Systems

This paper is concerned with the quantitative homogenization of $2m$-order elliptic systems with bounded measurable, rapidly oscillating periodic coefficients. We establish the sharp $O(\varepsilon)$ convergence rate in $W^{m-1, p_0}$ with $p_0=\frac{2d}{d-1}$ in a bounded Lipschitz domain in $\mathbb{R}^d$ as well as the uniform large-scale interior $C^{m-1, 1}$ estimate. With additional smoothness assumptions, the uniform interior $C^{m-1, 1}$, $W^{m,p}$ and $C^{m-1, α}$ estimates are also obtained. As applications of the regularity estimates, we establish asymptotic expansions for fundamental solutions.

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