arXiv subjects

Wenjia Jing

Publications and source records attributed to Wenjia Jing.

At least 19 recordsLinked to original sources

arXiv2026

Stochastic Homogenization of Non-local Hamilton-Jacobi-Bellman equations

In this paper we apply the method of Kosygina, Rezakhanlou and Varadhan (CPAM 2006) to establish stochastic homogenization of Hamilton-Jacobi-Bellman (HJB) equation with a vanishing non-local integro-differential operator and a convex super-linearly growing Hamiltonian in stationary ergodic random medium. Their method, first designed for homogenization of HJB equations with vanishing Laplacian operator, relies on stochastic optimal control representation of the solution, a key construction of approximate super-correctors and the technique of linking diffusion process to abstract diffusion in random media. We show how the procedures can be carried out for HJB equations with jump-diffusion. In particular, for the construction of approximate super-correctors, we represent the non-local integro-differential operator as the divergence of a regular integral operator acting on the gradient.

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arXiv2026

Minnaert resonances and higher-order acoustic modes for bubbles in a viscous fluid with surface tension

The aim of this paper is to account for viscosity, surface tension, and interactions between micro-bubbles in approximating their resonant behavior in the ultrasonic regime. Original asymptotic formulas for the resonance frequencies are derived in terms of the difference in the acoustic impedance at the interface between the gas and the fluid. Both low-frequency resonances (Minnaert resonances) and higher-frequency resonances, i.e., beyond the subwavelength regime, are considered. We also provide a resonant characterization for a system of several micro-bubbles.

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arXiv2026

Quantification of ergodicity for Hamilton--Jacobi equations in a dynamic random environment

We study quantitative large-time averages for Hamilton--Jacobi equations in a dynamic random environment that is stationary ergodic and has unit-range dependence in time. Our motivation comes from stochastic growth models related to the tensionless (inviscid) KPZ equation, which can be formulated as Hamilton--Jacobi equations with random forcing. Understanding the large-time behavior of solutions is closely connected to fundamental questions concerning fluctuations and scaling in such growth processes. In this article, we establish, up to slowly varying factors, convergence rates with exponent $1/2$ for the large-time averages of both the solutions and the associated metric problem toward their ergodic limits. Our proof relies crucially on a new almost-Lipschitz regularity theory for the metric problem, which is of independent interest.

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arXiv2025

Homogenization of the scattered wave and scattering resonances for periodic high-contrast subwavelength resonators

We study time-harmonic scattering by a periodic array of penetrable, high-contrast obstacles with small period, confined to a bounded Lipschitz domain. The strong contrast between the obstacles and the background induces subwavelength resonances. We derive a frequency-dependent effective model in the vanishing-period limit and prove quantitative convergence of the heterogeneous scattered wave to the effective scattered wave. We also identify the limiting set of scattering resonances and establish convergence rates. Finally, we establish convergence rates for the far-field pattern of the heterogeneous problem to that of the effective model.

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arXiv2025

Does Yakhot's growth law for turbulent burning velocity hold?

Using formal renormalization theory, Yakhot derived in ([32], 1988) an $O\left(\frac{A}{\sqrt{\log A}}\right)$ growth law of the turbulent flame speed with respect to large flow intensity $A$ based on the inviscid G-equation. Although this growth law is widely cited in combustion literature, there has been no rigorous mathematical discussion to date about its validity. As a first step towards unveiling the mystery, we prove that there is no intermediate growth law between $O\left(\frac{A}{\log A}\right)$ and $O(A)$ for two dimensional incompressible Lipschitz continuous periodic flows with bounded swirl sizes. In particular, we do not assume the non-degeneracy of critical points. Additionally, other examples of flows with lower regularity, Lagrangian chaos, and related phenomena are also discussed.

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arXiv2024

Unified quantitative analysis of the Stokes equations in dilute perforated domains via layer potentials

We develop a unified method to obtain the quantitative homogenization of Stokes systems in periodically perforated domains with no-slip boundary conditions on the perforating holes. The main novelty of our paper is a quantitative analysis of the asymptotic behavior of the two-scale cell correctors via periodic Stokes layer potentials. The two-scale cell correctors were introduced and analyzed qualitatively by Allaire in the early 90's. Thanks to our layer potential approach, we also provide a novel explanation of the conductivity matrix in Darcy's model, of the Brinkman term in Brinkman's model, and explain the special behavior for $d=2$. Finally, we also prove quantitative homogenization error estimates in various regimes of ratios between the size of the perforating holes and the typical distance between holes. In particular we handle a subtle issue in the dilute Darcy regime related to the non-vanishing of the Darcy velocity on the boundary.

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arXiv2024

Wave packets propagation in the subwavelength regime near the Dirac point

In [Ammari et al., SIAM J Math Anal., 52 (2020), pp. 5441--5466], the first author with collaborators proved the existence of Dirac dispersion cones at subwavelength scales in bubbly honeycomb phononic crystals. In this paper, we study the time-evolution of wave packets that are spectrally concentrated near such conical points. We prove that the wave packets dynamics is governed by a time-dependent effective Dirac system, which still depends, but in a simple way, on the subwavelength scale.

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arXiv2024

Quantitative homogenization of state-constraint Hamilton--Jacobi equations on perforated domains and applications

We study the periodic homogenization problem of state-constraint Hamilton--Jacobi equations on perforated domains in the convex setting and obtain the optimal convergence rate. We then consider a dilute situation in which the holes' diameter is much smaller than the microscopic scale. Finally, a homogenization problem with domain defects where some holes are missing is analyzed.

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arXiv2024

Convergence rate and uniform Lipschitz estimate in periodic homogenization of high-contrast elliptic systems

We consider the Dirichlet problem for elliptic systems with periodically distributed inclusions whose conduction parameter exhibits a significant contrast compared to the background media. We develop a unified method to quantify the convergence rates both as the periodicity of inclusions tends to zero and as the parameter approaches either zero or infinity. Based on the obtained convergence rates and a Campanato-type scheme, we also derive the regularity estimates that are uniform both in the periodicity and the contrast.

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arXiv2023

On the periodic homogenization of elliptic equations in non-divergence form with large drifts

We study the quantitative homogenization of linear second order elliptic equations in non-divergence form with highly oscillating periodic diffusion coefficients and with large drifts, in the so-called ``centered'' setting where homogenization occurs and the large drifts contribute to the effective diffusivity. Using the centering condition and the invariant measures associated to the underlying diffusion process, we transform the equation into divergence form with modified diffusion coefficients but without drift. The latter is in the standard setting for which quantitative homogenization results have been developed systematically. An application of those results then yields quantitative estimates, such as the convergence rates and uniform Lipschitz regularity, for equations in non-divergence form with large drifts.

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arXiv2022

Uniform convergence for linear elastostatic systems with periodic high contrast inclusions

We consider the Lame system of linear elasticity with periodically distributed inclusions whose elastic parameters have high contrast compared to the background media. We develop a unified method based on layer potential techniques to quantify three convergence results when some parameters of the elastic inclusions are sent to extreme values. More precisely, we study the incompressible inclusions limit where the bulk modulus of the inclusions tends to infinity, the soft inclusions limit where both the bulk modulus and the shear modulus tend to zero, and the hard inclusions limit where the shear modulus tends to infinity. Our method yields convergence rates that are independent of the periodicity of the inclusions array, and are sharper than some earlier results of this type. A key ingredient of the proof is the establishment of uniform spectra gaps for the elastic Neumann-Poincare operator associated to the collection of periodic inclusions that are independent of the periodicity.

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arXiv2022

Convergence rate for the homogenization of stationary diffusions in dilutely perforated domains with reflecting boundaries

We revisit the homogenization problem for the Poisson equation in periodically perforated domains with zero Neumann data at the boundary of the holes and prescribed Dirichlet data at the outer boundary. It is known that, if the periodicity of the holes goes to zero but their volume fraction remains fixed and positive, the limit problem is a Dirichlet boundary value problem posed in the domain without the holes, and the effective diffusion coefficients are non-trivially modified; if that volume fraction goes to zero instead, i.e. the holes are dilute, the effective operator remains the Laplacian (that is, unmodified). Our main results contain the study of a "continuity" in those effective models with respect to the volume fraction of the holes and some new convergence rates for homogenization in the dilute setting. Our method explores the classical two-scale expansion ansatz and relies on asymptotic analysis of the rescaled cell problems using layer potential theory.

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arXiv2021

Effective fronts of polygon shapes in two dimensions

We study the effective fronts of first order front propagations in two dimensions ($n=2$) in the periodic setting. Using PDE-based approaches, we show that for every $α\in (0,1)$, the class of centrally symmetric polygons with rational vertices and nonempty interior is admissible as effective fronts for given front speeds in $C^{1,α}(\mathbb T^2,(0,\infty))$. This result can also be formulated in the language of stable norms corresponding to periodic metrics in $\mathbb T^2$. Similar results were known long time ago when $n\geq 3$ for front speeds in $C^{\infty}(\mathbb T^n,(0,\infty))$. Due to topological restrictions, the two dimensional case is much more subtle. In fact, the effective front is $C^1$, which cannot be a polygon, for given $C^{1,1}(\mathbb T^2,(0,\infty))$ front speeds. Our regularity requirements on front speeds are hence optimal. To the best of our knowledge, this is the first time that polygonal effective fronts have been constructed in two dimensions.

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arXiv2020

Layer potentials for Lamé systems and homogenization of perforated elastic medium with clamped holes

We investigate Lamé systems in periodically perforated domains, and establish quantitative homogenization results in the setting where the domain is clamped at the boundary of the holes. Our method is based on layer potentials and it provides a unified proof for various regimes of hole-cell ratios (the ratio between the size of the holes and the size of the periodic cells), and, more importantly, it yields natural correctors that facilitate error estimates. A key ingredient is the asymptotic analysis for the rescaled cell problems, and this is studied by exploring the convergence of the periodic layer potentials for the Lamé system to those in the whole space when the period tends to infinity.

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arXiv2019

Effective fronts of polytope shapes

We study the periodic homogenization of first order front propagations. Based on PDE methods, we provide a simple proof that for $n \geq 3$, the class of centrally symmetric polytopes with rational coordinates and nonempty interior is admissible as effective fronts, which was also established in [1,10] in the form of stable norms as an extension of Hedlund's classical result [7]. Besides, we obtain the optimal convergence rate of the homogenization problem for this class.

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arXiv2019

Generalized ergodic problems: existence and uniqueness structures of solutions

We study a generalized ergodic problem (E), which is a Hamilton-Jacobi equation of contact type, in the flat $n$-dimensional torus. We first obtain existence of solutions to this problem under quite general assumptions. Various examples are presented and analyzed to show that (E) does not have unique solutions in general. We then study uniqueness structures of solutions to (E) in the convex setting by using the nonlinear adjoint method.

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arXiv2019

A unified homogenization approach for the Dirichlet problem in Perforated Domains

We revisit the periodic homogenization of Dirichlet problems for the Laplace operator in perforated domains, and establish a unified proof that works for different regimes of hole-cell ratios, that is the ratio between the scaling factor of the holes and that of the periodic cells. The approach is then made quantitative and it yields correctors and error estimates for vanishing hole-cell ratios. For positive volume fraction of holes, the approach is just the standard oscillating test function method; for vanishing volume fraction of holes, we study asymptotic behaviors of a properly rescaled cell problems and use them to build oscillating test functions. Our method reveals how the different regimes are intrinsically connected through the cell problems and the connection with periodic layer potentials.

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arXiv2018

A backscattering model based on corrector theory of homogenization for the random Helmholtz equation

This work concerns the analysis of wave propagation in random media. Our medium of interest is sea ice, which is a composite of a pure ice background and randomly located inclusions of brine and air. From a pulse emitted by a source above the sea ice layer, the main objective of this work is to derive a model for the backscattered signal measured at the source/detector location. The problem is difficult in that, in the practical configuration we consider, the wave impinges on the layer with a non-normal incidence. Since the sea ice is seen by the pulse as an effective (homogenized) medium, the energy is specularly reflected and the backscattered signal vanishes in a first order approximation. What is measured at the detector consists therefore of corrections to leading order terms, and we focus in this work on the homogenization corrector. We describe the propagation by a random Helmholtz equation, and derive an expression of the corrector in this layered framework. We moreover obtain a transport model for quadratic quantities in the random wavefield in a high frequency limit.

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