arXiv subjects

Haigang Li

Publications and source records attributed to Haigang Li.

At least 19 recordsLinked to original sources

arXiv2026

High-order finite element method for perfect conductivity and linear elasticity with nearly touching inclusions

In perfect conductivity and linear elasticity problems, the electric field and stress always become highly concentrated within narrow regions between adjacent perfect (rigid) inclusions, and blow up as the distance between inclusions approaches zero. The design of high-order numerical methods with rigorous error analysis for such concentration problems remains open. In this paper, we present the first high-order finite element method for solving these problems. Our approach is based on asymptotic estimates of high-order derivatives of solutions, employing a graded mesh and auxiliary basis functions specifically designed from these derivative estimates. We prove that the proposed method converges with an $H^1$-error bound of $O(h^p)$ and the error bound is independent of the distance (possibly approaching zero) between inclusions, where $h$ is the mesh size and $p$ is the degree of finite elements. Numerical examples in both two and three dimensions are presented to demonstrate the convergence rates of the numerical solutions. In particular, the blow-up behaviors of the gradients of solutions are demonstrated when the inclusions approach each other.

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arXiv2025

Higher derivative estimates for Stokes equations with closely spaced rigid inclusions in three dimensions

In this paper, we establish higher-order derivative estimates for the Stokes equations in a three-dimensional domain containing two closely spaced rigid inclusions. We construct a sequence of auxiliary functions via an inductive process to isolate the leading singular terms of higher-order derivatives within the narrow region between the inclusions. For a class of convex inclusions of general shapes, the construction of three-dimensional auxiliary functions -- unlike the two-dimensional case -- relies on the decay properties of solutions to a class of two-dimensional partial differential equations with singular coefficients. Taking advantage of this, we obtain pointwise upper bounds of derivatives up to the seventh order for general inclusions. Under additional symmetry conditions, we derive optimal estimates for derivatives of arbitrary order. Consequently, we obtain precise blow-up rates for the Cauchy stress and its higher-order derivatives in the narrow region between the inclusions.

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arXiv2025

Optimal gradient estimates for conductivity problems with imperfect low-conductivity interfaces

This paper studies field concentration between two nearly touching conductors separated by imperfect low-conductivity interfaces, modeled by Robin boundary conditions. It is known that for any sufficiently small interfacial bonding parameter $\gamma > 0$, the gradient remains uniformly bounded with respect to the separation distance $\varepsilon$. In contrast, for the perfect bonding case ($\gamma = 0$, corresponding to the perfect conductivity problem), the gradient may blow up as $\varepsilon \to 0$ at a rate depending on the dimension. In this work, we establish optimal pointwise gradient estimates that explicitly depend on both $\gamma$ and $\varepsilon$ in the regime where these parameters are small. These estimates provide a unified framework that encompasses both the previously known bounded case ($\gamma > 0$) and the singular blow-up scenario ($\gamma = 0$), thus furnishing a complete and continuous characterization of the gradient behavior throughout the transition in $\gamma$. The key technical achievement is the derivation of new regularity results for elliptic equations as $\gamma\to0$, along with a case dichotomy based on the relative sizes of $\gamma$ and a distance function $\delta(x')$. Our results hold for strictly relatively convex conductors in all dimensions $n \geq 2$.

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arXiv2025

Resonant frequencies distribution for multiple closely spaced subwavelength resonators

In this paper, we investigate a resonant system comprising $N$ closely packed spherical resonators ($N>2$). We analyze how the spatial arrangement of these resonators influences the distribution of resonant frequencies, focusing on leading-order terms. Furthermore, we characterize the asymptotic behavior of resonant modes linked to their respective frequencies. Our results demonstrate distinct trends across configurations: For single-row alignment, the system exhibits $N$ clearly separated resonant frequencies; For multi-row arrangements, the resonant frequency range broadens, though the total number of frequencies may diminish; while for ring configurations, comparable frequency ranges to chain arrangements emerge, but with fewer resonant frequencies. We derive explicit analytical expressions to quantify these frequency distributions. Regarding resonant modes, we identify that at specific frequencies, the gradient of these modes may exhibit different asymptotic behavior between different resonators.

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arXiv2024

Optimal higher derivative estimates for Stokes equations with closely spaced rigid inclusions

In this paper, we study the interaction between two closely spaced rigid inclusions suspended in a Stokes flow. It is well known that the stress significantly amplifies in the narrow region between the inclusions as the distance between them approaches zero. To gain deeper insight into these interactions, we derive high-order derivative estimates for the Stokes equation in the presence of two rigid inclusions in two dimensions. Our approach resonates with the method used to handle the incompressibility constraint in the standard convex integration scheme. Under certain symmetric assumptions on the domain, these estimates are shown to be optimal. As a result, we establish the precise blow-up rates of the Cauchy stress and its higher-order derivatives in the narrow region.

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arXiv2024

Stress concentration between two adjacent rigid particles in Navier-Stokes flow

In this paper we investigate the stress concentration problem that occurs when two convex rigid particles are closely immersed in a fluid flow. The governing equations for the fluid flow are the stationary incompressible Navier-Stokes equations. We establish precise upper bounds for the gradients and second-order derivatives of the fluid velocity as the distance between particles approaches zero, in dimensions two and three. The optimality of these blow-up rates of the gradients is demonstrated by deriving corresponding lower bounds. New difficulties arising from the nonlinear term in the Navier-Stokes equations is overcome. Consequently, the blow up rates of the Cauchy stress are studied as well.

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arXiv2024

Optimal higher derivative estimates for solutions of the Lam\'e system with closely spaced hard inclusions

We investigate higher derivative estimates for the Lam\'e system with hard inclusions embedded in a bounded domain in $\mathbb{R}^{d}$. As the distance $\varepsilon$ between two closely spaced hard inclusions approaches zero, the stress in the narrow regions between the inclusions increases significantly. This stress is captured by the gradient of the solution. The key contribution of this paper is a detailed characterization of this singularity, achieved by deriving higher derivative estimates for solutions to the Lam\'e system with partially infinite coefficients. These upper bounds are shown to be sharp in two and three dimensions when the domain exhibits certain symmetries. To the best of our knowledge, this is the first work to precisely quantify the singular behavior of higher derivatives in the Lam\'e system with hard inclusions.

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arXiv2024

Optimal boundary gradient estimates for the insulated conductivity problem

In this paper we study the boundary gradient estimate of the solution to the insulated conductivity problem with the Neumann boundary data when a convex insulating inclusion approaches the boundary of the matrix domain. The gradient of solutions may blow up as the distance between the inclusion and the boundary, denoted as $\varepsilon$, approaches to zero. The blow up rate was previously known to be sharp in dimension $n=2$ (see Ammari et al.\cite{AKLLL}). However, the sharp rates in dimensions $n\geq3$ are still unknown. In this paper, we solve this problem by establishing upper and lower bounds on the gradient and prove that the optimal blow up rates of the gradient are always of order $\epsilon^{-1/2}$ for general strictly convex inclusions in dimensions $n\geq3$. Several new difficulties are overcome and the impact of the boundary data on the gradient is specified. This result highlights a significant difference in blow-up rates compared to the interior estimates in recent works (\cites{LY,Weinkove,DLY,DLY2,LZ}), where the optimal rate is $\epsilon^{-1/2+\beta(n)}$, with $\beta(n)\in(0,1/2)$ varying with dimension $n$. Furthermore, we demonstrate that the gradient does not blow up for the corresponding Dirichlet boundary problem.

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arXiv2024

Higher order parabolic systems with piecewise DMO and H\"{o}lder continuous coefficients

In this paper, we are concerned with divergence form, higher-order parabolic systems in a cylindrical domain with a finite number of subdomains. We establish $L_\infty$ and Schauder estimates of solutions when the leading coefficients and the non-homogeneous term exhibit piecewise Dini mean oscillation and piecewise H\"{o}lder continuity, respectively. To the best of our knowledge, our results are new for higher-order elliptic and parabolic systems.

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arXiv2024

Optimal gradient estimates for the insulated conductivity problem with general convex inclusions case

We study the insulated conductivity problem which involves two adjacent convex insulators embedded in a bounded domain. It is known that the gradient of solutions may blow up as the distance between the two inclusions tends to zero. However, the sharpness of the blow up rate for general convex insulator case in dimension $n\geq3$ has remained open. The novelty of this paper is that we answer this problem affirmatively by establishing a pointwise upper bound of the gradient for general convex insulators, along with a corresponding lower bound that achieves optimal blow up rates. These rates are associated with the first nonzero eigenvalue of an elliptic operator determined by the geometry of insulators. Our results improve and make complete the previous result for ball insulators case studied in \cite{DLY}.

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arXiv2023

Estimates for stress concentration between two adjacent rigid inclusions in Stokes flow

In this paper, we establish the estimates for the gradient and the second-order partial derivatives for the Stokes flow in the presence of two closely located strictly convex inclusions in dimension three. Moreover, the blow-up rate of the gradient is showed to be optimal by a pointwise upper bound and a lower bound in the narrowest region. We also show the optimal blow-up rate of Cauchy stress tensor. In dimensions greater than three, the upper bounds of the gradient are established. These results answer the questions raised in [25].

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arXiv2023

On higher regularity of Stokes systems with piecewise H\"{o}lder continuous coefficients

In this paper, we consider higher regularity of a weak solution $({\bf u},p)$ to stationary Stokes systems with variable coefficients. Under the assumptions that coefficients and data are piecewise $C^{s,\delta}$ in a bounded domain consisting of a finite number of subdomains with interfacial boundaries in $C^{s+1,\mu}$, where $s$ is a positive integer, $\delta\in (0,1)$, and $\mu\in (0,1]$, we show that $D{\bf u}$ and $p$ are piecewise $C^{s,\delta_{\mu}}$, where $\delta_{\mu}=\min\big\{\frac{1}{2},\mu,\delta\big\}$. Our result is new even in the 2D case with piecewise constant coefficients.

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arXiv2023

Stress blow-up analysis when suspending rigid particles approach boundary in 3D Stokes flow

The stress concentration is a common phenomenon in the study of fluid-solid model. In this paper, we investigate the boundary gradient estimates and the second order derivatives estimates for the Stokes flow when the rigid particles approach the boundary of the matrix in dimension three. We classify the effect on the blow-up rates of the stress from the prescribed various boundary data: locally constant case and locally polynomial case. Our results hold for general convex inclusions, including two important cases in practice, spherical inclusions and ellipsoidal inclusions. The blow-up rates of the Cauchy stress in the narrow region are also obtained. We establish the corresponding estimates in higher dimensions greater than three.

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arXiv2023

The interaction between two close-to-touching convex acoustic subwavelength resonators

The Minneart resonance is a low frequency resonance in which the wavelength is much larger than the size of the resonators. It is interesting to study the interaction between two adjacent bubbles when they are brought close together. Because the bubbles are usually compressible, in this paper we mainly investigate resonant modes of two general convex resonators with arbitrary shapes to extend the results of Ammari, Davies, Yu in [4], where a pair of spherical resonators are considered by using bispherical coordinates. We combine the layer potential method for Helmholtz equation in [4,5] and the elliptic theory for gradient estimates in [26,30] to calculate the capacitance coefficients for the coupled $C^{2,\alpha}$ resonators, then show the leading-order asymptotic behaviors of two different resonant modes and reveal the dependance of the resonant frequencies on their geometric properties, such as convexity, volumes and curvatures. By the way, the blow-up rates of gradient of the scattered pressure are also presented.

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arXiv2023

Stress blow-up analysis when a suspending rigid particle approaches the boundary in Stokes flow: 2D case

It is an interesting and important topic to study the motion of small particles in a viscous liquid in current applied research. In this paper we assume the particles are convex with arbitrary shapes and mainly investigate the interaction between the rigid particles and the domain boundary when the distance tends to zero. In fact, even though the domain and the prescribed boundary data are both smooth, it is possible to cause a definite increase of the blow-up rate of the stress. This problem has the free boundary value feature due to the rigidity assumption on the particle. We find that the prescribed local boundary data directly affects on the free boundary value on the particle. Two kinds of boundary data are considered: locally constant boundary data and locally polynomial boundary data. For the former we prove the free boundary value is close to the prescribed constant, while for the latter we show the influence on the blow-up rate from the order of growth of the prescribed polynomial. Based on pointwise upper bounds in the neck region and lower bounds at the midpoint of the shortest line between the particle and the domain boundary, we show that these blow-up rates obtained in this paper are optimal. These precise estimates will help us understand the underlying mechanism of the hydrodynamic interactions in fluid particle model.

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arXiv2022

The Dirichlet problem for Hessian quotient equations on exterior domains

In this paper, we consider the exterior Dirichlet problem for Hessian quotient equations with the right hand side $g$, where $g$ is a positive function and $g=1+O(|x|^{-β})$ near infinity, for some $β>2$. Under a prescribed generalized symmetric asymptotic behavior at infinity, we establish an existence and uniqueness theorem for viscosity solutions, by using comparison principles and Perron's method. This extends the previous results for Monge--Ampère equations and Hessian equations.

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arXiv2022

Estimates for stress concentration between two adjacent rigid inclusions in two-dimensional Stokes flow

It is vital important in material sciences and fluid mechanics to study the field enhancements in the narrow region between two inclusions. Complex fluids including particle suspensions usually result in complicated flow behavior. In this paper we establish the pointwise upper bounds of the gradient and the second-order partial derivatives for the Stokes flow when two rigid particles are closely spaced suspending in an open bounded domain and away from the boundary in dimension two. Moreover, the lower bounds of the gradient estimates at the narrowest place of the neck region show the optimality of the blow-up rate. These results are valid for inclusions with arbitrary shape.

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arXiv2020

Blow-up analysis of hydrodynamic forces exerted on two adjacent $M$-convex particles

In a viscous incompressible fluid, the hydrodynamic forces acting on two close-to-touch rigid particles in relative motion always become arbitrarily large, as the interparticle distance parameter $\varepsilon$ goes to zero. In this paper we obtain asymptotic formulas of the hydrodynamic forces and torque in $2\mathrm{D}$ model and establish the optimal upper and lower bound estimates in $3\mathrm{D}$, which sharply characterizes the singular behavior of hydrodynamic forces. These results reveal the effect of the relative convexity between particles, denoted by index $m$, on the blow-up rates of hydrodynamic forces. Further, when $m$ degenerates to infinity, we consider the particles with partially flat boundary and capture that the largest blow-up rate of the hydrodynamic forces is $\varepsilon^{-3}$ both in 2D and 3D. We also clarify the singularities arising from linear motion and rotational motion, and find that the largest blow-up rate induced by rotation appears in all directions of the forces.

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