Serena Dipierro, Ovidiu Savin, Enrico Valdinoci
We show that any function can be locally approximated by solutions of prescribed linear equations of nonlocal type. In particular, we show that every function is locally $s$-caloric, up to a small error. The case of non-elliptic and non-parabolic operators is taken into account as well.
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We show that any viscosity solution to a general fully nonlinear nonlocal elliptic equation can be approximated by smooth ($C^\infty$) solutions.
Using a fixed point theorem in a proper Banach space, we prove existence and uniqueness results of positive solutions for a fractional Riemann-Liouville nonlocal thermistor problem on arbitrary nonempty closed subsets of the real numbers.
We study a nonlocal Venttsel' problem in a non-convex bounded domain with a Koch-type boundary. Regularity results of the strict solution are proved in weighted Sobolev spaces. The numerical approximation of the problem is carried out and optimal a priori error estimates are obtained.
In this paper we prove a nonlocal version of the Cordes-Niremberg estimates. We use it to extend our previous regularity results for fully nonlinear integro-differential equations to the variable coefficient case and several other settings where the equation can be thought of as a perturbation of the translation invariant case.
In this note we prove that appropriately scaled threshold dynamics-type algorithms corresponding to the fractional Laplacian of order $α\in (0,2)$ converge to moving fronts. When $α\geqq 1$ the resulting interface moves by weighted mean curvature, while for $α<1$ the normal velocity is nonlocal of ``fractional-type.'' The results easily extend to general nonlocal anisotropic threshold dynamics schemes.
A note on nonlocal approximations of sign-unrestricted solutions of conservation laws
本文研究非局部守恒律的奇异极限，证明当核趋于狄拉克分布时非局部解收敛到局部熵解，推广了符号限制情形，涵盖广义Burgers方程和Keyfitz--Kranzer系统。
Optimal Runge approximation for nonlocal wave equations and unique determination of polyhomogeneous nonlinearities
本文为线性非局部波动方程建立$L^2(0,T;\widetilde{H}^s(Ω))$中的Runge型逼近，扩展极弱解理论，并据此证明多重齐次非线性Calderón问题的唯一确定性结果。
We deal with the problem of approximating a scalar conservation law by a conservation law with nonlocal flux. As convolution kernel in the nonlocal flux, we consider an exponential-type approximation of the Dirac distribution. This enables us to obtain a total variation bound on the nonlocal term. By using this, we prove that the (unique) weak solution of the nonlocal problem converges strongly in $C(L^{1}_{\text{loc}})$ to the entropy solution of the local conservation law. We conclude with several numerical illustrations which underline the main results and, in particular, the difference bet…
Nonlocal approximation of an anisotropic cross-diffusion system
本文针对具有相分离现象的各向异性交叉扩散系统，建立其非局部到局部极限，并证明熵耗散恒等式对任意弱解成立，填补了各向异性系统非局部逼近的研究空白。
We show how the Stefan type free boundary problem with random diffusion in one space dimension can be approximated by the corresponding free boundary problem with nonlocal diffusion. The approximation problem is a slightly modified version of the nonlocal diffusion problem with free boundaries considered in [4,8]. The proof relies on the introduction of several auxiliary free boundary problems and constructions of delicate upper and lower solutions for these problems. As usual, the approximation is achieved by choosing the kernel function in the nonlocal diffusion term of the form $J_ε(x)=\fra…
We consider inertial manifolds and their approximation for a class of partial differential equations with a nonlocal Laplacian operator $-(-Δ)^{\fracα{2}}$, with $0<α<2$. The nonlocal or fractional Laplacian operator represents an anomalous diffusion effect. We first establish the existence of an inertial manifold and highlight the influence of the parameter $α$. Then we approximate the inertial manifold when a small normal diffusion $\varepsilon Δ$ (with $\varepsilon \in (0, 1)$) enters the system, and obtain the estimate for the Hausdorff semi-distance between the inertial manifolds with and…
Optimal Runge approximation for damped nonlocal wave equations and simultaneous determination results
本文针对阻尼非局部波动方程的Calderón型反问题，通过扩展非常弱解理论得到最优Runge逼近定理，进而证明了线性和半线性情形下的同时确定结果，建立了新的唯一性结果。
On the singular limit problem for nonlocal conservation laws: A general approximation result for kernels with fixed support
本文针对固定支撑且非单调的非局部核，证明当核趋近狄拉克分布时非局部守恒律解收敛至局部熵解，通过尖锐估计与熵条件分析完成收敛性证明。
We study a non local approximation of the Gaussian perimeter, proving the Gamma convergence to the local one. Surprisingly, in contrast with the local setting, the halfspace turns out to be a volume constrained stationary point if and only if the boundary hyperplane passes through the origin. In particular, this implies that Ehrhard symmetrization can in general increase the considered non local Gaussian perimeter.
Nonlocal Approximation Principle for Entropy Solutions of Scalar Conservation Laws
本文提出非局部近似原理，用于标量守恒律的熵解。通过非局部守恒律的弱星极限，证明非负初值的熵解，并探讨凸通量的收敛估计。
In this note, we develop Fourier approximation methods for the solutions of first-order nonlocal mean-field games (MFG) systems. Using Fourier expansion techniques, we approximate a given MFG system by a simpler one that is equivalent to a convex optimization problem over a finite-dimensional subspace of continuous curves. Furthermore, we perform a time-discretization for this optimization problem and arrive at a finite-dimensional saddle point problem. Finally, we solve this saddle-point problem by a variant of a primal dual hybrid gradient method.
Keller-Segel type approximation for nonlocal Fokker-Planck equations in one-dimensional bounded domain
本文研究一维有界域中非局部Fokker-Planck方程的Keller-Segel型逼近问题。通过奇异极限分析，证明该逼近对任意势函数均可行并确定了收敛速度，同时利用Chebyshev节点Lagrange插值多项式显式确定逼近所需的Keller-Segel系统参数。
Nonlocal balance equation: representation and approximation of solution
该研究针对描述无穷全同粒子系统演化的非局部平衡方程，证明叠加原理，基于源汇项的跳跃概率速率解释，用非线性马尔可夫链替代确定性动力学，以常微分方程组解逼近方程解并给出逼近速率，为数值解构造提供依据。
Nonlocal approximation of nonlinear diffusion equations
本文通过非局部偏微分方程的梯度流极限逼近退化非线性扩散方程，利用Jordan-Kinderlehrer-Otto格式构造弱解，覆盖慢扩散多孔介质方程，并给出定性粒子逼近。
We study a discrete approximation of functionals depending on nonlocal gradients. The discretized functionals are proved to be coercive in classical Sobolev spaces
Well-Posedness and Numerical Approximation of a Class of Nonlocal Elliptic Equations with Gaussian Kernels
本文研究非局部椭圆方程的适定性与数值逼近，采用有限差分与高斯核近似，并通过固定点迭代方案验证了理论结果。
We obtain some nonlocal characterizations for a class of variable exponent Sobolev spaces arising in nonlinear elasticity, in the theory of electrorheological fluids as well as in image processing for the regions where the variable exponent $p(x)$ reaches the value $1$.
We construct a deterministic, Lagrangian many-particle approximation to a class of nonlocal transport PDEs with nonlinear mobility arising in many contexts in biology and social sciences. The approximating particle system is a nonlocal version of the follow-the-leader scheme. We rigorously prove that a suitable discrete piece-wise density reconstructed from the particle scheme converges strongly in $L^1_{loc}$ towards the unique entropy solution to the target PDE as the number of particles tends to infinity. The proof is based on uniform BV estimates on the approximating sequence and on the ve…
On De Giorgi's Conjecture of Nonlocal approximations for free-discontinuity problems: The symmetric gradient case
本研究将De Giorgi关于自由不连续问题非局部逼近的猜想推广至对称梯度泛函情形，通过推广Fréchet-Kolmogorov方法证明能量等有界形变的紧性与Gamma收敛，借助GSBD空间新刻画确定极限空间，建立极限表示。
In this paper, we prove that it is always possible to define a realization of the Laplacian $Δ_{κ,θ}$ on $L^2(Ω)$ subject to nonlocal Robin boundary conditions with general jump measures on arbitrary open subsets of $\mathbb R^N$. This is made possible by using a capacity approach to define admissible pair of measures $(κ,θ)$ that allows the associated form $\mathcal E_{κ,θ}$ to be closable. The nonlocal Robin Laplacian $Δ_{κ,θ}$ generates a sub-Markovian $C_0-$semigroup on $L^2(Ω)$ which is not dominated by Neumann Laplacian semigroup unless the jump measure $θ$ vanishes. Finally, the converg…
In this note we analyse \emph{quantitative} approximation properties of a certain class of \emph{nonlocal} equations: Viewing the fractional heat equation as a model problem, which involves both \emph{local} and \emph{nonlocal} pseudodifferential operators, we study quantitative approximation properties of solutions to it. First, relying on Runge type arguments, we give an alternative proof of certain \emph{qualitative} approximation results from \cite{DSV16}. Using propagation of smallness arguments, we then provide bounds on the \emph{cost} of approximate controllability and thus quantify th…
Nonlocal-to-local limit for linear transport equations with measure initial data
研究线性输运方程的非局部到局部极限，引入非局部版本及适定性理论，推导相关方程作为极限，部分结果适用于任意维度和测度初始数据。
Approximation and uniqueness results for the nonlocal diffuse optical tomography problem
本文研究非局部漫射光学层析成像的逆问题，利用 DN 映射恢复扩散与吸收系数，并基于 Caffarelli-Silvestre 延拓和 Hahn-Banach 定理证明了方程解与 DN 映射的逼近性及系数的局部与全局唯一性。
Measure solutions, smoothing effect, and deterministic particle approximation for a conservation law with nonlocal flux
本文研究一类具有非局部通量的守恒律，首次在概率测度初值下建立存在唯一性理论，并通过分位数表述证明确定性粒子逼近定理及瞬时测度到L∞的光滑化效应。
We study an integral equation that extends the problem of anti-differentiation. We formulate this equation by replacing the classical derivative with a known nonlocal operator similar to those applied in fracture mechanics and nonlocal diffusion. We show that this operator converges weakly to the classical derivative as a nonlocality parameter vanishes. Using Fourier transforms, we find the general solution to the integral equation. We show that the nonlocal antiderivative involves an infinite dimensional set of functions in addition to an arbitrary constant. However, these functions converge …
The approximation of the quadratic porous medium equation via nonlocal interacting particles subject to repulsive Morse potential
本文提出一种基于排斥性Morse势的确定性粒子方法，证明其收敛到非局部方程弱测度解，并在Morse势局部化为Dirac delta的联合极限下逼近二次多孔介质方程，核心贡献在于利用粒子层面的$L^p$估计和熵耗散估计建立收敛性。
We construct numerical approximations for Mean Field Games with fractional or nonlocal diffusions. The schemes are based on semi-Lagrangian approximations of the underlying control problems/games along with dual approximations of the distributions of agents. The methods are monotone, stable, and consistent, and we prove convergence along subsequences for (i) degenerate equations in one space dimension and (ii) nondegenerate equations in arbitrary dimensions. We also give results on full convergence and convergence to classical solutions. Numerical tests are implemented for a range of different…
This paper is devoted to the analysis of non-negative solutions for the chemotaxis model with nonlocal source in bounded domain. The qualitative behavior of solutions is determined by the nonlinearity from the aggregation and the reaction. The nonlocal nonlinear Fisher-KPP reaction helps preventing blow-up phenomena in chemotaxis system. For appropriately chosen exponents, the global existence of classical solutions is proved with arbitrary initial data, where a modified Moser-Alikakos iteration method plays a key role in the a priori estimates.
Homogenization of nonlocal equations in randomly evolving media. Diffusion approximation
本文研究非局部方程在随机演变介质中的均质化问题，通过分析解的归一化差的极限行为，证明其收敛于线性随机偏微分方程的解。
Maximal $L_q$-regularity of nonlocal parabolic equations in higher order Bessel potential spaces
本文研究了变系数分数阶抛物型方程，在任意非负阶Bessel位势空间中建立了极大$L_q$-正则性，并应用于证明分数阶多孔介质方程与非局部Kirchhoff方程的高阶正则性和瞬时平滑性。
We consider conservation laws with nonlocal velocity and show for nonlocal weights of exponential type that the unique solutions converge in a weak or strong sense (dependent on the regularity of the velocity) to the entropy solution of the local conservation law when the nonlocal weight approaches a Dirac distribution. To this end, we establish first a uniform total variation estimate on the nonlocal velocity which enables it to prove that the nonlocal solution is entropy admissible in the limit. For the entropy solution, we use a tailored entropy flux pair which allows the usage of only one …
We study a multi-dimensional nonlocal active scalar equation of the form $u_t+v\cdot \nabla u=0$ in $\mathbb R^+\times \mathbb R^d$, where $v=Λ^{-2+α}\nabla u$ with $Λ=(-Δ)^{1/2}$. We show that when $α\in (0,2]$ certain radial solutions develop gradient blowup in finite time. In the case when $α=0$, the equations are globally well-posed with arbitrary initial data in suitable Sobolev spaces.
A uniform-in-time nonlocal approximation of the standard Fokker-Planck equation
本文提出了一种时间一致的非局部近似方法，用于研究福克-普朗克方程的收敛性，利用哈里斯定理证明了解的自适应保持性，并分析了平衡态的正则性性质。
Equivalence of weak and viscosity solutions for nonlocal p-Laplace type equations on the Heisenberg group
研究海森堡群上一类非局部p-拉普拉斯型方程弱解与粘性解的等价性，核心方法是修正下卷积的内在逼近、海森堡平滑化及适应次黎曼几何的非局部分部积分论证，克服了海森堡群非交换结构带来的困难。
We prove existence and Sobolev regularity of solutions of a nonlinear system of degenerate-parabolic PDEs with self- and cross-diffusion, transport/confinement and nonlocal interaction terms. The macroscopic system of PDEs is formally derived from a large particle system and models the evolution of an arbitrary number of species with quadratic porous-medium interactions in a bounded domain $Ω$ in any spatial dimension. The cross interactions between different species are scaled by a parameter $δ<1$, with the $δ= 0$ case corresponding to no interactions across species. A smallness condition on …
Nonlocal Approximation of Slow and Fast Diffusion
本文提出线性与非线性扩散方程的非局部粒子逼近，借助 2-Wasserstein 与对偶 Sobolev 梯度流结构证明其在慢、线性和快扩散情形的收敛性，并衍生出可采样广泛概率测度的新粒子方法。
Recently, we constructed a class of nonlocal Poisson model on manifold under Dirichlet boundary with global $\mathcal{O}(δ^2)$ truncation error to its local counterpart, where $δ$ denotes the nonlocal horizon parameter. In this paper, the well-posedness of such manifold model is studied. We utilize Poincare inequality to control the lower order terms along the $2δ$-boundary layer in the weak formulation of model. The second order localization rate of model is attained by combining the well-posedness argument and the truncation error analysis. Such rate is currently optimal among all nonlocal m…
Particle approximation of nonlocal interaction energies
本文研究Riesz型非局部能量的粒子离散化，证明其Γ-收敛于概率测度上的Riesz泛函，并给出粒子最小构型存在性的证明。
$Γ$-Limsup estimate for a nonlocal approximation of the Willmore functional
本文基于分数阶 Allen-Cahn 能量一阶变分提出 Willmore 泛函的非局部逼近，并在 Gamma-convergence 意义下证明了相应的 $Γ$-limsup 估计，是局部相场逼近结果的非局部推广。
Convergence of the numerical approximations and well-posedness: Nonlocal conservation laws with rough flux
本文研究具有不连续通量的非线性非局部守恒律，在无附加条件下证明了Godunov和Lax-Friedrichs逼近的强紧致性与熵解存在性，并首次给出该类问题的适定性结果。
In this paper we consider the numerical approximation of nonlocal integro differential parabolic equations via neural networks. These equations appear in many recent applications, including finance, biology and others, and have been recently studied in great generality starting from the work of Caffarelli and Silvestre. Based in the work by Hure, Pham and Warin, we generalize their Euler scheme and consistency result for Backward Forward Stochastic Differential Equations to the nonlocal case. We rely on Lèvy processes and a new neural network approximation of the nonlocal part to overcome the …
A nonlocal approximation of the area in codimension two
本文针对余维二情形引入分数阶 s-质量作为非局部面积泛函，并证明其在平坦拓扑下的 Γ-收敛及点态收敛到 (n-2) 维面积。
In the present study we prove rigorously that in the long-wave limit, the unidirectional solutions of a class of nonlocal wave equations to which the improved Boussinesq equation belongs are well approximated by the solutions of the Camassa-Holm equation over a long time scale. This general class of nonlocal wave equations model bidirectional wave propagation in a nonlocally and nonlinearly elastic medium whose constitutive equation is given by a convolution integral. To justify the Camassa-Holm approximation we show that approximation errors remain small over a long time interval. To be more …
On the rate of convergence of Yosida approximation for the nonlocal Cahn-Hilliard equation
针对非局部 Cahn-Hilliard 方程的 Yosida 逼近缺乏显式收敛速率的问题，本文利用极大单调算子理论和 Hilbert-Schmidt 类型算子性质，证明了显式收敛速率 $\sqrtλ$，并为 Galerkin 方法提供了误差估计。
On the accuracy of the finite volume approximations to nonlocal conservation laws
本文针对模拟交通流和人群动力学的非局部标量守恒律，在无核函数与通量单调性假设下，证明了单调有限体积格式以√Δt速率收敛到熵解，并给出数值实验验证。
A Deep BSDE approximation of nonlinear integro-PDEs with unbounded nonlocal operators
针对带无界非局部算子的高维非线性积分偏微分方程，提出基于正倒向随机微分方程表示的深度BSDE求解格式，结合小跳高斯近似、正向模拟与倒向神经网络回归，证明其在无限活动场景下的收敛性，规避维数灾难。
Nonlocal particle approximation for linear and fast diffusion equations
本文基于 2-Wasserstein 梯度流与 JKO 格式，用紧支或全局支磨光核构造线性及快速扩散方程的非局部确定性粒子近似，并证明其向局部弱解的收敛及粒子数相关收敛速率。
On the rate of the vanishing viscosity approximation for Mean Field Games with nonlocal coupling
研究具有正则化耦合的一阶时间相关平均场博弈的消失粘性逼近速率，在标准假设下改进收敛速率，为正向福克 - 普朗克方程收敛建立定量估计，获平均场博弈系统定量收敛速率，扩展加强现有理论。
Quasiparticle solutions to the 1D nonlocal Fisher--KPP equation with a fractal time derivative in the weak diffusion approximation
本文基于半经典近似与Maslov方法，提出弱扩散近似下带非整数阶分形时间导数的非局部Fisher--KPP方程的准粒子解构造方法，并通过示例验证分形参数α的影响。
We study the equation $u_t +uu_x +u-K*u=0$ in the case of an arbitrary $K \geq 0$, which is a generalization of a model for radiating gas, in which $K(y)={1/2}e^{-|y|}$. Using a monotone iteration scheme argument we establish the existence of traveling waves, which gives a solution to an open question raised by Denis Serre.
Efficient solutions for nonlocal diffusion problems via boundary-adapted spectral methods
本文提出了一种高效的边界适应谱方法，用于求解具有任意边界条件的peridynamic扩散问题，通过将卷积积分转换为傅里叶空间中的乘法运算，实现O(NlogN)的计算复杂度，并利用体积惩罚法消除常规谱方法对周期性问题的限制，从而实现高精度的边界条件或体积约束。
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