Yoshikazu Giga, Yves van Gennip, Jun Okamoto
This paper gives a framework to study a continuum limit of a gradient flow on a graph where the number of vertices increases in an appropriate way. As examples we prove the convergence of a discrete total variation flow and a discrete Allen--Cahn flow on discretised tori to their respective continuum limits.
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Internal modes of discrete solitons near the anti-continuum limit of the dNLS equation
离散非线性薛定谔（dNLS）方程的离散孤子在格点间零耦合的反连续极限下得到紧支撑。dNLS方程在离散孤子处线性化的特征值决定了其谱稳定性和线性化稳定性。在此模型中，反连续极限附近的离散孤子的所有不稳定本征值都已被刻画出来。本文分析了预解算子，证明了如果离散孤子在反连续极限内是单连通的，则预解算子在连续谱的邻域内是一致有界的。这一结果排除了存在的内部模式（中性稳定的本征值的离散谱）的反连续极限附近的离散孤子。
We demonstrate the validity of Murray's law, which represents a scaling relation for branch conductivities in a transportation network, for discrete and continuum models of biological networks. We first consider discrete networks with general metabolic coefficient and multiple branching nodes and derive a generalization of the classical 3/4-law. Next we prove an analogue of the discrete Murray's law for the continuum system obtained in the continuum limit of the discrete model on a rectangular mesh. Finally, we consider a continuum model derived from phenomenological considerations and show th…
Under suitable technical conditions we show that minimisers of the discrete interaction energy for attractive-repulsive potentials converge to minimisers of the corresponding continuum energy as the number of particles goes to infinity. We prove that the discrete interaction energy $Γ$-converges in the narrow topology to the continuum interaction energy. As an important part of the proof we study support and regularity properties of discrete minimisers: we show that continuum minimisers belong to suitable Morrey spaces and we introduce the set of empirical Morrey measures as a natural discrete…
Continuum limit of the discrete nonlinear Klein-Gordon equation
本文研究无穷格点上离散非线性Klein-Gordon方程在连续极限下的收敛性，借助Shannon插值双线性估计和离散Sobolev范数增长控制，并展望了格点上非线性波的均匀色散估计。
In the recent trend of extending discrete-to-continuum limit passages for gradient flows of single-species particle systems with singular and nonlocal interactions to particles of opposite sign, any annihilation effect of particles with opposite sign has been side-stepped. We present the first rigorous discrete-to-continuum limit passage which includes annihilation. This result paves the way to applications such as vortices, charged particles, and dislocations. In more detail, the discrete setting of our discrete-to-continuum limit passage is given by particles on the real line. Particles of t…
A discrete to continuum analysis of dislocations in nanowire heterostructures
外延生长的异质纳米线半径较大时，在相界面处会出现位错。我们考虑了一个相应的具有二次成对原子相互作用能的变分离散模型。通过使用伽玛收敛的概念和几何刚性估计，我们执行了一个离散到连续极限和降维到一维系统。此外，我们比较了一个无缺陷的模型和模型的界面处的位错，并表明，后者是积极方便，如果线的厚度是足够大的。
Motivated by recent physics papers describing the formation of biological transport networks we study a discrete model proposed by Hu and Cai consisting of an energy consumption function constrained by a linear system on a graph. For the spatially two-dimensional rectangular setting we prove the rigorous continuum limit of the constrained energy functional as the number of nodes of the underlying graph tends to infinity and the edge lengths shrink to zero uniformly. The proof is based on reformulating the discrete energy functional as a sequence of integral functionals and proving their $Γ$-co…
We consider discrete nonlinear Schrödinger equations (DNLS) on the lattice $h\mathbb{Z}^d$ whose linear part is determined by the discrete Laplacian which accounts only for nearest neighbor interactions, or by its fractional power. We show that in the continuum limit $h\to 0$, solutions to DNLS converge strongly in $L^2$ to those to the corresponding continuum equations, but a precise rate of convergence is also calculated. In particular cases, this result improves weak convergence in Kirkpatrick, Lenzmann and Staffilani \cite{KLS}. Our proof is based on a suitable adjustment of dispersive PDE…
Discrete and Continuum Area-Preserving Mean-Curvature Flow of Rectangles
本研究结合约束最小化运动格式与离散到连续分析，研究二维晶格晶体的保面积平均曲率型运动，刻画了连续与离散模型的控制方程，并证明时空尺度同速率趋于零时离散到连续演化由不可退化为ODE的微分包含系统描述。
We study the atomistic-to-continuum limit of a class of energy functionals for crystalline materials via Gamma-convergence. We consider energy densities that may depend on interactions between all points of the lattice and we give conditions that ensure compactness and integral-representation of the continuum limit on the space of special functions of bounded variation. This abstract result is complemented by a homogenization theorem, where we provide sufficient conditions on the energy densities under which bulk- and surface contributions decouple in the limit. The results are applied to long…
In this paper we rigorously investigate the emergence of defects on Nematic Shells with genus different from one. This phenomenon is related to a non trivial interplay between the topology of the shell and the alignment of the director field. To this end, we consider a discrete $XY$ system on the shell $M$, described by a tangent vector field with unit norm sitting at the vertices of a triangulation of the shell. Defects emerge when we let the mesh size of the triangulation go to zero, namely in the discrete-to-continuum limit. In this paper we investigate the discrete-to-continuum limit in te…
We prove a compactness and integral-representation theorem for sequences of families of lattice energies describing atomistic interactions defined on lattices with vanishing lattice spacing. The densities of these energies may depend on interactions between all points of the corresponding lattice contained in a reference set. We give conditions that ensure that the limit is an integral defined on a Sobolev space. A homogenization theorem is also proved. The result is applied to multibody interactions corresponding to discrete Jacobian determinants and to linearizations of Lennard-Jones energie…
Discrete-to-continuum crystalline curvature flows
针对晶体平均曲率流，本文提出一种基于新离散距离的完全离散隐式变分格式，以消除空间漂移与钉扎，并证明其在任意维度和广泛纯晶体各向异性下的严格收敛性。
We consider the dispersion managed nonlinear Schrödinger equation with power-law nonlinearity and its discrete version of equations with step size $h\in(0,1]$. We prove that the solutions of the discrete equations strongly converge in $L^2(\mathbb{R})$ to the solution of the dispersion managed NLS as $h\to 0$ after showing the global well-posedness of the discrete equations.
Microstructures in a two-dimensional frustrated spin system: Scaling regimes and a discrete-to-continuum limit
本文针对二维正方晶格 $J_1$-$J_3$ 自旋模型在不相容铁磁边界条件下的图案形成问题，推导了螺旋磁/铁磁转变点处的离散到连续 $\Gamma$ 极限，并研究了离散最小能量的标度律，证明在某些参数区域涡旋结构的形成在能量上更有利。
In materials science, wedge disclinations are defects caused by angular mismatches in the crystallographic lattice. To describe such disclinations, we introduce an atomistic model in planar domains. This model is given by a nearest-neighbor-type energy for the atomic bonds with an additional term to penalize change in volume. We enforce the appearance of disclinations by means of a special boundary condition. Our main result is the discrete-to-continuum limit of this energy as the lattice size tends to zero. Our proof method is relaxation of the energy. The main mathematical novelty of our pro…
Discrete Quantitative Isocapacitary Inequality: Fluctuation Estimates
本文研究离散等容量问题在集合规模趋于无穷时的波动估计，通过将变分问题从离散扩展到连续域，并结合精确的连续等容量不等式，揭示了离散情况下的最小化器特性。
We study an atomistic model that describes the microscopic formation of material voids inside elastically stressed solids under an additional curvature regularization at the discrete level. Using a discrete-to-continuum analysis, by means of a recent geometric rigidity result in variable domains [27] and Γ-convergence tools, we rigorously derive effective linearized continuum models for elastically stressed solids with material voids in three-dimensional elasticity.
We present a spatial hybrid discrete-continuum modelling framework for the interaction dynamics between tumour cells and cytotoxic T cells, which play a pivotal role in the immune response against tumours. In this framework, tumour cells and T cells are modelled as individual agents while chemokines that drive the chemotactic movement of T cells towards the tumour are modelled as a continuum. We formally derive the continuum counterpart of this model, which is given by a coupled system that comprises an integro-differential equation for the density of tumour cells, a partial differential equat…
We study the discrete-to-continuum limit of ferromagnetic spin systems when the lattice spacing tends to zero. We assume that the atoms are part of a (maybe) non-periodic lattice close to a flat set in a lower dimensional space, typically a plate in three dimensions. Scaling the particle positions by a small parameter $\varepsilon>0$ we perform a $Γ$-convergence analysis of properly rescaled interfacial-type energies. We show that, up to subsequences, the energies converge to a surface integral defined on partitions of the flat space. In the second part of the paper we address the issue of sto…
Discrete-to-continuum linearization in atomistic dynamics
本文将原子到连续与线性化极限的 Γ-收敛分析从静态推广到动态，证明原子运动方程解收敛到线性化弹性连续体动量方程解，并获得能量的逐时收敛。
From discrete to continuum in the helical XY-model: emergence of chirality transitions in the $S^1$ to $S^2$ limit
研究受挫S^2自旋系统在离散到连续极限下的手性转变，通过限制自旋于覆盖S^2的S^1副本上，确定了能量标度与发散速率阈值，使得Γ-极限在连续极限下能捕捉手性转变并保持S^2能量描述。
A low frequency approximation of the discrete Sommerfeld diffraction problems, involving the scattering of a time harmonic lattice wave incident on square lattice by a discrete Dirichlet or a discrete Neumann half-plane, is investigated. It is established that the exact solution of the discrete model converges to the solution of the continuum model, i.e. the continuous Sommerfeld problem, in certain discrete Sobolev space defined by W. Hackbusch. The proof of convergence has been provided for both types of boundary conditions when the imaginary part of incident wavenumber is positive.
On a discrete-to-continuum convergence result for a two dimensional brittle material in the small displacement regime
我们考虑了一个二维原子质量弹簧系统，并表明，在小位移制度的相应的离散能量可以涉及到一个连续的Griffith能量泛函的意义上的伽玛收敛。我们还分析了拉伸边界条件下的矩形杆的连续性问题，发现根据边界载荷的极小化是均匀的弹性变形或配置，是完全开裂一般沿结晶线。作为应用，我们讨论了应变晶体的解理性质和磁体的有效连续断裂能。
Remarks on discrete Dirac operators and their continuum limits
本文评述离散 Dirac 算子的定义，针对直接离散化导致的费米子倍增问题，分析 Wilson 项与 KS-fermion（交错费米子）模型的数学表述及其连续极限。
We prove discrete-to-continuum convergence of interaction energies defined on lattices in the Euclidean space (with interactions beyond nearest neighbours) to a crystalline perimeter, and we discuss the possible Wulff shapes obtainable in this way. Exploiting the "multigrid construction" of quasiperiodic tilings (which is an extension of De Bruijn's "pentagrid" construction of Penrose tilings) we adapt the same techniques to also find the macroscopical homogenized perimeter when we microscopically rescale a given quasiperiodic tiling.
On the continuum limit for discrete NLS with long-range lattice interactions
考虑网格步长为$h>0$的格点$h \mathbb{Z}$上的一类离散非线性Schroedinger方程（DNLS）.在$h \to 0$的连续极限下，我们证明了极限动力学是由$\mathbb{R}$上的一个非线性Schroedinger方程（NLS）给出的，该方程以分数Laplacian $(-Δ)^α$为色散符号.特别地，我们得到分数幂$1/2 < α< 1$在传递到连续极限时由长程晶格相互作用产生，而具有非分数拉普拉斯$-Δ$的NLS描述了短程晶格相互作用的连续极限中的色散（例如，最近邻相互作用）。我们的结果严格证明某些NLS模型方程与分数拉普拉斯在物理文献中提出的。此外，本文所给出的结论也可用于讨论其他长程相互作用晶格系统的连续极限。
Continuum limit of resonances for discrete Schrödinger operators
本文证明离散 Schrödinger 算子的复共振在连续极限下收敛到连续算子的共振，并借助推广的范数预解收敛与 Fourier 空间复畸变方法，为奇异势情形提供近似离散算子构造方案。
Enlarging on Parts I, II, and III we write more equations in the desired format of the extended abstract theory of composites. We focus on a multitude of equations involving higher order derivatives. The motivation is that results and methods in the theory of composites then extend to these equations.
We study a discrete approximation of functionals depending on nonlocal gradients. The discretized functionals are proved to be coercive in classical Sobolev spaces
Discrete-to-continuum convergence of the density of states for Mathieu's equation
本文研究了Mathieu方程的离散到连续态密度收敛性，证明了紧束缚模型在缓慢变化周期性势下的态密度收敛到Mathieu型方程的态密度。
Discrete-to-continuum models of pre-stressed cytoskeletal filament networks
本文提出离散到连续的细胞骨架力学模型，用于预测预应力蛋白丝网络的力学响应，并验证了模型在波形蛋白网络中的有效性，发现刚度随预应力和珠子尺寸呈特定标度关系。
Continuum limit for discrete NLS with memory effect
本文研究带长程相互作用和记忆效应的离散非线性薛定谔方程在网格尺寸趋于零时的连续极限，证明其解在L²中强收敛到连续NLS型方程的解并给出收敛速率，方法推广了Hong-Yang及Kenig-Ponce-Vega的经典估计框架。
Discrete-to-continuum limits of optimal transport with linear growth on periodic graphs
本文研究了在周期图上具有线性增长能量密度的动态最优传输的离散到连续极限，扩展了已知文献到一些重要的例子类，如1-沃瑟斯坦传输问题的缩放极限，并讨论了图的几何结构对极限成本函数结构的影响。
Discrete-to-continuum limit for nonlinear reaction-diffusion systems via EDP convergence for gradient systems
本文针对满足细致平衡条件的质量作用定律反应扩散系统，构造空间离散过程，证明解与梯度系统层面的离散到连续收敛，并通过链式法则不等式和变分方法实现。
Quantitative approximation of stochastic kinetic equations: from discrete to continuum
本文研究带奇异系数的动力学型随机微分方程，证明低正则漂移下系统在强弱意义下良定义，且通用驯服EM格式均以1/2速率收敛。
Dynamics of periodic fractional discrete NLS in the continuum limit
本文研究周期格点上分数阶离散非线性薛定谔方程在连续极限下的收敛性，证明其收敛到分数阶NLS并获得最优收敛速率，同时揭示调制不稳定性中最大增益对振幅依赖关系的转变。
We consider dynamics driven by interaction energies on graphs. We introduce graph analogues of the continuum nonlocal-interaction equation and interpret them as gradient flows with respect to a graph Wasserstein distance. The particular Wasserstein distance we consider arises from the graph analogue of the Benamou-Brenier formulation where the graph continuity equation uses an upwind interpolation to define the density along the edges. While this approach has both theoretical and computational advantages, the resulting distance is only a quasi-metric. We investigate this quasi-metric both on g…
Discrete and continuum links to a nonlinear coupled transport problem of interacting populations
研究种群相互作用的非线性耦合传输问题，结合粒子系统与连续模型，利用相对熵、Lyapunov函数和数值方法分析交叉扩散与热扩散问题。
Linking discrete and continuum diffusion models: Well-posedness and stable finite element discretizations
本文研究了连续与离散扩散模型的结合，证明了其良好定义性和稳定性，并通过数值实验展示了方法在处理不完整数据时的应用。
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