Valerio Pagliari
Taking up a variational viewpoint, we present some nonlocal-to-local asymptotic results for various kinds of integral functionals. The content of the thesis comprises the contributions first appeared in some research papers in collaboration with J. Berendsen, A. Cesaroni, A. Chambolle, and M. Novaga.
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Convergence of solutions of a rescaled evolution nonlocal cross-diffusion problem to its local diffusion counterpart
本文证明对可积核作适当重标度后，SKT 非局部交叉扩散问题的解序列收敛到局部扩散问题的解，从而为后者解的存在性提供了新证明。
We study the rate of convergence of some nonlocal functionals recently considered by Bourgain, Brezis and Mironescu. In particular we establish the $Γ$-convergence of the corresponding rate functionals, suitably rescaled, to a limit functional of second order.
We prove the local minimality of halfspaces in Carnot groups for a class of nonlocal functionals usually addressed as nonlocal perimeters. Moreover, in a class of Carnot groups in which the De Giorgi's rectifiability Theorem holds, we provide a lower bound for the $Γ$-liminf of the rescaled energy in terms of the horizontal perimeter.
We consider nonlocal curvature functionals associated with positive interaction kernels, and we show that local anisotropic mean curvature functionals can be retrieved in a blow-up limit from them. As a consequence, we prove that the viscosity solutions to the rescaled nonlocal geometric flows locally uniformly converge to the viscosity solution to the anisotropic mean curvature motion. The result is achieved by combining a compactness argument and a set-theoretic approach related to the theory of De Giorgi's barriers for evolution equations.
In this paper, we study some qualitative properties for an evolution problem that combines local and nonlocal diffusion operators acting in two different subdomains and, coupled in such a way that, the resulting evolution problem is the gradient flow of an energy functional. The coupling takes place at the interface between the regions in which the different diffusions take place. We prove existence and uniqueness results, as well as, that the model preserves the total mass of the initial condition. We also study the asymptotic behavior of the solutions. Finally, we show a suitable way to reco…
In this paper we consider a model that involves nonlocal diffusion and a classical convective term. Using a scaling argument and a new compactness argument we obtain the first term in the asymptotic behavior of the solutions.
In this paper we study the asymptotic behaviour of a nonlocal nonlinear parabolic equation governed by a parameter. After giving the existence of unique branch of solutions composed by stable solutions in stationary case, we gives for the parabolic problem $L^\infty $ estimates of solution based on using the Moser iterations and existence of global attractor. We finish our study by the issue of asymptotic behaviour in some cases when $t\to \infty$.
In this paper we study the asymptotic behaviour of a nonlocal nonlinear parabolic equation governed by a parameter. After giving the existence of unique branch of solutions composed by stable solutions in stationary case, we gives for the parabolic problem $L^\infty $ estimates of solution based on using the Moser iterations and existence of global attractor. We finish our study by the issue of asymptotic behaviour in some cases when $t\to \infty$.
In this paper we prove the existence of the global attractor for the wave equation with nonlocal weak damping, nonlocal anti-damping and critical nonlinearity.
For some spatially nonlocal diffusion models with a finite range of nonlocal interactions measured by a positive parameter $δ$, we review their formulation defined on a bounded domain subject to various conditions that correspond to some inhomogeneous data. We consider their consistency to similar inhomogeneous boundary value problems of classical partial differential equation (PDE) models as the nonlocal interaction kernel gets localized in the local $δ\to 0$ limit, and at the same time, for rescaled fractional type kernels, to corresponding inhomogeneous nonlocal boundary value problems of f…
In this note we study the behaviour as $s\to 0^+$ of some semigroup based Besov seminorms associated with a non-symmetric and hypoelliptic diffusion with a drift. Our results generalise a previous one of Maz'ya and Shaposhnikova for the classical fractional Sobolev spaces $W^{s,p}$, and they also underscore a new phenomenon caused by the presence of the drift.
Positivity and asymptotic behaviour of solutions to a generalized nonlocal fast diffusion equation
针对含类稳定算子与奇异非线性项的广义非局部快扩散方程，通过弱Harnack不等式分析解的正性并推导其渐近等价于标准分数阶快扩散方程的Barenblatt解，突破了算子一般性带来的方法限制。
Nonlocal Diffusion Elliptic System Modelling The Behaviour Of a Bacteria And a Living Nutrient
本文研究模拟细菌与活性养料行为的非局部椭圆系统，通过分支方法和隐函数定理证明了共存态正解的存在性与唯一性。
We study a class of integral functionals known as nonlocal perimeters, which, intuitively, express a weighted interaction between a set and its complement. The weight is provided by a positive kernel K, which might be singular. In the first part of the paper, we show that these functionals are indeed perimeters in a generalised sense and we establish existence of minimisers for the corresponding Plateau problem. Also, when K is radial and strictly decreasing, we prove that halfspaces are minimisers if we prescribe flat boundary conditions. A Gamma-convergence result is discussed in the second …
Long-time behaviour of a nonlocal model for electroporation
研究非局部电穿孔模型的长期行为，证明了渐近自相似解的局部稳定性，通过分析一阶矩方程和与传输问题的比较，揭示了传输项对长期行为的驱动作用。
Ole\uınik-type estimates for nonlocal conservation laws and applications to the nonlocal-to-local limit
研究带指数核的非局部守恒律，证明非局部项满足 Ole\uınik 型熵条件，并推导出核缩放至 Dirac delta 时非局部弱解收敛到局部守恒律的唯一熵容许解。
We study local (the heat equation) and nonlocal (convolution type problems with an integrable kernel) evolution problems on a metric connected finite graph in which some of the edges have infinity length. We show that the asymptotic behaviour of the solutions to both local and nonlocal problems is given by the solution of the heat equation, but on a star shaped graph in which there is only one node and as many infinite edges as in the original graph. In this way we obtain that the compact component that consists in all the vertices and all the edges of finite length can be reduced to a single …
On the asymptotic behaviour of the restricted Dirichlet Laplacian of real order $s>0$ on stretching tubes
该研究针对拉伸（变细）管上实阶$s>0$的受限Dirichlet拉普拉斯算子，分析其第一特征值的渐近行为，并推导缩放后算子的$G$-极限与对应二次型的$\u0393$-极限，涵盖分数阶与整数阶（多重调和算子）情形。
We study the large-time behaviour of the solutions of the evolution equation involving nonlinear diffusion and gradient absorption, $$ \partial_t u - Δ_p u + |\nabla u|^q=0 . $$ We consider the problem posed for $x\in \real^N$ and t>0 with nonnegative and compactly supported initial data. We take the exponent p>2 which corresponds to slow p-Laplacian diffusion. The main feature of the paper is that the exponent q takes the critical value q=p-1 which leads to interesting asymptotics. This is due to the fact that in this case both the Hamilton-Jacobi term $ |\nabla u|^q$ and the diffusive term $…
We obtain upper bounds for the quenching time of the solutions of the nonlocal parabolic MEMS equation $u_t=Δu+\lam/(1-u)^2(1+χ\int_Ω1/(1-u) dx)^2$ in $Ω\times (0,\infty)$, $u=0$ on $\1Ω\times (0,\infty)$, $u(x,0)=u_0$ in $Ω$, when $λ$ is large. We prove the compactness of the quenching set under a mild condition on the initial data. When $Ω=B_R$ and $u_0$ is radially symmetric and monotone decreasing in $0\le r\le R$, we prove that the point $x=0$ is the only possible quenching set. When $u_0$ also satisfies some strict concavity assumption, we prove that for any $β\in (2,3)$ the solution sat…
Positivity and long-term behaviour of a diffusion model with measure-valued nonlocal reaction term
研究了在实线上具有非局部和奇异反应项的扩散方程解的行为，即由原点处的狄拉克源或汇给出。通过简化表示例如一个能够远距离感知浓度但干预原点的控制系统。正解（对于正初始条件）不能在所有参数设置下保证。我们确定了参数范围和正初始条件的单调性和对称性条件，使得可以得出解在所有时间的正性。此外，我们提供了确保系统收敛到常数稳态（点wise）的条件，该稳态位于观测区域之外。技术上，我们广泛使用拉普拉斯变换论证来获得这些结果。
Nonlocal to Local Convergence of Stefan Problems Under Optimal Convergence Condition
针对非局部扩散自由边界可能不连续扩张的问题，本文提出不依赖核函数对称性或紧性的最优 Fourier 收敛条件，并利用变分不等式、先验估计和 Fourier 变换证明重尺度化非局部问题的解收敛到经典 Stefan 问题的解。
Gamma-convergence as $s\to1^-$ of anisotropic nonlocal fractional perimeter functionals
本文研究了各向异性非局部 $s$-分数阶周长泛函当 $s\to1^-$ 时的 $Γ(L^1_{loc})$-收敛渐近行为，证明了其重新缩放后收敛到合适的各向异性周长，且该结论在 $\mathbb{R}^n$ 和具有 Lipschitz 边界的有界区域上均成立。
Large time behaviour in the multidimensional nonlocal Fisher-KPP equation
本文研究多维非局部Fisher-KPP方程等值集的扩张，推导出渐近对数延迟及方向依赖的常数修正项，深化了Freidlin-Gärtner型公式的精度。
Large-time behaviour for anisotropic stable nonlocal diffusion problems with convection
研究带对流项的非局部热方程非负解的大时间行为，发现其大时间极限由对流与扩散相对强度决定的极限方程的源型解给出，并确定了各向异性扩散算子在正交子空间上的投影。
$Γ$-convergence for higher order nonlocal phase transitions
该文研究高阶非局部相变能量泛函的$\Gamma$-收敛，证明当$0<s<3/4$时一阶变分贡献消失，极限为经典或分数阶周长，推翻曲率项预期。
Long-time behaviour of a nonlocal stochastic fractional reaction--diffusion equation arising in tumour dynamics
本文研究了源于肿瘤动力学的非局部随机分数反应-扩散方程的长期行为，通过分析噪声对肿瘤扩散和灭绝的影响，揭示了异常扩散与分数噪声的相互作用机制。
Local behaviour of the mixed local and nonlocal problems with nonstandard growth
本文研究了具有非标准增长条件 $1<p\le sq$ 的混合局部与非局部泛函极小子的局部行为，通过正性扩展方法证明了其局部有界性、局部 Hölder 连续性和 Harnack 不等式，将现有文献中 $p>sq$ 条件下的结果进行了自然推广。
Existence and weak-strong uniqueness of measure solutions to Euler-alignment/Aw--Rascle--Zhang model of collective behaviour
本文研究具有矩阵值通信核的多维Euler-alignment系统，证明其与非局部Aw--Rascle--Zhang交通模型形式等价，并通过单一退化无压Navier--Stokes逼近建立两种表述的测度解全局存在性及弱-强唯一性原理，从而严格论证二者在无粘极限下的对应关系。
In this paper we study nonlocal problems that are analogous to the local ones given by the Laplacian or the p-Laplacian with dynamical boundary conditions. We deal both with smooth and with singular kernels and show existence and uniqueness of solutions and study their asymptotic behaviour as t goes to infinity.
The existence and nonexistence for time nonlocal evolution equations with superlinear sources
研究具有超线性源的时间非局部发展方程初值问题，通过从属和单调迭代方法，分析其可解行为，建立存在性与不存在性充分条件，结果与初值指标相关且有临界维数现象，还通过例子展示适用性。
We show that the boundary behaviour of solutions to nonlocal fractional equations posed in bounded domains strongly differs from the one of solutions to elliptic problems modelled upon the Laplace-Poisson equation with zero boundary data. In this classical case it is known that, at least in a suitable weak sense, solutions of non-homogeneous Dirichlet problem are unique and tend to zero at the boundary. Limits of these solutions then produce solutions of some non-homogeneous Dirichlet problem as the interior data concentrate suitably to the boundary. Here, we show that such results are false f…
We investigate quantitative properties of nonnegative solutions $u(x)\ge 0$ to the semilinear diffusion equation $\mathcal{L} u= f(u)$, posed in a bounded domain $Ω\subset {\mathbb R}^N$ with appropriate homogeneous Dirichlet or outer boundary conditions. The operator $\mathcal{L}$ may belong to a quite general class of linear operators that include the standard Laplacian, the two most common definitions of the fractional Laplacian $(-Δ)^s$ ($0<s<1$) in a bounded domain with zero Dirichlet conditions, and a number of other nonlocal versions. The nonlinearity $f$ is increasing and looks like a …
Long-time asymptotics for nonlocal porous medium equation with absorption or convection
本文针对带吸收或对流项的非局部多孔介质方程，在非局部扩散占优的参数区域内，通过调整熵方法推导出相似变量下解的相对熵呈指数收敛，明确了该类方程的长时间渐近行为。
We prove convergence of positive solutions to \[ u_t = uΔu + u\int_Ω |\nabla u|^2, \qquad u\rvert_{\partialΩ} =0, \qquad u(\cdot,0)=u_0 \] in a bounded domain $Ω\subset \mathbb{R}^n$, $n\ge 1$, with smooth boundary in the case of $\int_Ωu_0=1$ and identify the $W_0^{1,2}(Ω)$-limit of $u(t)$ as $t\to \infty$ as the solution of the corresponding stationary problem. This behaviour is different from the cases of $\int_Ωu_0<1$ and $\int_Ωu_0>1$ which are known to result in convergence to zero or blow-up in finite time, respectively. The proof is based on a monotonicity property of $\int_Ω |\nabla u…
In this paper we give a new compactness criterion in the Lebesgue spaces $L^p((0,T)\times Ω)$ and use it to obtain the first term in the asymptotic behaviour of the solutions of a nonlocal convection diffusion equation. We use previous results of Bourgain, Brezis and Mironescu to give a new criterion in the spirit of the Aubin-Lions-Simon Lemma.
We prove a general result about the behaviour of minimizing sequences for nonlocal shape functionals satisfying suitable structural assumptions. Typical examples include functions of the eigenvalues of the fractional Laplacian under homogeneous Dirichlet boundary conditions. Exploiting a nonlocal version of Lions' concentration-compactness principle, we prove that either an optimal shape exists, or there exists a minimizing sequence consisting of two "pieces" whose mutual distance tends to infinity. Our work is inspired by similar results obtained by Bucur in the local case.
Spectral theory of plasmonic resonances in the nonlocal regime
本文基于非局域介质的非局域 regime，利用流体动力学 Drude 模型的准静态近似，推导出描述共振行为的分析 Fredholm 运算符，揭示非局域模型具有离散实谱结构，改变传统局部理论中无限多个表面等离子体模的结论。
In this paper we consider a nonlocal viscous Burgers equation and study the well-posedness and asymptotic behaviour of its solutions. We prove that under the smallness assumption on the initial data the solutions behave as the self similar profiles of the Burgers equation with Dirac mass as the initial datum. The first term in the asymptotic expansion of the solutions is obtained by rescaling the solutions and proving the compactness of their trajectories.
The aim of this paper is to provide a comprehensive study of some linear nonlocal diffusion problems in metric measure spaces. These include, for example, open subsets in $\mathbb{R}^N$, graphs, manifolds, multi-structures or some fractal sets. For this, we study regularity, compactness, positiveness and the spectrum of the stationary nonlocal operator. Then we study the solutions of linear evolution nonlocal diffusion problems, with emphasis in similarities and differences with the standard heat equation in smooth domains. In particular prove weak and strong maximum principles and describe th…
We study the behaviour of the solutions to a dynamic evolution problem for a viscoelastic model with long memory, when the rate of change of the data tends to zero. We prove that a suitably rescaled version of the solutions converges to the solution of the corresponding stationary problem.
On the precise cusped behaviour of extreme solutions to Whitham-type equations
本文证明了两类非局部方程（包括单向和双向Whitham方程的最高稳态尖峰波）非平凡解在原点的精确首阶渐近行为，部分解决了相关猜想，且方法可推广至其他非局部方程并确定任意阶导数的渐近行为。
Asymptotic behaviour and existence of positive solutions for mixed local nonlocal elliptic equations with Hardy potential
本文研究带Hardy位势的混合局部-非局部椭圆方程，通过变换建立解的一致渐近估计，分情形证明正解存在性，并在次线性小扰动下得到多解结果。
On Some Nonlocal in Time and Space Parabolic Problem
本文研究非局部时间与空间的非线性抛物型问题，探讨解的存在性、唯一性及渐进行为。
Uniqueness for local-nonlocal elliptic equations
本文研究带变系数ρ的混合局部-非局部椭圆方程，在关于ρ无穷远处行为的最优假设下证明加权Lebesgue空间中解的唯一性，并指出假设失效时会出现非唯一性，同时考察了相应的抛物型方程。
We study the asymptotic behaviour of solutions of a class of linear non-local measure-valued differential equations with time delay. Our main result states that the solutions asymptotically exhibit a parabolic like behaviour in the large times, that is precisely expressed in term of heat kernel. Our proof relies on the study of a-self-similar-rescaled family of solutions. We first identify the asymptotic behaviour of the solutions by deriving a convergence result in the sense of the Young measures. Then we strengthen this convergence by deriving suitable fractional Sobolev compactness estimate…
We prove existence, uniqueness and several qualitative properties for evolution equations that combine local and nonlocal diffusion operators acting in different subdomains and coupled in such a way that the resulting evolution equation is the gradient flow of an energy functional. We deal with the Cauchy, Neumann and Dirichlet problems, in the last two cases with zero boundary data. For the first two problems we prove that the model preserves the total mass. We also study the behaviour of the solutions for large times. Finally, we show that we can recover the usual heat equation (local diffus…
We establish the asymptotic behaviour of the least energy solutions of the following nonlocal Neumann problem: \begin{align*} \left\{\begin{array}{l l} { d(-Δ)^{s}u+ u= \abs{u}^{p-1}u } \text{ in $Ω,$ } { \mathcal{N}_{s}u=0 } \text{ in $\mathbb{R}^{n}\setminus \overlineΩ,$} {u>0} \text{ in $Ω,$} \end{array} \right.\end{align*} where $Ω\subset \mathbb{R}^{n}$ is a bounded domain of class $C^{1,1}$, $1<p<\frac{n+s}{n-s},\,n> \max \left\{1, 2s \right\}, 0<s<1,\,d>0$ and $\mathcal{N}_{s}u$ is the nonlocal Neumann derivative. We show that for small $d,$ the least energy solutions $u_d$ of the above…
Singular limit for a class of nonlocal conservation laws via compensated compactness
针对一类建模交通流的非局部守恒律，在核集中为狄拉克δ分布时，不依赖全变差界等传统条件，通过补偿紧性理论证明弱解强收敛到对应局部熵容许解，解决了非凸核相关的长期开放问题。
In the present work we deal with the existence of solutions for optimal control problems associated to transport equations. The behaviour of a population of individuals will be influenced by the presence of a population of control agents whose role is to lead the dynamics of the individuals towards a specific goal. The dynamics of the population of individuals is described by a suitable nonlocal transport equation, while the role of the population of agents is designed by the optimal control problem. This model has been first studied in [12] for a class of continuous nonlocal potentials, while…
Characterisation of homogenisation for nonlocal diffusion by local topologies
本文研究了非局部扩散问题的均质化特征，通过H-收敛性和Schur拓扑分析局部系数收敛性，并应用于对称系数和分数热方程的均质化问题。
Let M be a compact Riemannian manifold with smooth boundary. We obtain the exact long time asymptotic behaviour of the heat kernel on abelian coverings of M with mixed Dirichlet and Neumann boundary conditions. As an application, we study the long time behaviour of the abelianized winding of reflected Brownian motions in M. In particular, we prove a Gaussian type central limit theorem showing that when rescaled appropriately, the fluctuations of the abelianized winding are normally distributed with an explicit covariance matrix.
We consider a class of nonlocal viscous Cahn-Hilliard equations with Neumann boundary conditions for the chemical potential. The double-well potential is allowed to be singular (e.g. of logarithmic type), while the singularity of the convolution kernel does not fall in any available existence theory under Neumann boundary conditions. We prove well-posedness for the nonlocal equation in a suitable variational sense. Secondly, we show that the solutions to the nonlocal equation converge to the corresponding solutions to the local equation, as the convolution kernels approximate a Dirac delta. Th…
We study metrics of constant $Q$-curvature in the Euclidean space with a prescribed singularity at the origin, namely solutions to the equation $$(-Δ)^\frac{n}{2}w=e^{nw}-cδ_{0} \text{ on } \mathbb R^n,$$ under a finite volume condition. We analyze the asymptotic behaviour at infinity and the existence of solutions for every $n\ge 3$ also in a supercritical regime. Finally, we state some open problems.
We characterize the long time behaviour of a discrete-in-time approximation of the volume preserving fractional mean curvature flow. In particular, we prove that the discrete flow starting from any bounded set of finite fractional perimeter converges exponentially fast to a single ball. As an intermediate result we establish a quantitative Alexandrov type estimate in the fractional setting for normal deformations of a ball. Finally, we provide existence for flat flows as limit points of the discrete flow when the time discretization parameter tends to zero.
Asymptotics and geometric flows for a class of nonlocal curvatures
本文研究了一类由对称径向核确定的非局部曲率在极限过程下的渐近行为，恢复了分数阶曲率等已知结果，并证明了相应水平集抛物型Cauchy问题粘性解的存在性、唯一性与稳定性。
We study the higher Hölder regularity of local weak solutions to a class of nonlinear nonlocal elliptic equations with kernels that satisfy a mild continuity assumption. An interesting feature of our main result is that the obtained regularity is better than one might expect when considering corresponding results for local elliptic equations in divergence form with continuous coefficients. Therefore, in some sense our result can be considered to be of purely nonlocal type, following the trend of various such purely nonlocal phenomena observed in recent years. Our approach can be summarized as …
On nonlocal Dirichlet problems with oscillating term
本文研究一类非局部分数阶 Dirichlet 问题，利用 Ricceri 变分原理，在非线性项于原点或无穷远处具有合适振荡行为时，证明了该问题存在无穷多个弱解。
We consider a system of $n$ nonlocal interaction evolution equations on $\mathbb{R}^d$ with a differentiable matrix-valued interaction potential $W$. Under suitable conditions on convexity, symmetry and growth of $W$, we prove $λ$-geodesic convexity for some $λ\in\mathbb{R}$ of the associated interaction energy with respect to a weighted compound distance of Wasserstein type. In particular, this implies existence and uniqueness of solutions to the evolution system. In one spatial dimension, we further analyse the qualitative properties of this solution in the non-uniformly convex case. We obta…
Nonlinear nonlocal reaction-diffusion problem with local reaction
本文研究一般度量测度空间中带局部反应项的非局部扩散问题，证明了解的全局存在性、比较原理等性质，分析了平衡解与渐近行为，并对比了其与局部扩散问题的差异。
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