Vu Hoang, Maria Radosz
In this paper, we introduce a nonlocal evolution equation inspired by the Córdoba-Córdoba-Fontelos nonlocal transport equation. The Córdoba-Córdoba-Fontelos equation can be regarded as a model for the 2D surface quasigeostrophic equation or the Birkhoff-Rott equation. We prove blowup in finite time, and more importantly, investigate conditions under which the solution forms a cusp in finite time.
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We consider the Whitham equation $u_t + 2u u_x+Lu_x = 0$, where L is the nonlocal Fourier multiplier operator given by the symbol $m(ξ) = \sqrt{\tanh ξ/ξ}$. G. B. Whitham conjectured that for this equation there would be a highest, cusped, travelling-wave solution. We find this wave as a limiting case at the end of the main bifurcation curve of $P$-periodic solutions, and give several qualitative properties of it, including its optimal $C^{1/2}$-regularity. An essential part of the proof consists in an analysis of the integral kernel corresponding to the symbol $m(ξ)$, and a following study of…
Stable $C^{7/9}$ cusp formation for the Novikov equation
该研究针对Novikov方程确定光滑初值开集，通过引入Galilean型变量变换克服困难，证明其会形成稳定的$C^{7/9}$尖点，揭示了非局部项与非线性结构耦合决定尖点正则性。
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。
Cusp Formation in Vortex Patches
针对涡旋斑块的尖点形成问题，证明了任何具锐角的初始涡旋斑块会瞬时形成尖点，验证了数值文献中的推测。
This paper is concerned with a class of multi-dimensional transport equations with nonlocal velocity. It is shown that the local smooth solution cannot exist globally in time via the De Giorgi iteration technique.
Parabolic problems for direction-dependent local-nonlocal operators
本文研究了由方向依赖的局部-非局部积分微分算子控制的抛物方程，基于能量法证明了弱 Harnack 不等式与 Hölder 正则性估计，且常数独立于算子可微阶数。
We study a 1D transport equation with nonlocal velocity and show the formation of singularities in finite time for a generic family of initial data. By adding a diffusion term the finite time singularity is prevented and the solutions exist globally in time.
The $α$-patch model is used to study aspects of fluid equations. We show that solutions of this model form singularities in finite time and give a characterization of the solution profile at the singular time.
We study a nonlocal evolution equation generalising a model introduced by Shigeru Kondo to explain colour patterns on a skin of a guppy fish. We prove the existence of stationary solutions using either the bifurcation theory or the Schauder fixed point theorem. We also present numerical studies of this model and show that it exhibits patterns similar to those modelled by well-known reaction-diffusion equations.
Coupling a reaction-diffusion equation with ordinary differential equations (ODE) may lead to diffusion-driven instability (DDI) which, in contrast to the classical reaction-diffusion models, causes destabilization of both, constant solutions and Turing patterns. Using a shadow-type limit of a reaction-diffusion-ODE model, we show that in such cases the instability driven by nonlocal terms (a counterpart of DDI) may lead to formation of unbounded spike patterns.
We investigate the singularity formation of a nonlinear nonlocal system. This nonlocal system is a simplified one-dimensional system of the 3D model that was recently proposed by Hou and Lei in [13] for axisymmetric 3D incompressible Navier-Stokes equations with swirl. The main difference between the 3D model of Hou and Lei and the reformulated 3D Navier-Stokes equations is that the convection term is neglected in the 3D model. In the nonlocal system we consider in this paper, we replace the Riesz operator in the 3D model by the Hilbert transform. One of the main results of this paper is that …
Global Existence of Solutions of Nonlocal Geirer-Meinhardt Model and Effect of Nonlocal Operator in Pattern Formation
研究非局部Geirer-Meinhardt系统解的整体存在性，利用半群理论和L^b泛函得到与核无关的界，并数值模拟展示模式形成，与传统局部模型对比。
Shock formation in traffic flow models with nonlocal look ahead and behind flux
本文研究带非局部通量的LWR型交通流模型，在Arrhenius前瞻-后顾动力学和凹-凸通量条件下给出激波形成的阈值判据。
Dissipative gradient nonlinearities prevent $δ$-formations in local and nonlocal attraction-repulsion chemotaxis models
本文研究具有非线性扩散、化学敏感性、产生率及梯度源项的吸引-排斥趋化模型，证明耗散梯度非线性可阻止δ-聚集的形成。
We study the existence, bifurcations, and stability of stationary solutions for the doubly-nonlocal Fisher-KPP equation. We prove using Lyapunov-Schmidt reduction that under suitable conditions on the parameters, a bifurcation from the non-trivial homogeneous state can occur. The kernel of the linearized operator at the bifurcation is two-dimensional and periodic stationary patterns are generated. Then we prove that these patterns are, again under suitable conditions, locally asymptotically stable. We also compare our results to previous work on the nonlocal Fisher-KPP equation containing a lo…
We study a 1D fluid mechanics model with nonlocal velocity. The equation can be viewed as a fractional porous medium flow, a 1D model of the quasi-geostrophic equation, and also a special case of Euler-Alignment system. For strictly positive smooth initial data, global regularity has been proved by Do, Kiselev, Ryzhik and Tan. We construct a family of non-negative smooth initial data so that solution loses $C^1$ regularity. Our result indicates that strict positivity is a critical condition to ensure global regularity of the system. We also extend our construction to the corresponding models i…
Of concern is the a priori symmetry of traveling wave solutions for a general class of nonlocal dispersive equations \[ u_t +(u^2 +Lu)_x =0, \] where $L$ is a Fourier multiplier operator with symbol $m$. Our analysis includes both homogeneous and inhomogeneous symbols. We characterize a class of symbols $m$ guaranteeing that periodic traveling wave solutions are symmetric under a mild assumption on the wave profile. Particularly, instead of considering waves with a unique crest and trough per period or a monotone structure near troughs as classically imposed in the water wave problem, we formu…
We propose some one-dimensional reduced models for the three-dimensional electron magnetohydrodynamics which involves a highly nonlinear Hall term with intricate structure. The models contain nonlocal nonlinear terms. Local well-posedness is obtained in certain circumstances. In contrast, for a model with nonlocal transport term, we show that finite time singularity may occur for some initial data.
In this paper we study pattern formation for a physical local/nonlocal interaction functional where the local attractive term is given by the $1$-perimeter and the nonlocal repulsive term is the Yukawa (or screened Coulomb) potential. This model is physically interesting as it is the $Γ$-limit of a double Yukawa model used to explain and simulate pattern formation in colloidal systems \cite{BBCH,CCA,IR,GCLW}. Following a strategy introduced in~\cite{DR} we prove that in a suitable regime minimizers are periodic stripes, in any space dimension.
In this paper we study the Burgers equation with a nonlocal term of the form $Hu$ where $H$ is the Hilbert transform. This system has been considered as a quadratic approximation for the dynamics of a free boundary of a vortex patch. We prove blow up in finite time for a large class of initial data with finite energy. Considering a more general nonlocal term, of the form $Λ^αHu$ for $0<α< 1$, finite time singularity formation is also shown.
In this paper, we study the Cauchy problem of the Poiseuille flow of full Ericksen-Leslie model for nematic liquid crystals. The model is a coupled system of a parabolic equation for the velocity and a quasilinear wave equation for the director. For a particular choice of several physical parameter values, we construct solutions with smooth initial data and finite energy that produce, in finite time, cusp singularities - blowups of gradients. The formation of cusp singularity is due to local interactions of wave-like characteristics of solutions, which is different from the mechanism of finite…
Jumps and cusps: a new revival effect in local dispersive PDEs
本文研究Airy方程与错位Laplacian Schrödinger方程的Dirichlet边值问题，证明初始跳跃间断会在适当时刻复兴为跳跃间断与对数尖点奇点，并阐明其成因。
We investigate a class of nonlocal conservation laws with the nonlinear advection coupling both local and nonlocal mechanism, which arises in several applications such as the collective motion of cells and traffic flows. It is proved that the $C^1$ solution regularity of this class of conservation laws will persist at least for a short time. This persistency may continue as long as the solution gradient remains bounded. Based on this result, we further identify sub-thresholds for finite time shock formation in traffic flow models with Arrhenius look-ahead dynamics.
Some quantities in the reaction-diffusion models from cellular biology or ecology depend on the spatial average of density functions instead of local density functions. We show that such nonlocal spatial average can induce instability of constant steady state, which is different from classical Turing instability. For a general scalar equation with spatial average, the occurrence of the steady state bifurcation is rigorously proved, and the formula to determine the bifurcation direction and the stability of the bifurcating steady state is given. For the two-species model, spatially non-homogene…
Finite time singularity formation for a nonlocal Boussinesq system arising in cold-plasma dynamics
研究冷等离子体动力学中非局部Boussinesq系统的有限时间奇点形成，证明特定初始数据导致爆破，并给出生命跨度上界及速度梯度无界性。
We discuss fattening phenomenon for the evolution of planar curves according to their nonlocal curvature. More precisely, we consider a class of generalized curvatures which correspond to the first variation of suitable nonlocal perimeter functionals, defined in terms of an interaction kernel $K$, which is symmetric, nonnegative, possibly singular at the origin, and satisfies appropriate integrability conditions. We prove a general result about uniqueness of the geometric evolutions starting from regular sets with positive $K$-curvature and we discuss the fattening phenomenon for the evolution…
Local asymptotics for the nonlocal Swift-Hohenberg equation
本文研究非局部 Swift-Hohenberg 方程，证明了其适定性，并在齐次 Neumann 边界条件下利用能量估计分析了包含一项和两项非局部贡献的非局部到局部渐近性。
Formation of multiple flocks in a simple nonlocal aggregation model
该研究针对具有“势阱-势垒”核的非局部聚集模型，利用集中紧性原理的二分情形求解能量极小值，解释了大种群中多鸟群形成的机制。
Pseudo-differential noise and nonlocal singularity formation in the stochastic Córdoba--Córdoba--Fontelos equation
该研究针对乘性Stratonovich噪声驱动的随机Córdoba--Córdoba--Fontelos方程，建立了极大经典解的局部时间理论，证明有限时间爆破的相关性质并确定了爆破速率，为该类方程的奇异性研究提供了理论支撑。
A rigorous approach to pattern formation for isotropic isoperimetric problems with competing nonlocal interactions
本文针对竞争性局部/非局部各向同性泛函，利用变分法与几何测度论证明临界状态附近全局极小元呈平坦边界图案，并揭示非局部曲率量与平坦性的关系。
Bifurcation structure and mesa pattern formation in a one-component nonlocal adhesion model with population pressure and degenerate mobility
该研究分析含种群压力与退化迁移率的单组分非局部粘附模型的模式形成，通过线性与弱非线性分析推导分岔特性，结合数值模拟确认台地模式形成，将其与容量受限结构关联。
From an open set of initial data, we construct a family of classical solutions to the 1D nonisentropic compressible Euler equations which form $C^{0,ν}$ cusps as a first singularity, for any $ν\in [1/2,1)$. For this range of $ν$, this is the first result demonstrating the stable formation of such $C^{0,ν}$ cusp-type singularities, also known as pre-shocks. The proof uses a new formulation of the differentiated Euler equations along the fast acoustic characteristic, and relies on a novel set of $L^p$ energy estimates for all $1 < p < \infty$, which may be of independent interest.
Opinion formation on evolving network. The DPA method applied to a nonlocal cross-diffusion PDE-ODE system
研究网络上观点密度的非局部交叉扩散PDE-ODE系统，应用DPA方法证明解的存在性，并建立演化网络上的观点形成模型，分析开放程度和交互半径对群体激进化、极化和碎片化的影响。
Pattern formation of a nonlocal, anisotropic interaction model
研究非局部各向异性相互作用模型的图案形成，分析各向异性参数对模型从各向同性到各向异性转变的影响，通过解析和数值方法探讨平衡态及图案生成。
The lingering phenomenon and pattern formation in a nonlocal population model with cognitive map
本文提出非局部种群模型，研究认知地图对种群空间分布和持续现象的影响，揭示了学习与遗忘速率对种群动态的调控作用。
Kernel-Dependent Pattern Formation in a Population Model with Nonlocal Facilitation and Competition
本文研究含非局部竞争与促进的单物种反应-扩散模型，发现核函数选择影响模式形成分岔，且不同核函数下种群灭绝预警信号存在差异，揭示了空间模式的恢复力作用。
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