Yingdong Lu
We study an initial value problem with fractional Laplacian and a singular drift term, and obtain local and global existence theorems similar to the results in Jourdain et al.(2005).
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We prove an existence result for nonlinear diffusion equations in the presence of a nonlocal density-dependent drift which is not necessarily potential. The proof is constructive and based on the Helmholtz decomposition of the drift and a splitting scheme. The splitting scheme combines transport steps by the divergence-free part of the drift and semi-implicit minimization steps à la Jordan-Kinderlherer-Otto to deal with the potential part.
We investigate the existence of self-similar solutions for a coagulation equation with nonlocal drift. In addition to explicitly given exponentially decaying solutions we establish the existence of self-similar profiles with algebraic decay.
Potential theory for nonlocal drift-diffusion equations
本文通过逐点位势估计证明了非局部漂移-扩散方程的精细正则性结果，仅需对无散度漂移项作极小假设，并推导出耗散表面准地转方程的新估计。
We investigate the differentiability issue of the drift-diffusion equation with nonlocal Lévy-type diffusion at either supercritical or critical type cases. Under the suitable conditions on the drift velocity and the forcing term in terms of the spatial Hölder regularity, we prove that the vanishing viscosity solution is differentiable with some Hölder continuous derivatives for any positive time.
We study the free boundary of the porous medium equation with nonlocal drifts in dimension one. Under the assumption that the initial data has super-quadratic growth at the free boundary, we show that the solution is smooth in space and $C^{2,1}_{\loc}$ in time, and then the free boundary is $C^{2,1}_{\loc}$. Moreover if the drift is local, both the solution and the free boundary are smooth.
Harnack estimates for nonlocal drift-diffusion equations
本文研究带漂移项的非局部扩散方程，采用 DeGiorgi 类式测度论方法，首次建立其 Harnack 估计，并改进 Hölder 正则性结果，同时得到椭圆情形的非局部弱 Harnack 估计。
We obtain Dini and Schauder type estimates for concave fully nonlinear nonlocal parabolic equations of order $σ\in (0,2)$ with rough and non-symmetric kernels, and drift terms. We also study such linear equations with only measurable coefficients in the time variable, and obtain Dini type estimates in the spacial variable. This is a continuation of the work [10, 11] by the first and last authors.
In this paper we address the regularity issues of drift-diffusion equation with nonlocal diffusion, where the diffusion operator is in the realm of stable-type Lévy operator and the velocity field is defined from the considered quantity by a zero-order pseudo-differential operator. Through using the method of nonlocal maximum principle in a unified way, we prove the eventual regularity result in the supercritical type cases and the global regularity at some logarithmically supercritical cases. The feature of these results is that the time after which the solution is smoothly regular in the sup…
This paper is devoted to existence and uniqueness results for classes of nonlinear diffusion equations (or systems) which may be viewed as regular perturbations of Wasserstein gradient flows. First, in the case. where the drift is a gradient (in the physical space), we obtain existence by a semi-implicit Jordan-Kinderlehrer-Otto scheme. Then, in the nonpotential case, we derive existence from a regularization procedure and parabolic energy estimates. We also address the uniqueness issue by a displacement convexity argument.
We consider a one-dimensional nonlocal nonlinear equation of the form: $\partial_t u = (Λ^{-α} u)\partial_x u - νΛ^βu$ where $Λ=(-\partial_{xx})^{\frac 12}$ is the fractional Laplacian and $ν\ge 0$ is the viscosity coefficient. We consider primarily the regime $0<α<1$ and $0\le β\le 2$ for which the model has nonlocal drift, fractional dissipation, and captures essential features of the 2D $α$-patch models. In the critical and subcritical range $1-α\le β\le 2$, we prove global wellposedness for arbitrarily large initial data in Sobolev spaces. In the full supercritical range $0 \le β<1-α$, we …
Lack of uniqueness for an elliptic equation with nonlinear and nonlocal drift posed on a torus
研究了在环面上的非线性和非局部椭圆方程的非唯一性，通过分支定理证明了非常数周期解的存在，并构造了特定一维形式的显式多重解。
In this paper, we prove existence results of a one-dimensional periodic solution to equations with the fractional Laplacian of order $s\in(1/2,1)$, singular nonlinearity, and gradient term under various situations, including nonlocal contra-part of classical Lienard vector equations, as well other nonlocal versions of classical results know only in the context of second-order ODE. Our proofs are based on degree theory and Perron's method, so before that we need to establish a variety of priori estimates under different assumptions on the nonlinearities appearing in the equations. Besides, we o…
We study a class of nonlocal-diffusion equations with drifts, and derive a priori $Φ$-Hölder estimate for the solutions by using a purely probabilistic argument, where $Φ$ is an intrinsic scaling function for the equation.
In \cite{CordobaCordobaFontelos05}, Córdoba, Córdoba, and Fontelos proved that for some initial data, the following nonlocal-drift variant of the 1D Burgers equation does not have global classical solutions \[ \partial_t θ+u \; \partial_x θ= 0, \qquad u = H θ, \] where $H$ is the Hilbert transform. We provide four essentially different proofs of this fact. Moreover, we study possible Hölder regularization effects of this equation and its consequences to the equation with diffusion \[ \partial_t θ+ u \; \partial_x θ+ Λ^γθ= 0, \qquad u = H θ, \] where $Λ= (-Δ)^{1/2}$, and $1/2 \leq γ<1$. Our res…
Existence and Nonlocal-to-Local Convergence for Singular, Anisotropic Nonlocal Cahn-Hilliard Equations
本文研究了具有各向异性和奇异核的非局部Cahn-Hilliard方程的非局部到局部收敛性，并证明了在特定条件下非局部方程弱解的存在性。
We investigate existence and uniqueness of solutions for a class of nonlinear nonlocal problems involving the fractional $p$-Laplacian operator and singular nonlinearities.
This paper deals with a singular nonlocal phase field system of conserved type.Colli--K.\ [Nonlinear Anal.\ 190 (2020)] have derived existence of solutions to a singular phase field system of conserved type. On the other hand, Davoli--Scarpa--Trussardi [Arch. Ration. Mech. Anal.\ 239 (2021)] have studied nonlocal to local convergence of Cahn-Hilliard equations. In this paper we prove existence of solutions to a nonlocal singular phase field system of conserved type whose kernel is not $W^{1, 1}$ and focus on nonlocal to local convergence of singular phase field systems of conserved type.
We provide a suitable variational approach for a class of nonlocal problems involving the fractional laplacian and singular nonlinearities for which the standard techniques fail. As a corollary we deduce a characterization of the solutions.
This article is concerned with the following spectral problem: to find a positive function $Φ$ $\in$ C 1 ($Ω$) and $λ$ $\in$ R such that q(x)$Φ$ (x) + ^ $Ω$ J(x, y)$Φ$(y) dy + a(x)$Φ$(x) + $λ$$Φ$(x) = 0 for x $\in$ $Ω$, where $Ω$ $\subset$ R is a non-empty domain (open interval), possibly unbounded, J is a positive continuous kernel, and a and q are continuous coefficients. Such a spectral problem naturally arises in the study of nonlocal population dynamics models defined in a space-time varying environment encoding the influence of a climate change through a spatial shift of the coefficient.…
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 …
Liouville theorem for singular solutions to nonlocal equations
研究分数阶 Laplace 方程及一般非局部线性方程的奇异解，通过构造基本解和建立局部化比较原理，证明了 Bôcher 型结果与 Liouville 定理。
Positive Ground State Solutions to a Nonlocal Singular Elliptic Problem
研究Dirichlet边界条件下非局部奇异椭圆问题，在Musielak-Sobolev空间中证明正基态解的存在性与唯一性。
The theory of elliptic equations involving singular nonlinearities is well studied topic but the interaction of singular type nonlinearity with nonlocal nonlinearity in elliptic problems has not been investigated so far. In this article, we study the very singular and doubly nonlocal singular problem $(P_λ)$(See below). Firstly, we establish a very weak comparison principle and the optimal Sobolev regularity. Next using the critical point theory of non-smooth analysis and the geometry of the energy functional, we establish the global multiplicity of positive weak solutions.
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.
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.
Singularities of solutions of nonlocal nonlinear equations
本文研究非局部非线性方程弱解的局部奇异性，证明零容量集合在可积性条件下可去，并用基本解刻画孤立奇点附近的渐近行为。
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.
On mixed local-nonlocal Sobolev-type inequalities and their connection with singular equations in the Heisenberg group
本文在Heisenberg群中建立混合局部-非局部Sobolev型不等式，证明其与奇异p-Laplace方程的解关系，并提供了一种统一框架，连接不同情形下奇异方程的存在理论。
We study local behavior of positive solutions to the fractional Yamabe equation with a singular set of fractional capacity zero.
We investigate a system of nonlocal transport equations in one spatial dimension. The system can be regarded as a model for the 3D Euler equations in the hyperbolic flow scenario. We construct blowup solutions with control up to the blowup time.
We provide an informal overview of recent developments concerning the singular local limit of nonlocal conservation laws. In particular, we discuss some counterexamples to convergence and we highlight the role of numerical viscosity in the numerical investigation of the nonlocal-to-local limit. We also state some open questions and describe recent related progress.
Existence Theory for a class of semilinear mixed local and nonlocal equations involving variable singularities and singular measures
本文针对含可变奇异性和奇异测度的半线性局部-非局部混合方程，通过处理可变奇异指数与测度值数据（含新的奇异测度数据），建立了其弱解的存在性。
We prove existence, uniqueness and regularity of solutions of nonlocal heat equations associated to anisotropic stable diffusion operators. The main features are that the right-hand side has very few regularity and that the spectral measure can be singular in some directions. The proofs require having good enough estimates for the corresponding heat kernels and their derivatives.
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.
Regularizing effects of absorption terms in local-nonlocal mild singular problems
该研究针对局部-非局部算子驱动的带吸收项奇异问题，证明了能量解的存在唯一性，建立了Talenti型比较原理并得到解的可和性提升结果。
In this paper we deal with a singular nonlocal phase field system with inertial term. The system has the logarithm of the absolute temperature $θ$ under time derivative. Although the system has a difficult mathematical point caused by the combination of $(\ln θ)_{t}$, the inertial term and the nonlocal diffusion term for the order parameter $φ$ (see Section 1.1), we can establish existence of solutions by a key estimate (see Remark 1.1).
Singular limit for nonlocal conservation laws with non-locality in density and velocity
研究具有密度和速度非局部性的非局部守恒律的奇异极限，通过推导得出基础Burgers方程的经典解作为奇异极限，此推导在激波形成前有效，回答了相关文献提出的问题。
We study a class of elliptic problems with homogeneous Dirichlet boundary condition and a nonlinear reaction term $f$ which is nonlocal depending on the $L^{p}$-norm of the unknown function. The nonlinearity $f$ can make the problem degenerate since it may even have multiple singularities in the nonlocal variable. We use fixed point arguments for an appropriately defined solution map to produce multiplicity of classical positive solutions with ordered norms.
Multiplicity of positive solutions for mixed local-nonlocal singular critical problems
本文证明混合局部-非局部奇异临界椭圆问题至少存在两个正弱解，推广了奇异与临界问题的现有结果。
We study the existence of global-in-time solutions for a nonlinear heat equation with nonlocal diffusion, power nonlinearity and suitably small data (either compared pointwisely to the singular solution or in the norm of a critical Morrey space). Then, asymptotics of subcritical solutions is determined. These results are compared with conditions on the initial data leading to a finite time blowup.
In this article, we consider mixed local and nonlocal Sobolev $(q,p)$-inequalities with extremal in the case $0<q<1<p<\infty$. We prove that the extremal of such inequalities is unique up to a multiplicative constant that is associated with a singular elliptic problem involving the mixed local and nonlocal $p$-Laplace operator. Moreover, it is proved that the mixed Sobolev inequalities are necessary and sufficient condition for the existence of weak solutions of such singular problems. As a consequence, a relation between the singular $p$-Laplace and mixed local and nonlocal $p$-Laplace equati…
We establish existence results for a class of mixed anisotropic and nonlocal $p$-Laplace equation with singular nonlinearities. We consider both constant and variable singular exponents. Our argument is based on an approximation method. To this end, we also discuss the necessary regularity properties of weak solutions of the associated non-singular problems. More precisely, we obtain local boundedness of subsolutions, Harnack inequality for solutions and weak Harnack inequality for supersolutions.
On a class of mixed local and nonlocal semilinear elliptic equation with singular nonlinearity
该研究利用变分法与逼近方法，考察兼具局部与非局部 Laplace 算子的奇异半线性椭圆方程，证明参数依赖情形下弱解存在及扰动情形下多解存在。
In this contribution we study the singular limit problem of a nonlocal conservation law with a discontinuity in space. The specific choice of the nonlocal kernel involving the spatial discontinuity as well enables it to obtain a maximum principle for the nonlocal equation. The corresponding local equation can be transformed diffeomorphically to a classical scalar conservation law where the well-know Kružkov theory can be applied. However, the nonlocal equation does not scale that way which is why the study of convergence is interesting to pursue. For exponential kernels in the nonlocal operato…
Continuity of solutions to equations with weakly singular nonlocal operators
本文研究涉及几乎无可微性阶积分微分算子的线性非局部方程，证明了有界弱解的连续性并给出了一致的连续性模先验估计，同时允许算子高度各向异性、弱奇异且核函数在奇点附近消失。
Mixed local-nonlocal $p$-Laplace equation with variable singular nonlinearity in the Heisenberg group
本文研究了海森堡群中混合局部-非局部 $p$-拉普拉斯方程的弱解存在性、唯一性和正则性，首次在非交换设定中处理此类问题。
Regularity and existence for semilinear mixed local-nonlocal equations with variable singularities and measure data
本文证明了一类具有可变奇异指数和测度数据的混合局部-非局部奇异问题弱解的存在性与正则性，其中源项可同时为测度，该现象在常数奇异指数情形下也是新的。
Obstacle problems for nonlocal operators with singular kernels
本文研究具有奇异核的非局部障碍问题，针对一类非常一般的积分微分算子建立了最优正则性估计与自由边界平滑性，突破了以往要求核与分数阶拉普拉斯算子核逐点可比的限制。
Mixed local and nonlocal semilinear elliptic equation with strongly singular and critical Choquard nonlinearity
本文研究混合局部与非局部椭圆方程，结合奇异与临界非线性，通过变分方法证明正解的存在性与多重性。
We prove continuity for bounded weak solutions of a nonlinear nonlocal parabolic type equation associated to a Dirichlet form with a rough kernel. The equation is allowed to be singular at the level zero, and solutions may change sign. If the nonlinearity in the equation does not oscillate too much at the origin, the solution is proved to be moreover Hölder continuous. The results are new even when the Dirichlet form is the one corresponding to the fractional Laplacian.
Existence and uniqueness of solutions in fractional Orlicz-Sobolev spaces for a nonlocal singular elliptic problem
本文研究了一类带边界条件的奇异非线性椭圆问题，证明了其正解的存在性与唯一性，且该结果在分数阶 Orlicz-Sobolev 空间中成立。
Multiplicity results for mixed local-nonlocal equations with singular and critical exponential nonlinearity in R^2
本文研究R^2中带可变奇异指数与临界指数非线性项的混合局部-非局部方程，利用下解-上解技巧与变分方法证明了至少两个正弱解的存在性。
On the singular limit problem in nonlocal balance laws: Applications to nonlocal lane-changing traffic flow models
本文研究了非局部平衡律系统的奇异极限问题，证明了当非局部算子收敛到Dirac分布时，系统向局部熵解收敛，并应用于非局部换道交通流模型。
We provide symmetrization results in the form of mass concentration comparisons for fractional singular elliptic equations in bounded domains, coupled with homogeneous external Dirichlet conditions. Two types of comparison results are presented, depending on the summability of the right-hand side of the equation. The maximum principle arguments employed in the core of the proofs of the main results offer a nonstandard, flexible alternative to the ones described in [18, Theorem 31]. Some interesting consequences are Lp regularity results and nonlocal energy estimates for solutions.
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…
Piecewise regular solutions to scalar balance laws with singular nonlocal sources
本文研究含卷积型奇异非局部源项的标量平衡律，建立了带单一激波的分段正则解的局部适定性，并刻画了激波曲线邻域内解的结构。
We prove existence of solutions and study the nonlocal-to-local asymptotics for nonlocal, convective, Cahn-Hilliard equations in the case of a W^{1,1} convolution kernel and under homogeneous Neumann conditions. Any type of potential, possibly also of double-obstacle or logarithmic type, is included. Additionally, we highlight variants and extensions to the setting of periodic boundary conditions and viscosity contributions, as well as connections with the general theory of evolutionary convergence of gradient flows.
We show that weak solutions of degenerate Navier-Stokes equations converge to the strong solutions of the pressureless Euler system with linear drag term, Newtonian repulsion and quadratic confinement. The proof is based on the relative entropy method using the artificial velocity formulation for the one-dimensional Navier-Stokes system.
Symmetry, existence and regularity results for a class of mixed local-nonlocal semilinear singular elliptic problem via variational characterization
本文利用变分刻画和分解性质，结合比较原理与移动平面方法，研究一类混合局部-非局部奇异椭圆问题弱解的对称性、存在性、不存在性与正则性，并用非光滑临界点理论分析跳跃问题。
We prove a Harnack inequality for functions which, at points of large gradient, are solutions of elliptic equations with unbounded drift.
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