Uniqueness of coexistence state with small perturbation

Uniqueness for the fractional Calderón problem with quasilocal perturbations

We study the fractional Schrödinger equation with quasilocal perturbations. These are a family of nonlocal perturbations vanishing at infinity, which include e.g. convolutions against Schwartz functions. We show that the qualitative unique continuation and Runge approximation properties hold in the assumption of sufficient decay. Quantitative versions of both results are also obtained via a propagation of smallness analysis for the Caffarelli-Silvestre extension. The results are then used to show uniqueness in the inverse problem of retrieving a quasilocal perturbation from DN data under suita…

Stability and uniqueness of self-similar profiles in $L^1$ spaces for perturbations of the constant kernel in Smoluchowski's coagulation equation

In this work, we consider self-similar profiles for Smoluchowski's coagulation equation for kernels which are possibly unbounded perturbations of the constant one. For this model, we show that the self-similar solutions for the perturbed kernel are close in weighted $L^1$ spaces to the profile of the unperturbed equation, i.e. the profiles are stable with respect to the perturbation. Additionally, we revisit the problem of uniqueness for these coagulation kernels. In fact, we will improve a corresponding result (see arXiv:1510.03361 and ref. [22]) by relaxing the conditions on the perturbation…

Uniqueness of fat-tailed self-similar profiles to Smoluchowski's coagulation equation for a perturbation of the constant kernel

This article is concerned with the question of uniqueness of self-similar profiles for Smoluchowski's coagulation equation which exhibit algebraic decay (fat tails) at infinity. More precisely, we consider a rate kernel $K$ which can be written as $K=2+\varepsilon W$. The perturbation is assumed to have homogeneity zero and might also be singular both at zero and at infinity. Under further regularity assumptions on $W$, we will show that for sufficiently small $\varepsilon$ there exists, up to normalisation of the tail behaviour at infinity, at most one self-similar profile.

Uniqueness of smooth stationary black holes in vacuum: small perturbations of the Kerr spaces

We prove that a regular stationary black-hole solution of the Einstein vacuum equations which is "close" to some Kerr solution is, in fact, isometric to that Kerr solution.

Quantitative uniqueness for zero-order perturbations of generalized Baouendi-Grushin operators

Based on a variant of the frequency function approach of Almgren, we establish an optimal bound on the vanishing order of solutions to stationary Schrödinger equations associated to a class of subelliptic equations with variable coefficients whose model is the so-called Baouendi-Grushin operator. Such bound provides a quantitative form of strong unique continuation that can be thought of as an analogue of the recent results of Bakri and Zhu for the standard Laplacian.

Uniqueness of the pendent drop of infinite length

We prove the uniqueness of the infinite length axisymmetric solution to the capillary equation. We observe that capillary equation can be viewed, at large depth, as a perturbation of an integrable two-dimensional differential system. Uniqueness is then proved by an elementary perturbation argument.

带非线性局部扰动的高阶分数阶薛定谔方程：唯一性

The higher-order fractional Schrödinger equation with nonlinear local perturbations: Uniqueness

本文研究带局部非线性扰动的高阶分数阶薛定谔方程，建立正问题的 Sobolev 与 Hölder 适定性估计，并利用高阶线性化和分数阶 Laplace 算子的唯一延拓性证明反问题中非线性扰动可由 Dirichlet-to-Neumann 映射唯一确定。

On weak-strong uniqueness for compressible Navier-Stokes system with general pressure laws

The goal of the present paper is to study the weak--strong uniqueness problem for the compressible Navier--Stokes system with a general barotropic pressure law. Our results include the case of a hard sphere pressure of Van der Waals type with a non--monotone perturbation and a Lipschitz perturbation of a monotone pressure. Although the main tool is the relative energy inequality, the results are conditioned by the presence of viscosity and do not seem extendable to the Euler system.

非局部扩散椭圆系统模拟细菌与活性养料的行为

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。

Uniqueness and Nondegeneracy of positive solutions to Kirchhoff equations and its applications in singular perturbation problems

In the present paper, we establish the uniqueness and nondegeneracy of positive energy solutions to the Kirchhoff equation \begin{eqnarray*} -\left(a+b\int_{\mathbb{R}^{3}}|\nabla u|^{2}\right)Δu+u=|u|^{p-1}u & & \text{in }\mathbb{R}^{3}, \end{eqnarray*} where $a,b>0$, $1<p<5$ are constants. Then, as applications, we derive the existence and local uniqueness of solutions to the perturbed Kirchhoff problem \begin{eqnarray*} -\left(ε^2a+εb\int_{\mathbb{R}^{3}}|\nabla u|^{2}\right)Δu+V(x)u=|u|^{p-1}u & & \text{in }\mathbb{R}^{3} \end{eqnarray*} for $ε>0$ sufficiently small, under some mild assump…

The higher order fractional Calderón problem for linear local operators: uniqueness

We study an inverse problem for the fractional Schrödinger equation (FSE) with a local perturbation by a linear partial differential operator (PDO) of order smaller than the order of the fractional Laplacian. We show that one can uniquely recover the coefficients of the PDO from the Dirichlet-to-Neumann (DN) map associated to the perturbed FSE. This is proved for two classes of coefficients: coefficients which belong to certain spaces of Sobolev multipliers and coefficients which belong to fractional Sobolev spaces with bounded derivatives. Our study generalizes recent results for the zeroth a…

On some non linear evolution systems which are perturbations of Wasserstein gradient flows

This paper presents existence and uniqueness results for a class of parabolic systems with non linear diffusion and nonlocal interaction. These systems can be viewed as regular perturbations of Wasserstein gradient flows. Here we extend results known in the periodic case to the whole space and on a smooth bounded domain. Existence is obtained using a semi-implicit Jordan-Kinderlehrer-Otto scheme and uniqueness follows from a displacement convexity argument.

Unique determination of anisotropic perturbations of a polyharmonic operator from partial boundary data

We study an inverse problem involving the unique recovery of several lower order anisotropic tensor perturbations of a polyharmonic operator in a bounded domain from the knowledge of the Dirichlet to Neumann map on a part of boundary. The uniqueness proof relies on the inversion of generalized momentum ray transforms (MRT) for symmetric tensor fields, which we introduce for the first time to study Calderón-type inverse problems. We construct suitable complex geometric optics (CGO) solutions for the polyharmonic operators that reduces the inverse problem to uniqueness results for a generalized …

扩张喷管中二维定常可压缩欧拉流跨声速激波解的唯一性

Uniqueness of Transonic Shock Solutions for Two-Dimensional Steady Compressible Euler Flows in an Expanding Nozzle

本文针对扩张喷管中的二维定常可压缩欧拉流，通过提出跨激波压力条件、建立亚声速流先验估计并利用压缩论证，证明了在喷管扰动足够小时跨声速激波解的唯一性。

Navier-Stokes-Fourier-Korteweg系统的粘性-弥散激波剖面的存在性与唯一性

Existence and uniqueness of viscous-dispersive shock profiles for the Navier-Stokes-Fourier-Korteweg system

该研究针对满足Rankine-Hugoniot与Lax熵条件的小强度激波，证明了一维NSFK系统粘性-弥散激波剖面的存在唯一性，通过几何奇异摄动理论构造剖面并给出逐点估计，为后续黎曼解稳定性分析提供基础。

外圆柱中轴对称稳态 Navier-Stokes 方程经典解的存在性与非唯一性

Existence and non-uniqueness of classical solutions to the axially symmetric stationary Navier-Stokes equations in an exterior cylinder

研究外周期圆柱内轴对称稳态 Navier-Stokes 方程，构造满足边界与无穷远条件、具临界及超临界衰减的经典解，并证明径向汇足够强时解非唯一。

Uniqueness of the minimizer for a random nonlocal functional with double-well potential in $d\le2$

We consider a small random perturbation of the energy functional $$ [u]^2_{H^s(Λ, R^d)} + \int_ΛW(u(x)) dx $$ for $s \in (0,1),$ where the non-local part $ [u]^2_{H^s(Λ,R^d)}$ denotes the total contribution from $Λ\subset R^d$ in the $H^s (R^d)$ Gagliardo semi-norm of $u$ and $W$ is a double well potential. We show that there exists, as $Λ$ invades $ R^d$, for almost all realizations of the random term a minimizer under compact perturbations, which is unique when $d=2$, $s \in (\frac 12,1)$ and when $d=1$, $s \in [\frac 14, 1).$ This uniqueness is a consequence of the randomness. When the rand…

从柯西数据确定二维薛定谔方程中的无界势

Determination of separable perturbations of an unbounded potential in the two-dimensional Schrödinger equation

利用狄利克雷-诺伊曼映射和卡尔曼不等式，证明了从边界数据唯一确定二维薛定谔方程中无界势的结果，并给出了定量版本。

The fractional in time Schrödinger equation with a Hartree perturbation

The aim of this work is to show existence, uniqueness and regularity properties of nonlinear fractional Schrödinger equation with fractional time derivative of order $α\in (0,1)$ and with a Hartree-type nonlinear term.

两物种Lotka-Volterra竞争-扩散系统行波的存在性与渐近稳定性：几何奇异摄动方法

Existence and Asymptotic Stability of Traveling Waves of Two-species Lotka-Volterra Competition-Diffusion Systems via Geometric Singular Perturbations

本文通过几何奇异摄动理论，证明了两物种Lotka-Volterra竞争-扩散系统中连接共存态与平凡态的行波存在性，并分析了其单调性与渐近稳定性。

$\boldsymbol{\text{R}}^N$中带幂扰动的对数薛定谔方程的基态：存在性、唯一性与非退化性

Ground states of a logarithmic Schrödinger equation with a power perturbation in $\mathbb{R}^N$: existence, uniqueness and nondegeneracy

本文研究$\boldsymbol{\text{R}}^N$中带幂扰动的对数薛定谔方程的正基态解，证明其对任意正参数存在、小参数下唯一且非退化，通过Nehari流形极小化等方法完成证明。

Pathwise regularization of the stochastic heat equation with multiplicative noise through irregular perturbation

Existence and uniqueness of solutions to the stochastic heat equation with multiplicative spatial noise is studied. In the spirit of pathwise regularization by noise, we show that a perturbation by a sufficiently irregular continuous path establish wellposedness of such equations, even when the drift and diffusion coefficients are given as generalized functions or distributions. In addition we prove regularity of the averaged field associated to a Lévy fractional stable motion, and use this as an example of a perturbation regularizing the multiplicative stochastic heat equation.

拓扑绝缘体中波导的反散射问题

Inverse scattering for waveguides in topological insulators

本文研究了拓扑非平凡波导分隔二维拓扑绝缘体的反散射问题，采用Dirac系统模型，证明在线性化和有限维设定下，短程扰动可通过散射数据完全重建，并在适当拓扑下提供稳定性结果，通过标准共轭方法数值求解并验证理论结果。