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1. 分层仿积

Hierarchical paraproducts

作者: Oluwadamilola Fasina

类别: math.AP

发布日期: 2026-02-18

arXiv ID: 2602.16644v1

链接: http://arxiv.org/abs/2602.16644v1

摘要: We outline an extension of paraproduct decompositions for compositions of the form $A(f)$ where $A \in C^{d}(\mathbb{R}), f \in Λ_α([0,1]^d)$ developed in [arXiv:2503.12629] and [arXiv:2508.13322] to settings where $(A \in C^1(\mathbb{R}),f \in Λ_α(X))$ and $ (A \in C^2(\mathbb{R}),f \in Λ_α(X \times Y))$. To do so, we construct partition trees on $X$ and $X \times Y$ such that analysis with respect to scale is sensible. We obtain results resembling those of [arXiv:2503.12629] and [arXiv:2508.13322], but with the finite sets $X$ and $X \times Y $ as support. In particular we construct the paraproduct $Π_{A',A''}^{L,S}: f \to \tilde{A}_{L,S}(f) + Δ_{L,S}(A,f)$ such that $Δ_{L,S}(A,f) \in Λ_{2α}(X \times Y)$ and $\lVert Δ_{L,S}(A,f) \rVert_{Λ_{2α}(X \times Y)} \leq C_A \lVert f \rVert_{Λ_α(X \times Y)}$. Analogous results are obtained when the support is just one finite set, $X$. This extension is motivated by situations where one wishes to separate the singular and smooth components of such compositions in graph signal processing environments.

2. 次球扇形区域中混合边值问题正解的对称性

Symmetry properties for positive solutions of mixed boundary value problems in a sub-spherical sector

作者: Ruofei Yao

类别: math.AP

发布日期: 2026-02-18

arXiv ID: 2602.16630v1

链接: http://arxiv.org/abs/2602.16630v1

摘要: In this paper, we investigate the symmetry properties of positive solutions $u$ to a semilinear elliptic equation under mixed Dirichlet-Neumann boundary conditions in symmetric domains. First, we establish a maximum principle tailored to mixed-boundary problems in domains of either small volume or narrow width, thereby enabling the application of the moving plane method. Secondly, in contrast to the purely Dirichlet case, a key challenge is to establish the non-vanishing of the tangential derivative of $u$ along the Neumann boundary. To address this, we employ local analysis techniques of angular derivatives, as introduced by Hartman and Wintner [Amer. J. Math., 1953]. Thirdly, we identify the signs of directional derivatives of $u$ along sections of the moving line. Using a planar sub-spherical sector as an example, we illustrate how these new innovative techniques and the moving plane method can be combined to derive symmetry and monotonicity results, particularly when the amplitude is less than or equal to $2π/3$.

3. 面向多状态最优设计问题的混合优化技术

Hybrid Optimization Techniques for Multi-State Optimal Design Problems

作者: Marko Erceg, Petar Kunštek, Marko Vrdoljak

类别: math.OC, math.AP

发布日期: 2026-02-18

arXiv ID: 2602.16592v1

链接: http://arxiv.org/abs/2602.16592v1

摘要: This paper addresses optimal design problems governed by multi-state stationary diffusion equations, aiming at the simultaneous optimization of the domain shape and the distribution of two isotropic materials in prescribed proportions. Existence of generalized solutions is established via a hybrid approach combining homogenization-based relaxation in the interior with suitable restrictions on admissible domains. Based on this framework, we propose a numerical method that integrates homogenization and shape optimization. The domain boundary is evolved using a level set method driven by the shape derivative, while the interior material distribution is updated via an optimality criteria algorithm. The approach is demonstrated on a representative example.

4. 含重力薄膜方程自相似剖面的适定性与稳定性

Well-posedness and stability of the self-similar profile for a thin-film equation with gravity

作者: Manuel V. Gnann, Slim Ibrahim

类别: math.AP, math-ph

发布日期: 2026-02-18

arXiv ID: 2602.16549v1

链接: http://arxiv.org/abs/2602.16549v1

摘要: We consider the thin-film equation with linear mobility and a stabilizing second-order porous-medium type term modeling gravity. The model admits self-similar solutions, and our goal is to analyze their stability. We reformulate the problem in mass-Lagrangian coordinates and exploit the underlying gradient-flow structure of the equation with respect to a weighted $L^2$ inner product, where the weight is given by the self-similar source-type profile. This framework allows us to establish a coercivity result for the Hessian (the linearization around the self-similar solution) in a suitably weighted inner product. As a consequence, we prove the convergence of perturbations toward the self-similar profile at an algebraic rate of order $t^{-\frac 1 5}$, in arbitrary scales of weighted Sobolev norms. The analysis relies on maximal-regularity estimates for the linearized evolution, combined with appropriate estimates for the nonlinear terms. Notably, beyond perturbative regimes and in contrast to previous results for the thin-film equation (convergence to the Smyth-Hill profile) or the porous-medium equation (convergence to the Barenblatt-Pattle solution), our analysis does not rely on an explicit (algebraic) representation of the self-similar profile. Instead, it is based solely on a systematic use of the ordinary differential equation satisfied by the self-similar solution, together with a careful analysis of its boundary asymptotics. As a result, we expect that the approach developed here can serve as a flexible toolbox for the study of more general classes of equations and for the stability analysis of special solutions in future work.

5. 洛伦兹等变指数定理

A Lorentzian Equivariant Index Theorem

作者: Onirban Islam, Lennart Ronge

类别: math.DG, math-ph, math.AP

发布日期: 2026-02-18

arXiv ID: 2602.16547v1

链接: http://arxiv.org/abs/2602.16547v1

摘要: We develop a formula for the equivariant index of a twisted Dirac operator on a compact globally hyperbolic spacetime with timelike boundary on which a group acts isometrically, subject to APS boundary conditions. The formula is the same as in the Riemannian case: the equivariant index for a group element is an integral over the fixed point set of that element plus some boundary terms. The proof uses a surprisingly simple technique for reducing from the equivariant to the non-equivariant regime in order to show an equivariant version of the Lorentzian “index $=$ spectral flow” formula.

6. 具有临界平方反比势的非线性薛定谔方程

Nonlinear Schrödinger equations with a critical, inverse-square potential

作者: Bartosz Bieganowski, Adam Konysz, Simone Secchi

类别: math.AP

发布日期: 2026-02-18

arXiv ID: 2602.16524v1

链接: http://arxiv.org/abs/2602.16524v1

摘要: We study the existence of solutions of the following nonlinear Schrödinger equation

$$ -Δu+V(x)u-\frac{(N-2)^2}{4|x|^2}u=f(x,u) $$

where $V:\mathbb{R}^N\to\mathbb{R}$ and $f:\mathbb{R}^N\times \mathbb{R}\to \mathbb{R}$ are periodic with respect to $x\in\mathbb{R}^N.$ We assume that $V$ has positive essential infimum, $f$ satisfies weak growth conditions and $N\geq 3$. The approach to the problem uses variational methods with nonstandard functional setting. We obtain the existence of the ground state solution using the new profile decomposition.

7. 全变号 Nehari 约束与竞争型薛定谔方程组的变号解

Fully sign-changing Nehari constraint vs sign-changing solutions of a competitive Schrödinger system

作者: Xuejiao Fu, Fukun Zhao

类别: math.AP

发布日期: 2026-02-18

arXiv ID: 2602.16506v1

链接: http://arxiv.org/abs/2602.16506v1

摘要: We study a competitive nonlinear Schrödinger system in $\mathbb{R}^N$ whose nonlinear potential is localized in small regions that shrink to isolated points. Within a variational framework based on a fully sign-changing Nehari constraint and Krasnosel’skii genus, we construct, for all $\varepsilon>0$, a sequence of sign-changing solutions with increasing and unbounded energies, and after suitable translations they converge to a sequence of sign-changing solutions of the associated limiting system as $\varepsilon\to 0$ in $H^1$-norm. Moreover, these sign-changing solutions concentrate around the prescribed attraction points both in $H^1$-norm and $L^q$-norm for $q\in [1,\infty]$.

8. 偏微分方程概率解的正则性与路径界

Regularity and Pathwise bounds for probabilistic solutions of PDEs

作者: Mouhamadou Sy

类别: math.AP, math.PR

发布日期: 2026-02-18

arXiv ID: 2602.16478v1

链接: http://arxiv.org/abs/2602.16478v1

摘要: In this paper, we build a procedure that allows to establish regularity and controls in time for probabilistic solutions to PDEs. Probabilistic approaches to global wellposedness problems usually provide ensemble bounds on the solutions. These bounds are the main tools to ensure convergence procedures yielding the existence and uniqueness of global solutions. A question of interest consists in transforming such ensemble bounds into individual controls on the flow ; this, among other uses, gives valuable information on the long-time behavior of the solutions. Toward such question of bounds transformation, Bourgain initiated a successful procedure that exploited the local wellposedness of the PDE, with an estimate of the time of size-doubling. In this note, we construct an estimation procedure which relies on a different local requirement. It turns out that this substitute is flexible enough to be possible to fulfill with the help of the ensemble bound itself. For applications of the procedure, we are able to provide new pathwise controls on solutions to NLS equations.

9. 关于任意通量值下通道内库埃特 - 泊肃叶流动的唯一性与结构稳定性

On the uniqueness and structural stability of Couette-Poiseuille flow in a channel for arbitrary values of the flux

作者: Giovanni P. Galdi, Filippo Gazzola, Mikhail V. Korobkov, Xiao Ren, Gianmarco Sperone

类别: math.AP

发布日期: 2026-02-18

arXiv ID: 2602.16460v1

链接: http://arxiv.org/abs/2602.16460v1

摘要: We establish uniqueness and structural stability of a class of parallel flows in a 2D straight, infinite channel, under perturbations with either globally or locally bounded Dirichlet integrals. The significant feature of our result is that it does not require any restriction on the size of the flux characterizing the flow. Precisely, by extending and refining an approach initially introduced by J.B. McLeod, we demonstrate the continuous invertibility of the linearized operator at a generic Couette-Poiseuille solution that does not exhibit flow reversal. We then deduce local uniqueness of these solutions as well as their nonlinear structural stability under small external forces. Moreover, we prove the uniqueness of certain class of Couette-Poiseuille solutions ``in the large," within the set of solutions possessing natural symmetry. Finally, we bring an example showing that, in general, if the flow reversal assumption is violated, the linearized operator is no longer invertible.

10. 非均匀裂隙多孔介质中非线性反应输运的多尺度双曲 - 抛物模型

Multiscale Hyperbolic-Parabolic Models for Nonlinear Reactive Transport in Heterogeneously Fractured Porous Media

作者: Taras Mel’nyk, Sorin Pop, Christian Rohde

类别: math.AP

发布日期: 2026-02-18

arXiv ID: 2602.16439v1

链接: http://arxiv.org/abs/2602.16439v1

摘要: We study nonlinear reactive transport in a layered porous medium separated by an $\varepsilon$-thin, highly heterogeneous fracture whose aperture and obstacle pattern vary periodically. Species transport in the bulk is governed by parabolic reaction–diffusion equations, coupled to a convection-diffusion-reaction problem in the fracture with nonlinear wall and obstacle reactions and Peclet number of order $O(\varepsilon^{-1})$. Via multiscale analysis as $\varepsilon \to 0$, when the fracture collapses to a flat interface, we derive a new type of homogenized model consisting of bulk diffusion–reaction equations coupled through nonlinear interface conditions and a first-order semilinear hyperbolic system on the interface. We prove well-posedness and regularity of the limit system, construct a multiscale approximation with boundary-layer correctors, and derive quantitative error estimates in suitable energy norms.

11. 关于受限集中阶梯状层合的注记

A remark on staircase laminates in restricted sets

作者: Igor Buchowiec, Pholphum Kamthorntaksina, Katarzyna Mazowiecka, Armin Schikorra, Akshara Vincent

类别: math.AP

发布日期: 2026-02-18

arXiv ID: 2602.16407v1

链接: http://arxiv.org/abs/2602.16407v1

摘要: We slightly extend the convex integration via staircase laminate toolbox recently developed by Kleiner, Müller, Székelyhidi, and Xie. As an example we revisit the proof by Astala-Faraco-Székelyhidi on optimal Meyers’ regularity theory via this framework.

12. 非紧流形上对合系统的全局盖夫雷次椭圆性

Global Gevrey Hypoellipticity of Involutive Systems on Non-Compact Manifolds

作者: Sandro Coriasco, Alexandre Kirilov, Wagner Augusto Almeida de Moraes, Pedro Meyer Tokoro

类别: math.AP

发布日期: 2026-02-18

arXiv ID: 2602.16366v1

链接: http://arxiv.org/abs/2602.16366v1

摘要: We investigate the global Gevrey hypoellipticity of a class of first-order differential operators associated with tube-type involutive structures on $M\times\mathbb{T}^m$, where $M$ is a non-compact manifold diffeomorphic to the interior of a compact manifold with boundary and $\mathbb{T}^m$ is the $m$-dimensional torus. For $s>1$, we work in Gevrey classes of Roumieu and Beurling type. A key step is the construction, on $M$, of a scattering metric whose coefficients are Gevrey of order $s$ in every analytic chart; this allows us to use Hodge theory and obtain Gevrey regularity for the harmonic forms. Under a natural condition on the defining closed $1$-forms, we obtain a sharp criterion for global Gevrey hypoellipticity in terms of rationality and (Roumieu/Beurling) exponential Liouville behavior.

13. 由具有非正则空间依赖系数的混合局部 - 非局部算子驱动的热方程

Heat Equation driven by mixed local-nonlocal operators with non-regular space-dependent coefficients

作者: Arshyn Altybay, Michael Ruzhansky

类别: math.AP

发布日期: 2026-02-18

arXiv ID: 2602.16348v1

链接: http://arxiv.org/abs/2602.16348v1

摘要: In this paper, we study the Cauchy problem for a heat equation governed by a mixed local–nonlocal diffusion operator with spatially irregular coefficients. We first establish classical well-posedness in an energy framework for bounded, measurable coefficients that satisfy uniform positivity, and we derive an a priori estimate ensuring uniqueness and continuous dependence on the initial data. We then extend the notion of solution to distributional coefficients and initial data by a Friedrichs-type regularisation procedure. Within this very weak framework, we establish the existence and uniqueness of solution nets and prove consistency with the classical weak solution whenever the coefficients are regular.

14. 格与一般有限子图上热方程及分数阶热方程的长时间渐近性

Large-Time Asymptotics for Heat and Fractional Heat Equations on the Lattice and General Finite Subgraphs

作者: Rui Chen, Bo Li

类别: math.AP

发布日期: 2026-02-18

arXiv ID: 2602.16339v1

链接: http://arxiv.org/abs/2602.16339v1

摘要: In this paper, we study large-time asymptotics for heat and fractional heat equations in two discrete settings: the full lattice $\mathbb Z^d$ and finite connected subgraphs with Dirichlet boundary condition. These results provide a unified discrete theory of long-time asymptotics for local and nonlocal diffusions. For $d\ge1$ and $s\in(0,1]$, we consider on $\mathbb Z^d$ the Cauchy problem

$$ \partial_t u+(-Δ)^s u=0,\qquad u(0)=u_0\in \ell^1(\mathbb Z^d), $$

and derive a precise first-order asymptotic expansion toward the lattice fractional heat kernel $G_t^{(s)}$. The main technical input is a pair of sharp translation-increment bounds for $G_t^{(s)}$: a pointwise estimate and an $\ell^1$-estimate. As consequences, under finite first moment we obtain the optimal decay rate $t^{-1/(2s)}$ in $\ell^p$-asymptotics ($1\le p\le\infty$), and we prove sharpness by explicit shifted-kernel examples. Without moment assumptions, we still establish convergence in the full $\ell^1$-class, and we show that no universal quantitative rate can hold in general. We also analyze fractional Dirichlet diffusion on finite connected subgraphs (restricted fractional setting, including $s=1$ as the local case). In this finite-dimensional framework, solutions admit spectral decomposition and exhibit exponential large-time behavior governed by the principal eigenvalue and the spectral gap. In addition, we study positivity improving properties of the associated semigroups for both the lattice and Dirichlet evolutions.

15. 具有 Riesz 势幂非线性及伪测度空间初值的半线性阻尼波动方程的临界阈值

Critical thresholds for semilinear damped wave equations with Riesz potential power nonlinearity and initial data in pseudo-measure spaces

作者: Tang Trung Loc, Duong Dinh Van, Phan Duc An

类别: math.AP

发布日期: 2026-02-18

arXiv ID: 2602.16293v1

链接: http://arxiv.org/abs/2602.16293v1

摘要: In this paper, our primary objective is to establish the time decay properties of solutions $u(t, x)$ with initial data $u_0, u_1 \in \mathcal{Y}^q$ (pseudo-measure spaces) to the linear damped wave equation in the spaces $L^2\left(\mathbb{R}^n\right)$ and $\dot{H}^s\left(\mathbb{R}^n\right)$ (for $s \leq 0$ or $s \geq 0$). Our subsequent aim is to investigate the semilinear damped wave equation with a Riesz potential-type power nonlinearity $\mathcal{I}_γ\left(|u|^p\right)$, where $γ\in [0, n)$, and initial data belonging to pseudo-measure spaces $\mathcal{Y}^q$. In addition, we derive a new critical exponent $p=p_{\mathrm{crit}}(n, q, γ):=1+\frac{2+γ}{n-q}$ for some $q \in\left(0, \frac{n}{2}\right)$ and in low spatial dimensions, within the framework of pseudo-measure spaces $\mathcal{Y}^q$. Specifically, we prove the global (in time) existence of small-data Sobolev solutions with low regularity when $p \geq p_{\mathrm{crit}}(n, q, γ)$, and the finite-time blow-up of weak solutions, even for small initial data, whenever $1< p<p_{\mathrm{crit}}(n, q, γ)$. Moreover, in order to characterize the blow-up time more precisely, we establish sharp upper and lower bound estimates for the lifespan of solutions in the subcritical regime.

16. 有界域上具有组合非线性项的非线性薛定谔方程归一化解的存在性与非存在性

Existence and nonexistence of normalized solutions for nonlinear Schrödinger equation involving combined nonlinearities in bounded domain

作者: Zhen-Feng Jin, Weimin Zhang

类别: math.AP

发布日期: 2026-02-18

arXiv ID: 2602.16263v1

链接: http://arxiv.org/abs/2602.16263v1

摘要: In this paper, we consider the existence, multiplicity and nonexistence of solutions for the following equation \begin{equation*} \begin{cases} \begin{aligned} &-Δu+ωu=μu^{p-1}+u^{q-1},~ u>0 \quad &&\text { in } Ω, \ &u=0 &&\text { on } \partialΩ, \ \end{aligned} \end{cases} \end{equation*} with prescribed $L^2$-norm $\|u\|_2^2=ρ$, where $N\ge 1$, $ρ>0$, $μ\in \mathbb{R}$, $1<p\le q$, and $Ω\subset\mathbb{R}^N$ is a bounded smooth domain. The parameter $ω\in\mathbb{R}$ arises as a Lagrange multiplier. Firstly, when $2<p\le q\le \frac{2N}{(N-2)^+}$ and $ρ$ is small, we establish the existence of a local minimizer of energy. Furthermore, when $μ\ge 0$ and $Ω$ is a star-shaped domain, using the monotonicity trick and the Pohozaev identity, we show that there exists a second solution which is of mountain pass type. Secondly, when $μ\ge 0$, $N\ge 3$, $1<p\le 2$, $q\ge \max\left\{\frac{2N}{N-2}, 3\right\}$ and $Ω$ is a convex domain, using the moving-plane method, we prove the nonexistence of normalized solutions for large $ρ$. Finally, when $μ=0$, $N\ge 3$, $q=\frac{2N}{N-2}$ and $Ω$ is a ball, we give a dichotomy result of normalized solutions for the Brézis-Nirenberg problem by continuation arguments.

17. 不连续非线性与惩罚方法下 $(p,N)$-拉普拉斯方程的集中现象

Concentration Phenomena for $(p,N)$-Laplace Equation Under Discontinuous Nonlinearities and Penalization Method

作者: Ankit, Giovany M. Figueiredo, Abhishek Sarkar

类别: math.AP

发布日期: 2026-02-18

arXiv ID: 2602.16254v1

链接: http://arxiv.org/abs/2602.16254v1

摘要: In this paper, we investigate the existence and concentration of solutions to a $(p,N)$-Laplace equation in $\mathbb{R}^N$ involving a discontinuous nonlinearity and critical exponential growth. To establish the existence of solutions, we employ a penalization technique in the sense of Del Pino and Felmer adapted to a locally Lipschitz functional. Furthermore, by combining variational methods with Moser-type iteration techniques, we obtain the concentration behavior of the solutions. Our results contribute to the study of nonlinear elliptic problems with irregular nonlinearities and critical growth phenomena.

18. 流体中接触线的全局动态稳定性：二维液滴问题

Global dynamic stability of contact lines in fluids: 2-D droplet problem

作者: Xiaoding Yang

类别: math.AP

发布日期: 2026-02-18

arXiv ID: 2602.16164v1

链接: http://arxiv.org/abs/2602.16164v1

摘要: In this paper, we investigate the dynamics of an incompressible viscous Navier-Stokes fluid evolving above a one-dimensional flat surface. The fluid is subject to a uniform gravitational field and capillary forces acting along the free boundary. The interface between the fluid and the surrounding air is a free surface whose motion is driven by gravity, surface tension, and the fluid velocity field. The triple-phase intersections where the fluid, the air above the vessel, and the solid vessel wall meet are referred to as contact points, and the angles formed there are called contact angles. The model under consideration incorporates boundary conditions that allow for full motion of the contact points and dynamic contact angles. Under these conditions, \cite{Yang} established the existence of equilibrium configurations for the model. These equilibria consist of a quiescent fluid occupying a domain whose upper boundary can be represented as the graph of a function in polar coordinates, minimizing a gravity-capillary energy functional subject to a fixed mass constraint. The equilibrium contact angles may take any value in $(0,π)$ depending on the choice of capillary parameters. In the present work, we develop a framework of a priori estimates for this model. We prove that, for initial data sufficiently close to equilibrium, the system admits global solutions that converge exponentially fast to a (horizontally) shifted equilibrium state.

19. 一维具组合非线性的非线性薛定谔方程

The nonlinear Schrödinger equation with combined nonlinearities in 1D

作者: Oscar Riaño, Alex D Rodriguez, Svetlana Roudenko

类别: math.AP, math.NA

发布日期: 2026-02-18

arXiv ID: 2602.16122v1

链接: http://arxiv.org/abs/2602.16122v1

摘要: We consider the one-dimensional nonlinear Schrödinger equation

$$ iu_t + u_{xx} + \mathcal{N}(u)u=0, \quad x,t \in \mathbb R, $$

with the nonlinearity term that is expressed as a sum of powers, possibly infinite:

$$ \mathcal{N}(u) = \sum d_k |u|^{α_k}, \quad α_k > 0. $$

We first investigate the local well-posedness of this equation for any positive powers of $α_k$ in a certain weighted class of initial data, subset of $H^1 (\mathbb R)$. For that we use an approach of Cazenave-Naumkin [19], thus, avoiding any Strichartz estimates. Then, using the pseudo-conformal transformation, we extend the local result to the global one for the initial data with a quadratic phase. Furthermore, we investigate the asymptotic behavior of such global solutions and prove scattering for data with the quadratic phase $e^{ib|x|^2}$ with sufficiently large positive $b$, in $H^1(\mathbb R)$. One of the advantages of considering an infinite sum in the nonlinearity term is being able to consider exponential nonlinearities, such as $e^{γ|u|^{k}} u$, as well as sine or cosine nonlinearities, and obtain well-posedness in those cases, the first such result for most of those nonlinearities. To conclude, we show numerical simulations for various examples of combined nonlinearities, including the double nonlinearity and an exponential one, then investigate the behavior of solutions with positive or negative initial $b$ in a quadratic phase data. Furthermore, we also show that a ground state in the NLS equation with combined nonlinearities no longer provides a sharp threshold for global behavior such as scattering vs. finite time blow-up, instead the equation has a much richer dynamics.

20. 一种纳维 - 斯托克斯方程伽辽金近似的内禀展开方法

An intrinsic expansion approach to the Galerkin approximations for the Navier-Stokes equations

作者: Luan Hoang, Michael S. Jolly

类别: math.AP, math.NA

发布日期: 2026-02-17

arXiv ID: 2602.16064v1

链接: http://arxiv.org/abs/2602.16064v1

摘要: We study the Galerkin approximation of the three-dimensional Navier-Stokes equations. In particular, we examine the convergence of these solutions in a sequence of finite dimensional spaces as the dimension goes to infinity. For any sequence of steady state or, respectively, time dependent Galerkin solutions that converges to a solution of the Navier-Stokes equations, we obtain a subsequence with an intrinsic asymptotic expansion in appropriate nested function spaces. Consequently, an induced asymptotic expansion is obtained in a more standard spatial Sobolev or, respectively, spatiotemporal Sobolev-Lebesgue space. In the case of steady states, we establish certain relations among leading terms of this expansion.

21. 狄拉克算子的定量 Landis 型结果

Quantitative Landis-type result for Dirac operators

作者: Ujjal Das, Luca Fanelli, Luz Roncal

类别: math.AP, math-ph, math.SP

发布日期: 2026-02-17

arXiv ID: 2602.16049v1

链接: http://arxiv.org/abs/2602.16049v1

摘要: We study quantitative unique continuation at infinity for Dirac equations with bounded matrix-valued potentials. For the massless Dirac operator $\mathcal{D}_n$ in $\mathbb{R}^n$, we establish a Landis-type estimate showing that the vanishing order of any nontrivial bounded solution of $( \mathcal{D}_n + \mathbb{V} ) \varphi = 0$ satisfies a lower bound of order $\exp(-κR^{2} (\log R)^{2})$ as $|x|=R\to \infty$; the quadratic growth in the exponent is sharp, in view of previous known results. Our proof follows a Bourgain–Kenig type approach based on a Carleman inequality for Dirac operators which relies on a local Hölder regularity result, which we also prove. In two dimension, we obtain improved quantitative estimates under symmetry assumptions on the potential $\mathbb{V}$ and for real-valued solutions. Finally, we also derive qualitative Landis-type results for Dirac equations with decaying potentials, including critical decay rates.

22. 关于高维极小超曲面的卡拉比 - 丘猜想

On the Calabi–Yau Conjectures for Minimal Hypersurfaces in Higher Dimensions

作者: Shrey Aryan, Alexander McWeeney

类别: math.DG, math.AP

发布日期: 2026-02-17

arXiv ID: 2602.16048v1

链接: http://arxiv.org/abs/2602.16048v1

摘要: In this paper, we study the Calabi–Yau conjectures for complete minimal hypersurfaces $Σ^{n}\subset \mathbb{R}^{n+1}$ in dimensions $n\ge 3$. These conjectures ask whether a complete minimal hypersurface must be unbounded, and more strongly, whether it must be proper. For the unboundedness question, we prove a chord–arc estimate for an embedded minimal disk with bounded curvature, showing that intrinsic distance is controlled by a polynomial of the extrinsic distance. On the other hand, using gluing techniques, we construct a complete, improperly embedded minimal hypersurface in $\mathbb{R}^{n+1}$ for every $n\ge 3$. This example shows that the properness conjecture suggested by the deep work of Colding Minicozzi [CM08] in the case $n=2$ fails in higher dimensions.

23. 具有认知地图的非局部种群模型中的滞留现象与模式形成

The lingering phenomenon and pattern formation in a nonlocal population model with cognitive map

作者: Kyung-Han Choi, Thomas Hillen

类别: math.AP, q-bio.PE

发布日期: 2026-02-17

arXiv ID: 2602.16022v1

链接: http://arxiv.org/abs/2602.16022v1

摘要: The rates at which individuals memorize and forget environmental information strongly influence their movement paths and long-term space use. To understand how these cognitive time scales shape population-level patterns, we propose and analyze a nonlocal population model with a cognitive map. The population density moves by a Fokker–Planck type diffusion driven by a cognitive map that stores a habitat quality information nonlocally. The map is updated through local presence with learning and forgetting rates, and we consider both truncated and normalized perception kernels. We first study the movement-only system without growth. We show that finite perceptual range generates spatial heterogeneity in the cognitive map even in nearly homogeneous habitats, and we prove a lingering phenomenon on unimodal landscapes: for the fixed high learning rate, the peak density near the best location is maximized at an intermediate forgetting rate. We then couple cognitive diffusion to logistic growth. We establish local well-posedness and persistence by proving instability of the extinction equilibrium and the existence of a positive steady state, with uniqueness under an explicit condition on the motility function. Numerical simulations show that lingering persists under logistic growth and reveal a trade-off between the lingering and total population size, since near the strongest-lingering regime the total mass can fall below the total resource, in contrast to classical random diffusive–logistic models.

24. 具有代数封闭关系的双流体可压缩系统的弱 - 强唯一性

Weak–strong uniqueness for bi-fluid compressible system with algebraic closure

作者: Yang Li, Mária Lukáčová-Medvid’ová, Milan Pokorný, Ewelina Zatorska

类别: math.AP

发布日期: 2026-02-17

arXiv ID: 2602.15970v1

链接: http://arxiv.org/abs/2602.15970v1

摘要: We consider a real two-fluid system of compressible viscous fluids with a common velocity field and algebraic closure for the pressure law. The constitutive relation involves densities of both fluids through an implicit function. The existence of global-in-time finite energy weak solutions to this system is known since the work of Novotný and Pokorný [Arch. Rational Mech. Anal., 2020]. On the other hand, existence of local-in-time strong solutions is due to Piasecki and Zatorska [J. Math Fluid Mech.,, 2022]. In this paper, we establish the weak–strong uniqueness principle using the relative entropy method. In sharp contrast to the two-phase model of Baer-Nunziato type, the volume fraction of phase $+$ obeys a transport equation with an additional nonlinear term. This gives rise to troublesome terms in the relative entropy inequality. We are able to close the estimate by making an elaborate use of the structure of the system.

25. 有效能量 - 拟能扩散过程与凝聚界

Effective energy-enstrophy diffusion process and condensation bound

作者: Alain-Sol Sznitman, Klaus Widmayer

类别: math.PR, math-ph, math.AP

发布日期: 2026-02-17

arXiv ID: 2602.15810v1

链接: http://arxiv.org/abs/2602.15810v1

摘要: In this article we use Gaussian measure on $\mathbb{R}^N$ to define the coefficients of an elliptic diffusion on an open cone of $\mathbb{R}^2$. We prove the existence and uniqueness of a stationary distribution for this diffusion. In a companion article, we show that the diffusion constructed in this work is the inviscid limit of the laws of the ``enstrophy-energy’’ process of a stationary $N$-dimensional Galerkin-Navier-Stokes type evolution with Brownian forcing and random stirring (the strength of which can be made to go to zero in the inviscid limit). In the present work, owing to the special properties of the coefficients constructed with the Gaussian measure, we bound the distance to $1$ of the ratio of the expected energy to the expected enstrophy (this ratio is at most $1$ with our normalization). Together with our companion article, this shows that for suitable Brownian forcings an inviscid condensation inducing an attrition of all but the lowest modes takes place.

26. 无粘极限与有效能量 - 拟能扩散过程

Inviscid limit and an effective energy-enstrophy diffusion process

作者: Alain-Sol Sznitman, Klaus Widmayer

类别: math.PR, math-ph, math.AP

发布日期: 2026-02-17

arXiv ID: 2602.15805v1

链接: http://arxiv.org/abs/2602.15805v1

摘要: In this article we consider a stationary $N$-dimensional Galerkin-Navier-Stokes type evolution with Brownian forcing and random stirring (of arbitrarily small strength). We show that the stationary diffusion in an open two-dimensional cone constructed in a companion article, stands as the inviscid limit of the laws of the ``enstrophy-energy’’ process of the $N$-dimensional diffusion process considered here, this regardless of the strength of the stirring. With the help of the quantitative condensation bounds of the companion article, we infer quantitative inviscid condensation bounds, which for suitable forcings show an attrition of all but the lowest modes in the inviscid limit.

27. 带电零尘埃坍缩中的类时反弹超曲面

Timelike bounce hypersurfaces in charged null dust collapse

作者: David Bick

类别: gr-qc, math-ph, math.AP, math.DG

发布日期: 2026-02-17

arXiv ID: 2602.15786v1

链接: http://arxiv.org/abs/2602.15786v1

摘要: We establish results on the dynamics of interacting charged null fluids in general relativity, specifically in the context of the bouncing continuation proposed in [Ori91]. In this model - the setting for a number of prominent case studies on black hole formation - charged massless particles may instantaneously change direction (bounce) after losing all their 4-momentum due to electrostatic repulsion. We initiate the study of timelike bounce hypersurfaces in spherical symmetry: scenarios in which an incoming beam of charged null dust changes direction along a timelike surface $\mathcal{B}$, which is the (free) boundary of an interacting 2-dust region. We identify a novel decoupling of the equations of motion in this region. First, it is shown that every timelike curve segment $γ$ in the spherically symmetric quotient of Minkowski or Reissner-Nordström spacetimes arises as the bounce hypersurface $\mathcal{B}$ of a charged null dust beam incident from past null infinity $\mathcal{I}^-$. We construct a spacetime $(\mathcal{M},g_{μν})$ describing the full trajectory of the beam, which includes gluing to Reissner-Nordström and Vaidya regions. Across $\mathcal{B}$ the metric has regularity $g_{μν}\in C^{2,1}$ and satisfies Einstein’s equation classically, while $C^\infty$ gluing may be achieved across all other interfaces. We also obtain examples of timelike bounce hypersurfaces terminating in a null point. Since these constructions are teleological, we secondly consider a given charged incoming beam from past null infinity. We formulate and solve a free boundary problem which represents the formation of a timelike bounce hypersurface. The result is conditional, applying only in the exterior region of a Reissner-Nordström spacetime, and subject to a technical regularity condition.