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1. 一维空间中具有独立对流的交叉扩散系统的整体解
Global solutions to cross-diffusion systems with independent advections in one dimension
作者: Jakub Skrzeczkowski
类别: math.AP
发布日期: 2026-03-20
arXiv ID: 2603.20153v1
链接: http://arxiv.org/abs/2603.20153v1
摘要: We consider cross-diffusion systems describing evolution of two species $u$ and $v$ moving according to the Darcy’s law with the pressure law $p(s) = \frac{1}{α-1} s^{α-1}$ where $s=u+v$. One of the most challenging questions in the field is the construction of solutions to the problem in the presence of additional advection fields, without imposing any artificial structure (for instance, preventing the species from mixing). Although advection arises naturally in these models, it breaks the symmetry of the system and prevents application of techniques developed in recent years. Here, we provide a new approach to construct solutions in one space dimension that works in a unified manner for all pressure exponents $α\in (0,\infty)$ and for arbitrary initial data (segregated, mixed, or partially mixed). In particular, in the regime $α> 1$, this yields the first existence result of its kind, obtained without any structural assumptions. We construct the solutions as a limit of a vanishing viscosity approximation $(u_{\varepsilon}, v_{\varepsilon})$. The main challenge is to identify the limit of $u_{\varepsilon} \, \partial_x p(s_{\varepsilon})$, where $s_{\varepsilon} = u_{\varepsilon} + v_{\varepsilon}$. The key new insight is that possible oscillations of $u_\varepsilon$ and $\partial_x p(s_\varepsilon)$ are correlated, simplifying the Young measure analysis in the compensated compactness argument and allowing identification of the limit.
2. 分数阶Orlicz-Sobolev空间中不带Δ₂条件的非局部拟线性各向异性方程的特征值问题
An eigenvalue problem for a nonlocal quasilinear anisotropic equation in fractional Orlicz Sobolev spaces without the $Δ_2$–condition
作者: Julian Fernandez Bonder, Martin Guzman, Juan F. Spedaletti
类别: math.AP
发布日期: 2026-03-20
arXiv ID: 2603.20095v1
链接: http://arxiv.org/abs/2603.20095v1
摘要: In this paper we analyze an eigenvalue problem associated to fractional operators of the form
which represents a generalization model for nonlocal, nonstandard growth diffusion problems. We study this problem in the context of the fractional Orlicz Sobolev spaces without assuming the so-called $Δ_2$–condition on the Young functions involved. We show existence of a sequence of eigenpairs $(u_k,λ_k)\to (0,+\infty)$.
3. 二维欧拉方程在Yudovich正则性下的剪切流渐近稳定性
Asymptotic stability of shear flows for 2D Euler equations at Yudovich regularity
作者: Dengjun Guo, Xiaoyutao Luo
类别: math.AP
发布日期: 2026-03-20
arXiv ID: 2603.20065v1
链接: http://arxiv.org/abs/2603.20065v1
摘要: The nonlinear asymptotic stability of shear flows in the 2D Euler equations has traditionally been linked to inviscid damping in the periodic setting. Since Gevrey regularity is required to suppress the ``echo’’ phenomenon, asymptotic stability is known to be impossible in Sobolev spaces. In this paper, we identify a distinct stabilizing mechanism available in the infinite channel: the advection of vorticity to spatial infinity. We establish nonlinear asymptotic stability for the 2D Euler equations in the infinite channel $\mathbb{R}\times[0,1]$ at the minimal regularity of the Yudovich class ($L^{\infty}$ vorticity). Specifically, for a class of non-negative shear flows with a curvature bound, any $L^\infty$-small, compactly supported vorticity perturbation leads to decay on compact subsets and weak convergence to zero.
4. 三维广义牛顿流体的正则性准则
Regularity criterion for the 3D generalized Newtonian fluids
作者: Qiao Liu, Xincheng Shi
类别: math.AP
发布日期: 2026-03-20
arXiv ID: 2603.19941v1
链接: http://arxiv.org/abs/2603.19941v1
摘要: In this paper, we prove that a weak solution of the Cauchy problem for 3D unsteady flows of a generalized Newtonian fluid becomes a strong solution for $\frac{5}{3} <p<\frac{11}{5} $ provided that the gradient of velocity $\nabla \boldsymbol{u}$ belongs to the critical space $L^{\frac{2}{2-(3-p)a}}(0,T;\dot{B}^{-a}_{\infty,\infty}(\mathbb{R}^3))$, where $a\in(\frac{3}{2},\frac{2}{3-p})$ if $p\in(\frac{5}{3},2)$ and $a\in(\frac{1}{p},\frac{2}{3-p})$ if $p\in[2,\frac{11}{5})$.
5. $\mathbb{R}^N$ 中临界指数情形下分数阶薛定谔方程多峰解的存在性、非退化性和局部唯一性
Existence, non-degeneracy and local uniqueness of multi-peak solutions to the fractional Schrödinger equation with nearly critical exponent in $\mathbb{R}^N$
作者: Yanyan Guo, Ying Li, Zhongyuan Liu, Pingping Yang
类别: math.AP
发布日期: 2026-03-20
arXiv ID: 2603.19868v1
链接: http://arxiv.org/abs/2603.19868v1
摘要: In this paper, we consider the following fractional Schrödinger equation \begin{equation*} \left{ \begin{array}{lcl} (-Δ)^{s}u+V(x)u=u^{{p_s}-ε}\ \ \ &\hbox{in}\ \mathbb{R}^N,\ u>0\ \ \ &\hbox{in}\ \mathbb{R}^N, \end{array} \right. \end{equation*} where $0<s<1$, $ε>0$, $p_s=(N+2s)/(N-2s)$, $N>4s$ and $V(x)\in C^1(\mathbb{R}^N)\cap L^\infty (\mathbb{R}^N)$ is non-negative. We first use the Lyapunov-Schmidt reduction method to construct multi-peak solutions to the above equation provided that $V(x)$ possesses $k$ stable critical points. Then we prove the non-degeneracy and local uniqueness of the multi-peak solutions, for $\frac{1}{2}<s<1$, $N\geq 6s$, via the blow-up argument based on various local Pohozaev identities. Due to the nonlocal property of the fractional Laplacian, we need to make delicate analysis of the approximate solutions and establish the local Pohozaev identities for the corresponding harmonic extension instead of $u$. This approach not only requires to develop refined estimates for several integrals in the local Pohozaev identities, but also to apply Pohozaev identities through a markedly different way.
6. 闭可定向黎曼流形上椭圆算子主特征值的渐近分析
Asymptotics of the principal eigenvalue of an elliptic operator on closed and orientable Riemannian manifolds
作者: Xin Xu, Kexin Zhang
类别: math.AP
发布日期: 2026-03-20
arXiv ID: 2603.19861v1
链接: http://arxiv.org/abs/2603.19861v1
摘要: This paper investigates the asymptotic behavior of the principal eigenvalue $λ(s)$, as $s\to+\infty$, for the following elliptic eigenvalue problem \begin{equation*}\label{E} -Δ_{M}u-s\langle \nabla_M f, \nabla_M u\rangle_g +c u=λ(s)u, \end{equation*} defined on an orientable and closed Riemannian manifold $(M,g)$. Assuming $f$ is a Morse function defined on $M$, we find that the limit $\lim\limits_{s\to+\infty} λ(s)$ is determined by the minimum value of the function $c$ over the set of the maximum points of $f$, a result that is independent of the curvature of manifold.
7. 两相老化偏微分方程模型的稳定性分析与长时间收敛
Stability analysis and long-time convergence of a partial differential equation model of two-phase ageing
作者: Luce Breuil
类别: math.AP, math.DS, q-bio.PE
发布日期: 2026-03-20
arXiv ID: 2603.19814v1
链接: http://arxiv.org/abs/2603.19814v1
摘要: Recent biological evidence suggests the presence of a two-phase ageing process in several species. We introduce a system of two age-structured partial differential equations (PDE) representing two phases of ageing of a wild population. The model includes a coupling of both equations through birth and transition between phases and non-linearities due to competition. We show the existence, positivity and uniqueness of weak solutions in a general setting. For a simplified system of ordinary differential equations (ODE), we show existence and uniqueness of a strictly positive steady state attracting all trajectories. We study another simplification, a coupled PDE-ODE model, for which we prove existence, uniqueness and local asymptotic stability of a strictly positive steady state. Under further assumptions, but without assuming weak non-linearities, we show the global asymptotic stability of that steady state. The uniqueness of steady states and absence of oscillations in these systems show that the proportion of individuals in each phase at equilibrium is a unique feature of the model. This paves the way to ecological applications as the experimental measure of such a proportion could help gain some insight on the health of a wild population.
8. 双时间尺度学习动力学：神经网络训练的总体视角
Two-Time-Scale Learning Dynamics: A Population View of Neural Network Training
作者: Giacomo Borghi, Hyesung Im, Lorenzo Pareschi
类别: cs.LG, math.AP, stat.ML
发布日期: 2026-03-20
arXiv ID: 2603.19808v1
链接: http://arxiv.org/abs/2603.19808v1
摘要: Population-based learning paradigms, including evolutionary strategies, Population-Based Training (PBT), and recent model-merging methods, combine fast within-model optimisation with slower population-level adaptation. Despite their empirical success, a general mathematical description of the resulting collective training dynamics remains incomplete. We introduce a theoretical framework for neural network training based on two-time-scale population dynamics. We model a population of neural networks as an interacting agent system in which network parameters evolve through fast noisy gradient updates of SGD/Langevin type, while hyperparameters evolve through slower selection–mutation dynamics. We prove the large-population limit for the joint distribution of parameters and hyperparameters and, under strong time-scale separation, derive a selection–mutation equation for the hyperparameter density. For each fixed hyperparameter, the fast parameter dynamics relaxes to a Boltzmann–Gibbs measure, inducing an effective fitness for the slow evolution. The averaged dynamics connects population-based learning with bilevel optimisation and classical replicator–mutator models, yields conditions under which the population mean moves toward the fittest hyperparameter, and clarifies the role of noise and diversity in balancing optimisation and exploration. Numerical experiments illustrate both the large-population regime and the reduced two-time-scale dynamics, and indicate that access to the effective fitness, either in closed form or through population-level estimation, can improve population-level updates.
9. 燃烧反应扩散方程的弯曲波前
Curved fronts of combustion reaction-diffusion equations
作者: Wei-Jie Sheng, Xin-Tian Zhang
类别: math.AP
发布日期: 2026-03-20
arXiv ID: 2603.19767v1
链接: http://arxiv.org/abs/2603.19767v1
摘要: This paper is concerned with curved fronts of combustion reaction-diffusion equations in $\mathbb{R}^N$ $(N\geq2)$. By mixing finite planar fronts and constructing suitable super- and subsolutions, we prove the existence, uniqueness and stability of polytope-like curved fronts in $\mathbb{R}^N$. Besides, we show that these curved fronts are transition fronts.
10. 贝克曼参数最优传输问题的正则性
Regularity of Solutions to Beckmann’s Parametric Optimal Transport
作者: Hanno Gottschalk, Tobias J. Riedlinger
类别: math.AP, stat.ML
发布日期: 2026-03-20
arXiv ID: 2603.19755v1
链接: http://arxiv.org/abs/2603.19755v1
摘要: Beckmann’s problem in optimal transport minimizes the total squared flux in a continuous transport problem from a source to a target distribution. In this article, the regularity theory for solutions to Beckmann’s problem in optimal transport is developed utilizing an unconstrained Lagrangian formulation and solving the variational first order optimality conditions. It turns out that the Lagrangian multiplier that enforces Beckmann’s divergence constraint fulfills a Poisson equation and the flux vector field is obtained as the potential’s gradient. Utilizing Schauder estimates from elliptic regularity theory, the exact Hölder regularity of the potential, the flux and the flow generating is derived on the basis of Hölder regularity of source and target densities on a bounded, regular domain. If the target distribution depends on parameters, as is the case in conditional (``promptable’’) generative learning, we provide sufficient conditions for separate and joint Hölder continuity of the resulting vector field in the parameter and the data dimension. Following a recent result by Belomnestny et al., one can thus approximate such vector fields with deep ReQu neural networks in C^(k,alpha)-Hölder norm. We also show that this approach generalizes to other probability paths, like Fisher-Rao gradient flows.
11. 极值原理与2-Hessian问题的先验估计
Minimum principles and a priori estimates for 2-Hessian problems
作者: Cristian Enache
类别: math.AP
发布日期: 2026-03-20
arXiv ID: 2603.19740v1
链接: http://arxiv.org/abs/2603.19740v1
摘要: In this paper we investigate a class of $2$-Hessian equations and establish a minimum principle for a $P$-function in the sense of L.E. Payne (see R. Sperb \cite{Sp81}). The analysis is based on a sharp matrix inequality providing an estimate for a suitable combination of second-order partial derivatives of the solution. Exploiting this estimate, we derive a differential inequality for the associated $P$-function and obtain a minimum principle in higher dimensions under a convexity assumption. As an application of our results, together with convexity results established in X.-N. Ma and L. Xu \cite{MX08}, P. Liu, X.-N. Ma and L. Xu \cite{LMX10}, P. Salani \cite{Sa12}, and Y. Ye \cite{Ye13}, we derive a priori bounds for solutions of several classical $2$-Hessian boundary value problems.
12. 小间隙区间内反向旋转圆柱的库艾特-泰勒不稳定性
Couette Taylor instabilities for counter-rotating cylinders in the small-gap regime
作者: Dongfen Bian, Emmanuel Grenier, Gérard Iooss, Zhuolun Yang
类别: math.AP
发布日期: 2026-03-20
arXiv ID: 2603.19704v1
链接: http://arxiv.org/abs/2603.19704v1
摘要: We study the Couette Taylor instabilities for an incompressible viscous fluid between two coaxial cylinders of nearly equal radii, allowing counter-rotation with the ratio of rotation rate $μ\in [-1,1]$. Working in a rotating frame and in a small-gap and small-viscosity regime, we derive the corresponding limiting Navier Stokes system and analyze the linear stability of the Couette flow. In particular, we numerically compute the critical Taylor number for general perturbations and identify a transition near $μ_c \approx -0.8$: for $μ> μ_c$ the most unstable mode is axisymmetric, whereas for $μ< μ_c$ the most unstable mode is non-axisymmetric. Near criticality, slowly varying traveling waves are governed by a time-independent Ginzburg Landau equation. The nonlinear coefficient changes sign near $\hatμ_c \approx -0.65$, yielding a supercritical regime for $μ> \hatμ_c$ and a subcritical regime for $μ_c < μ< \hatμ_c$. In the subcritical range, we classify small-amplitude steady states, including Taylor vortex flows, wavy vortices, a two-parameter family of quasi-periodic flows, and a localized traveling perturbation of the Couette flow.
13. 大振幅稳态孤立水波与一般涡量
Large-Amplitude Steady solitary water waves with general vorticity
作者: Jifeng Chu, Zihao Wang, Yong Zhang
类别: math.AP
发布日期: 2026-03-20
arXiv ID: 2603.19699v1
链接: http://arxiv.org/abs/2603.19699v1
摘要: In this paper, we study two-dimensional steady solitary gravity waves propagating along the surface of a fluid of finite depth. In particular, we can deal with general vorticity distributions and overhanging wave profiles. By conformal mappings, we reformulate the problem into an overdetermined elliptic system coupled with an elliptic boundary value problem in a fixed strip domain. To avoid imposing extra constraints on vorticity function, we further reformulate the problem into the form of an abstract operator. Based on the formulations, the existence of small-amplitude solitary waves is proved by the center manifold reduction method, while the large-amplitude waves are obtained based on the analytic global bifurcation theorem.
14. 双相拟凸泛函及其部分正则性理论
Double phase quasiconvex functionals and their partial regularity theory
作者: Sunwoo Jeong, Jihoon Ok
类别: math.AP
发布日期: 2026-03-20
arXiv ID: 2603.19696v1
链接: http://arxiv.org/abs/2603.19696v1
摘要: We consider degenerate nonautonomous energies
for vector-valued functions $v \in W^{1,1}(Ω, \mathbb{R}^N)$, where the integrand $f(x,P)$ satisfies growth and weak uniform quasiconvexity assumption associated with the double phase function $H(x,t)=t^p + a(x)t^q$. We establish partial Hölder regularity for the gradients of minimizers under suitable, and possibly minimal, regularity assumptions on $H$ and $f$. Our approach relies on two approximation results: $\mathcal{A}$-harmonic approximation and a variational version of the $φ$-harmonic approximation.
15. 临界 $p$-拉普拉斯 Patlak-Keller-Segel 模型中的自相似动力学：收缩爆破与扩展传播
Self-similar Dynamics in the Critical $p$-Laplacian Patlak-Keller-Segel Model: Shrinking Blow-up and Expanding Propagation
作者: Chunhua Jin, Fengqing Zhang
类别: math.AP
发布日期: 2026-03-20
arXiv ID: 2603.19679v1
链接: http://arxiv.org/abs/2603.19679v1
摘要: In this paper, we study the following Patlak-Keller-Segel model with $p$-Laplacian diffusion \begin{align*} \left{ \begin{aligned} &ρ_t=\nabla \cdot \left( \left| \nabla ρ\right|^{p-2}\nabla ρ\right) -χ\nabla \cdot \left( ρ\nabla c \right), &0=\varDelta c+ρ^m, \end{aligned}\right. \end{align*} and the exponent $m>0$ is chosen as
This relation ensures the scale invariance of the system and is conjectured to be the critical exponent that separates global boundedness from finite-time blow-up. We prove that, at the critical threshold $m=\frac{(p-2)N + p}{N}$, the system indeed admits finite-time blow-up solutions. More precisely, in the slow diffusion regime $p>2$, there exist backward self-similar blow-up solutions that are radially decreasing, compactly supported, and concentrate into a Dirac $δ$-measure at the blow-up time $T$; and their supports shrink toward the origin at the rate $(T-t)^{\frac1{mN}}$. For the fast diffusion case $1<p\le 2$, we show that there are no backward self-similar blow-up solutions with finite-mass. Additionally, we also explore forward self-similar solutions in both the slow diffusion and fast diffusion cases. These solutions also carry finite mass and exhibit a Dirac $δ$-singularity at the initial moment. Specifically, in the slow diffusion case, the support expands at the rate $t^{\frac1{mN}}$, whereas in the fast diffusion case, the solution becomes strictly positive for all positive times. Our work provides the first blow up analysis for the $p$-Laplacian Keller-Segel system when $p\ne 2$, and it confirms that the exponent $m$ given above is indeed the sharp threshold between global existence and finite time singularity formation.
16. 具信号消耗的Keller-Segel模型中大质量自相似奇异解的分类
Classification of self-similar singular solutions with large mass for Keller-Segel model with signal consumption
作者: Chunhua Jin
类别: math.AP
发布日期: 2026-03-20
arXiv ID: 2603.19663v1
链接: http://arxiv.org/abs/2603.19663v1
摘要: In this paper, we concentrate on investigating the self-similar singular solutions of Keller-Segel model with signal consumption ($-uv^α$) and singular sensitivity. We perform a detailed exploration into the existence and decay rate of self-similar solutions, particularly, the permissibility of arbitrary mass for these solutions across all possible cases. Based on these findings, we can delve deeper into verifying that these self-similar solutions $(u, v)$ exhibit varying degrees of singularity depending on the value of $α$ and the spatial dimension. Our analysis reveals that the component $u$ (with arbitrary mass) of the solution consistently behaves analogous to heat kernel, that is, $u$ exhibiting a Dirac $δ$ initial singularity identical to that of the fundamental solution, and converges to $0$ in the sense of the $L^p$-norm ($p>1$) as time approaches infinity. However, the initial behavior of the other component $v$ varies significantly based on the value of $α$ and the spatial dimension, exhibiting regularity (not singular), less singularity, or strong singularity (more singular than fundamental solution). Moreover, both $u$ and $v$ undergo instantaneous smoothing, becoming smooth immediately after $t>0$. This phenomenon reveals the adaptive strategies of cells in high-density aggregation environments to prevent resource depletion, reflecting an optimization process of self-organizing behavior.
17. 关于一维欧拉-泊松方程小非平凡紧解不存在性的注记
A note on the non-existence of small non-trivial compact solutions for Euler-Poisson equation in 1D
作者: Masaya Maeda, Tetsu Mizumachi
类别: math.AP
发布日期: 2026-03-20
arXiv ID: 2603.19662v1
链接: http://arxiv.org/abs/2603.19662v1
摘要: In this short note, we prove the non-existence of slow and fast small nontrivial compact solutions for the Euler-Poisson system in $1$D. The proof is based on the virial estimate which provides local in space average decay of bounded small solutions.
18. 接触型哈密顿-雅可比方程的半群交换性质
Commutation Properties of Semi-groups for Contact Type Hamilton-Jacobi Equation
作者: Guyu Jin
类别: math.AP
发布日期: 2026-03-20
arXiv ID: 2603.19650v1
链接: http://arxiv.org/abs/2603.19650v1
摘要: We know that there exist semi-groups for contact type Hamilton-Jacobi equations, which refers to \cite{KLJ2}. Guy Barles and Agnès Tourin give a proof of the commutation properties for normal Hamilton-Jacobi equations at \cite{GA}. In this article, we provide a proof of the commutation property of semi-groups for contact type Hamilton-Jacobi equations.
19. Cartan-Hadamard 3-空间中的Minkowski型精确不等式
Sharp Minkowski Type Inequality in Cartan-Hadamard 3-Spaces
作者: Fang Hong
类别: math.DG, math.AP
发布日期: 2026-03-20
arXiv ID: 2603.19646v1
链接: http://arxiv.org/abs/2603.19646v1
摘要: In this paper, we proved a sharp Minkowski type inequality in Cartan-Hadamard 3-spaces by harmonic mean curvature flow and improves the known estimates for total mean curvature in hyperbolic 3-space. In particular, we sharpened Ghomi-Spruck’s result. As a corollary, we also get a comparison theorem between total mean curvature in Cartan-Hadamard 3-spaces with that of the geodesic sphere in hyperbolic 3-space with constant curvature.
20. 具临界幂次非线性的Euler-Poisson-Darboux方程解的爆破问题
Blow-up of solutions to the Euler-Poisson-Darbox equation with critical power nonlinearity
作者: Mengting Fan, Ning-An Lai, Hiroyuki Takamura
类别: math.AP
发布日期: 2026-03-20
arXiv ID: 2603.19614v1
链接: http://arxiv.org/abs/2603.19614v1
摘要: In our recent precious work, we established the finite time blow up result and upper bound of lifespan estimate to the singular Cauchy problem of semilinear Euler-Poisson-Darboux equation in R^n with subcritical power type nonlinearity. By introducing an improved test function, we obtain an enhanced lower bound for the functional including the spacetime integral of the nonlinear term with an additional logarithmic growth, which finally yields the blow up result and upper bound of lifespan estimate for the corresponding Cauchy problem with “critical” nonlinear power. And this gives some partial answer to the open problem 1 posed by D’Abbicco (J. Differential Equations 286 (2021), 531-556).
21. 非自治退化抛物型方程的扩散估计
Diffusion bounds for non-autonomous degenerate parabolic equations
作者: Marius Lemm, Israel Michael Sigal, Jingxuan Zhang
类别: math.AP, math.DG
发布日期: 2026-03-20
arXiv ID: 2603.19540v1
链接: http://arxiv.org/abs/2603.19540v1
摘要: We prove the Davies-Gaffney (i.e., integrated Nash-Aronson) type diffusive upper bounds on the propagators of parabolic equations in $L^p$-sense for all $1\le p\le\infty$. Our approach is based on a simple exponential deformation argument that does not require hypoellipticity. It provides a unified approach to diffusive upper bounds that covers a wide class of problems including degenerate, non-autonomous, and non-linear equations.
22. 深度Hilbert–Galerkin方法求解无限维偏微分方程与最优控制
Deep Hilbert–Galerkin Methods for Infinite-Dimensional PDEs and Optimal Control
作者: Samuel N. Cohen, Filippo de Feo, Jackson Hebner, Justin Sirignano
类别: cs.LG, math.AP, math.NA, math.OC, math.PR
发布日期: 2026-03-19
arXiv ID: 2603.19463v1
链接: http://arxiv.org/abs/2603.19463v1
摘要: We develop deep learning-based approximation methods for fully nonlinear second-order PDEs on separable Hilbert spaces, such as HJB equations for infinite-dimensional control, by parameterizing solutions via Hilbert–Galerkin Neural Operators (HGNOs). We prove the first Universal Approximation Theorems (UATs) which are sufficiently powerful to address these problems, based on novel topologies for Hessian terms and corresponding novel continuity assumptions on the fully nonlinear operator. These topologies are non-sequential and non-metrizable, making the problem delicate. In particular, we prove UATs for functions on Hilbert spaces, together with their Fréchet derivatives up to second order, and for unbounded operators applied to the first derivative, ensuring that HGNOs are able to approximate all the PDE terms. For control problems, we further prove UATs for optimal feedback controls in terms of our approximating value function HGNO. We develop numerical training methods, which we call Deep Hilbert–Galerkin and Hilbert Actor-Critic (reinforcement learning) Methods, for these problems by minimizing the $L^2_μ(H)$-norm of the residual of the PDE on the whole Hilbert space, not just a projected PDE to finite dimensions. This is the first paper to propose such an approach. The models considered arise in many applied sciences, such as functional differential equations in physics and Kolmogorov and HJB PDEs related to controlled PDEs, SPDEs, path-dependent systems, partially observed stochastic systems, and mean-field SDEs. We numerically solve examples of Kolmogorov and HJB PDEs related to the optimal control of deterministic and stochastic heat and Burgers’ equations, demonstrating the promise of our deep learning-based approach.
23. 一类超定问题的构造性方法
A Constructive Approach to a Class of Overdetermined Problems
作者: Alessandro Fortunati, Filomena Pacella
类别: math.AP, math.FA
发布日期: 2026-03-19
arXiv ID: 2603.19446v1
链接: http://arxiv.org/abs/2603.19446v1
摘要: In this paper we study an overdetermined problem which is directly related to the well known torsion problem studied by J. Serrin. A perturbed version of the latter is tackled by using asymptotic series as well as tools borrowed from the celebrated Nekhoroshev Theorem. In a similar fashion to this class of results, we establish the existence of infinitely many approximants for the perturbed problem’s solution, whose approximation error is so small which can be regarded as negligible for practical applications. The approach is fully constructive and this feature is demonstrated via an example in the final section.
24. 伪卡莱流正则性结果的一个改进
An improvement of regularity result for pseudo Calabi flow
作者: Jingrui Cheng, Junhao Tian
类别: math.DG, math.AP
发布日期: 2026-03-19
arXiv ID: 2603.19405v1
链接: http://arxiv.org/abs/2603.19405v1
摘要: In this paper, we observe that if the initial data of pseudo Calabi flow has volume form $C^0$ close to a smooth one, then the flow is immediately smooth for $t>0$. As an application, we show that if the initial data has volume form $C^0$ close to that of a cscK metric, then the pseudo Calabi flow exists for $t\in (0,+\infty)$. We also prove similar improvement of regularity and long time existence result for pseudo Calabi flow on a Fano manifold when the volume form is bounded and the class is close to $c_1(M)$.
25. 耦合3D-1D输运问题的数值分析
Numerical Analysis of a Coupled 3D-1D Transport Problem
作者: Alyssa Taylor-LaPole, Uzochi Gideon, Beatrice Riviere, Duygu Vargun
类别: math.NA, math.AP
发布日期: 2026-03-19
arXiv ID: 2603.19395v1
链接: http://arxiv.org/abs/2603.19395v1
摘要: A finite element solution coupled with an interior penalty discontinuous Galerkin solution are defined for the approximation of the coupled 3D-1D solute transport problem. Under sufficient regularity for the weak solutions, optimal error bounds are derived for the 3D concentration and 1D concentration, that are optimal with respect to the time step size and the mesh sizes. Numerical results verify the theoretical results.