Sophie Moufawad, Nabil Nassif, Faouzi Triki
We consider the mathematical model of gas trapping in deep polar ice (firns), which consists of a parabolic partial differential equation, that can degenerate at one boundary extreme. In [1], we considered all the coefficients to be constants, except the diffusion coefficient D(z) that is to be reconstructed. In this paper, we assume both the diffusion coefficient D(z) and the volume fraction f(z) are functions. The difficulty in this problem, both theoretically and computationally, arises from the fact that D(z) and f(z) may be zero at bottom of the firn. To handle such degeneracy, we defined appropriate weighted Sobolev spaces and used Lion's theorem to prove existence and uniqueness of the semi-variational formulation of the Firn PDE. A full discrete system is obtained through a P1 Finite element Galerkin procedure in space and an Euler-Implicit scheme in time. Sufficient conditions for the existence and uniqueness of the solution for the discrete system are obtained.
Gossrin Jean-Marc Bomisso, Ali Ouattara Kouma, Marie Esther Anassé
We study a nonlinear coupled system of partial differential equations arising from thermo--reaction--phase models. The system combines a heat diffusion equation, temperature-dependent chemical reactions of Arrhenius type, and a phase variable, and is formulated as a strongly coupled parabolic problem with homogeneous Neumann boundary conditions. We first establish a maximum principle ensuring the positivity of the temperature on a suitable time interval, as well as the invariance of the physically admissible domain. In particular, we prove that the internal variables remain in the interval [0,1]. We then analyse the asymptotic behaviour of the system in the free regime, that is, in the absence of external forcing. By introducing a relative energy functional and exploiting the structure of the coupling terms, we obtain local asymptotic stability of a homogeneous stationary state. The model belongs to a broader class of coupled diffusion--reaction--phase systems.
Pêdra Andrade, Verena Bögelein, Frank Duzaar, Kristian Moring
We establish local Calderón-Zygmund type estimates for weak solutions to nonlinear parabolic systems with $p$-growth and VMO coefficients. In particular, we prove that if the right-hand side belongs locally to $L^{μs}$, where the exponent $μ$ depends explicitly on $p$, $N$, and a prescribed target exponent $s>p$, then the spatial gradient of the solution enjoys improved integrability $Du \in L^s_{\rm{loc}}$. The result provides a sharp transfer of integrability from the data to the gradient, consistent with the natural parabolic scaling, and recovers the optimal exponents in the linear case $p=2$. The proof combines intrinsic scaling techniques with a Calderón-Zygmund type iteration scheme.
Phan Thành Nam, Rongchan Zhu, Xiangchan Zhu
We derive the classical Gibbs measure on $\mathbb{T}^2$ associated with the fractional Bessel interaction potential $\widehat{v}_β(k)=\langle k\rangle^{-β}$ from a renormalized grand-canonical quantum Bose gas with the same interaction. Our result covers the whole range $\frac32<β\leq2$, where $\widehat{v}_β(k)$ is not summable and the quantum model cannot be written in the usual density-square form, as the associated self-energy diverges. We therefore need to renormalize the zero mode by a centered number-fluctuation term and then develop a detailed analysis for the high-frequency remainders. All this allows us to implement a low-frequency localization and obtain the convergence of the quantum relative free energy to the classical fractional-Bessel free energy, as well as the convergence of the reduced density matrices to the limiting Gibbs measure.
Manzi Huang, Panu Lahti, Jiang Li, Zhuang Wang
For $0<δ,τ<1$ and $1\le s\le \frac{n}{n-δ}$, we prove that for a given $s$-John domain $Ω\subset \mathbb{R}^n$, the following Boxing inequality holds for every Lebesgue measurable set $U\subsetΩ$ with $|U|/|Ω|\leγ<1$: \[ \mathcal{H}^{s(n-δ)}_{\infty}(U\setminus\mathcal{N}_U)\le C(1-δ)\int_Ω\int_{|x-y|<τ\operatorname{dist}(y,\partialΩ)}\frac{|χ_U(x)-χ_U(y)|}{|x-y|^{n+δ}}\,dx\,dy, \] where $\mathcal{H}^{s(n-δ)}_{\infty}(U)$ denotes the $s(n-δ)$-dimensional Hausdorff content of $U$, $\mathcal{N}_U$ is a set of Lebesgue measure zero and the constant $C$ depends only on $n,τ,s,γ$, the John constant and the diameter of $Ω$. Moreover, we establish the functional formulation of the above Boxing inequality and discuss the equivalence between these two formulations. Based on the Boxing inequality, we prove the fractional Poincaré--Wirtinger trace inequality on $s$-John domains, of which the fractional Sobolev--Poincaré inequality and fractional Hardy-type inequality are special cases. Notably, we prove all of the aforementioned inequalities with the Bourgain--Brezis--Mironescu (BBM) factor $1-δ$. Furthermore, with the aid of the Bourgain--Brezis--Mironescu formula, we recover the Poincaré--Wirtinger trace inequality. Finally, by showing that, under the separation property, any domain supporting the Boxing inequality is necessarily a John domain, we conclude that the John domain condition is essentially sharp for the above inequalities. All the above inequalities with the BBM factor are new even for Lipschitz domains.
Renjun Duan, Jinkai Ni
The time-periodic problem on the Boltzmann equation with a given time-periodic external force in the three-dimensional whole space has remained open since it was first studied in [13] for only spatial dimensions not less than five. The goal of this paper is to give an affirmative answer to this problem provided that the external force is sufficiently small in the function space $\mathcal{C}(\mathbb{R};\dot{B}^{-3/2}_{2,\infty}\cap\dot{H}^N)$ with $N\geq 4$. The proof is based on Serrin's method through studying the global-in-time stability of the Cauchy problem with time-periodic external forces. As a direct consequence, the result also yields the existence and stability of stationary solutions to the physically realistic three-dimensional Boltzmann equation when the external force is time-independent.
Fayziev Yusuf, Jumaeva Shakhnoza
In this paper, we investigate a fractional differential equation involving sequential Caputo derivatives, motivated by recent research on fractional models with multiple memory effects. Using techniques inspired by earlier works on sequential fractional operators, we derive the exact analytic solution of the problem in terms of the bivariate Mittag-Leffler function. Additionally, several useful properties of the bivariate Mittag-Leffler function are formulated to support the solution construction. Furthermore, we develop a numerical scheme using a sequential reformulation and the L1-finite element method.
Priyank Oza, Vishvesh Kumar, Durvudkhan Suragan
We study a class of inhomogeneous parabolic equations on the Heisenberg group $\mathbbm{H}^N$ with Hardy-type singular potentials, nonlocal memory terms, and a space-time forcing term: \begin{align} \partial_tu-Δ_{H}u=λ\frac{ψu}{\|\cdot\|^{2}_{H}}+\frac{1}{Γ(γ)}\int_0^t(t-τ)^{γ-1}|u(τ)|^{p}dτ+t^αf \text{ in } \,\mathbbm{H}^N\times (0,T). \end{align} Here, $γ\in [0,1),$ $α\in (-1,\infty),$ $p>1,$ $λ>0,$ and $ψ(\cdot)=|\nabla_{H}\|\cdot\|_{H}|^2,$ where $\nabla_H$ is the horizontal gradient associated to $Δ_H.$ Also, $\|\cdot\|_{H}$ and $Δ_{H}$ denote the Korányi norm and sub-Laplacian associated with the sub-Riemannian geometry of $\mathbbm{H}^N,$ respectively. The combination of a singular Hardy potential and a memory kernel introduces significant analytical challenges. Using a Harnack-type inequality adapted to the Heisenberg group setting, we obtain quantitative positivity estimates that enable a detailed blow-up analysis. We identify parameter regimes depending on $p,γ,α$ leading to finite-time blow-up or instantaneous blow-up, and establish local well-posedness in the absence of the Hardy potential. These results reveal an interplay between the spatial singularity, temporal nonlocality and a time-dependent forcing term. Finally, under a suitable lower bound on the forcing term $f,$ we derive an explicit lifespan estimate for local-in-time solutions.
Benjamin Ingimarson, Igor Kukavica
Given that a solution to the 3D incompressible Euler equations on a bounded domain blows up at a time $T_\ast$ and that $T_\ast$ is the first such time, we provide pointwise-in-time lower bounds on $\|D^kω\|_{L^\infty(Ω)}$ for $k \geq 1$. We also show that the Gronwall-type inequality satisfied by $\|ω(t)\|_{L^\infty}$, in the cases that $Ω= \mathbb{R}^3$, $\mathbb{T}^3$, or a bounded domain, exhibits wildly oscillating solutions.
Tomoaki Yoshizawa
We consider temporal decay estimates for global solutions of the Navier-Stokes equations with the Coriolis force. We show that under several conditions including the smallness of the initial data, the solution decays as fast as the corresponding linearized solutions, and its decay rate is higher than expected from the flow of the heat equation. The estimates are derived for all $L^p$-norms with $p\in[2, \infty].$
Rishad Shahmurov
This manuscript assembles the full axisymmetric-with-swirl large-data program in a single self-contained master file. The paper fixes the lifted five-dimensional formulation, the extraction score, the coherent-versus-noncoherent branch structure, the geometric elimination of fragmented, vertically thinned, displaced-only, and off-axis thin-ring channels, and the local packet-window architecture for the residual axis-centered regime. The final analytic task is reduced to a localized proximal diffuse estimate on a finite packet window. We record the complete theorem stack and the exact local operator identities needed for that final verification, in a form suitable for direct journal submission and final checking.
Maria Deliyianni, Boris Muha, Andrej Novak
We study a diffuse-interface model for thermally driven phase separation in viscous incompressible mixtures. The system couples a convective Cahn-Hilliard equation for the order parameter with a Stokes subsystem for the velocity-pressure field and a heat equation for the temperature. Temperature enters the bulk free energy through a Landau-type coefficient, while the phase field feeds back on the flow through concentration-dependent density and viscosity, yielding a phenomenological temperature-coupled Cahn-Hilliard-Stokes-Heat system. We motivate the chemical potential by a temperature-dependent Landau free energy and derive a priori estimates for the regularized subproblems. On the analytical side, we prove local-in-time existence of weak solutions for a regularized coupled system. On the numerical side, we propose a fully discrete finite element scheme combining a convex-splitting time discretization for the Cahn-Hilliard equation with an implicit treatment of viscous and thermal diffusion terms and a an implicit Stokes solve. Under impermeable velocity boundary conditions, the Cahn-Hilliard substep conserves mass, in the purely diffusive isothermal case, the convex-splitting discretization is unconditionally energy-stable for the Cahn-Hilliard free energy. Numerical experiments in two dimensions illustrate thermally driven spinodal decomposition, wall-induced phase separation near cooled walls, and phase separation in narrow channels under imposed thermal gradients. The simulations show the qualitative influence of key nondimensional parameters (such as the mass and thermal Péclet numbers, the Cahn number, the density and viscosity ratios, and the gravitational parameter $G$) on pattern formation, interface motion, and flow structure, and confirm that the proposed framework is a robust tool for studying thermally driven phase separation in confined geometries.
Sergio Moroni, Ramón G. Plaza
This paper is devoted to the analysis of the following nonlinear wave equation \[ u_{tt} - u_{xx} + (1 + qδ_0(x)) \sin u = 0, \] where $δ_0 = δ_0(x)$ is the Dirac delta function centered at the origin and $q \in \mathbb{R}$ is a constant. Equations of this form arise in the study of propagating solitons in the presence of a localized inhomogeneity. It is proved that the Cauchy problem for this equation is globally well-posed in the energy space $H^1_{\sin} \times L^2$. A complete characterization of stationary waves in the energy space, based on the parameter $q$, is also provided. Finally, a criterion to determine the stability or instability of the stationary waves, which depends upon the sign of the parameter $q$, is established.
Amir Noorizadegan, Sifan Wang
The Gaussian scale parameter \(ε\) is central to the behavior of Gaussian Kolmogorov--Arnold Networks (KANs), yet its role in deep edge-based architectures has not been studied systematically. In this paper, we investigate how \(ε\) affects Gaussian KANs through first-layer feature geometry, conditioning, and approximation behavior. Our central observation is that scale selection is governed primarily by the first layer, since it is the only layer constructed directly on the input domain and any loss of distinguishability introduced there cannot be recovered by later layers. From this viewpoint, we analyze the first-layer feature matrix and identify a practical operating interval, \[ ε\in \left[\frac{1}{G-1},\frac{2}{G-1}\right], \] where \(G\) denotes the number of Gaussian centers. For the standard shared-center Gaussian KAN used in current practice, we interpret this interval not as a universal optimality result, but as a stable and effective design rule, and validate it through brute-force sweeps over \(ε\) across function-approximation problems with different collocation densities, grid resolutions, network architectures, and input dimensions, as well as a physics-informed Helmholtz problem. We further show that this range is useful for fixed-scale selection, variable-scale constructions, constrained training of \(ε\), and efficient scale search using early training MSE. Finally, using a matched Chebyshev reference, we show that a properly scaled Gaussian KAN can already be competitive in accuracy relative to another standard KAN basis. In this way, the paper positions scale selection as a practical design principle for Gaussian KANs rather than as an ad hoc hyperparameter choice.
Anh Tuan Duong, Yao Liu, Nguyên Công Minh, Dao Trong Quyet, Yuhua Sun
In this paper, we study the following quasi-linear elliptic inequality $Δ_m u +u^p |\nabla u|^q \leqslant 0$ on weighted graphs, where $(m,p,q)\in (1,\infty)\times\mathbb{R}\times\mathbb{R}$. According to the ranges of parameters $(m, p, q)$, we establish the non-existence of nontrivial positive solutions under the corresponding sharp volume growth conditions. Our results can be viewed as a discrete generalization of their counterparts on Riemannian manifolds established by [Sun, Yuhua; Xiao, Jie; Xu, Fanheng, Math. Ann. 384 (2022), no. 3-4, 1309--1341.]. However, this generalization is far from trivial, many results exhibit significant differences from the manifold setting, highlighting the distinct behaviors and challenges that arise in the discrete weighted graph framework.
Ahmed Bou-Rabee, Paul Dario
We consider the long-range random conductance model on $\mathbb{Z}^d$ at the critical exponent: the jump rate between sites $x$ and $y$ decays as $\mathbf{a}(x,y) |x-y|^{-(d+2)}$, where $\mathbf{a}(x,y)$ are i.i.d. uniformly elliptic conductances. Below the critical exponent $(d+2)$ the walk converges to a stable process; above it, to Brownian motion with diffusive $\sqrt{t}$ scaling. At criticality the second moment of the jump kernel diverges logarithmically. We establish quantitative homogenization of the associated elliptic equation to the Laplacian at the rate $1/\sqrt{|\ln\varepsilon|}$. As a consequence, we deduce quenched convergence of the random walk to Brownian motion under the anomalous $\sqrt{t \log t}$ scaling. Unlike in standard homogenization, the effective diffusivity is determined by the mean conductance alone, with no corrector contribution at leading order.
Yingying Cai, Xavier Tolsa
We provide quantitative estimates for the dimension drop of harmonic measure. We show that for a domain $Ω= \mathbb{R}^{n+1} \setminus E$ where $E$ is an $s$-Ahlfors regular compact set satisfying a uniform $L^2$-based non-flatness condition $β_2 \ge δ_0$, the dimension of its harmonic measure is strictly less than $s$ for $s \in (n - cδ_0^2, n]$. For planar domains, we establish an analogous quantitative threshold $s_0 = 1 - cδ_0^2$ under Azzam's uniform non-flatness condition $β_\infty + β_{\operatorname{hole}} \ge δ_0$.
Roberto Feola, Riccardo Montalto, Federico Murgante
We study the long-time dynamics of small-amplitude solutions to the three-dimensional gravity-capillary water waves equations for an inviscid and irrotational fluid with periodic boundary conditions. We prove that, for almost all values of the surface tension parameter, solutions with initial size $\varepsilon$ exist and remain small over time intervals of order $\varepsilon^{-2}$. A major difficulty arises from the loss of derivatives caused by the quasilinear nature of the equations combined with severe quadratic and cubic small-divisor interactions in high space dimensions. Classical normal form methods applied to 3D water waves system typically fail to prevent derivative loss due to the accumulation of near-resonances. To overcome this obstruction, we develop a new analytical strategy that combines a sharp frequency partition with a quasi-resonant normal form transformation acting only on selected interaction scales. Our microlocal analysis reveals that the potentially dangerous interactions terms exhibit a block-diagonal structure, which stems from both the geometric properties of the quasi-resonant frequency sets and the Hamiltonian structure of the water waves system. As a consequence, these operators preserve Sobolev norms and do not produce energy growth. This structural insight, together with the quasi-resonant normal-form transformation, allows us to prevent derivative-loss mechanisms while avoiding the accumulation of harmful small denominators.
Clifford Taubes, Yingying Wu
We describe novel local singularity models for $\mathbb Z/2$ harmonic 1-forms, self-dual 2-forms and spinors in dimension 4. These models are homogeneous versions on $\mathbb{R}^4$ whose singular sets are cones on the 1-skeletal of certain regular 4-dimensional polytopes.
Charles L. Epstein, Yoichiro Mori, Han Zhou
We study a bulk-surface coupled Laplace system involving an embedded open boundary. The problem is reformulated as an integro-differential equation using boundary integral representations, for which we establish existence and uniqueness of the solution. A Wiener-Hopf technique is employed to study the solution regularity and derive asymptotic expressions for the edge singularity. Building on these results, we develop a finite element method that incorporates the singularity structure and provide a rigorous error analysis. Numerical experiments confirm the theoretical convergence rates.