Papers

In this paper, motivated by study on universal inequalities for eigenvalues of the Dirichlet Laplacian, we prove some new inequalities for eigenvalues of the Dirichlet Laplacian on the hyperbolic space. In particular, we verify Cheng's conjecture (Adv. Lect. Math. 37, 2017) up to loss of $\epsilon$ for two special kinds of bounded domains in the hyperbolic space.

We study a two-layer one-dimensional energy balance model, which allows for vertical energy exchanges between a surface layer and the atmosphere, as well as meridional energy transport across latitudes via a diffusion law. The evolution equations of the surface temperature and the atmospheric temperature are coupled by exchange of infrared radiation as well as other non-radiative energy exchanges. The energy enters the system as solar radiation, which is partially absorbed and partially reflected by the two layers. The system is then composed of two degenerate parabolic equations coupled by nonlinear terms, the growth of these terms being crucial for the choice of the functional setting. An essential parameter is the absorptivity of the atmosphere, denoted $\varepsilon _a$, whose value depends critically on greenhouse gases. We prove that blow up in finite time occurs if $\varepsilon _a >2$, while global existence of solutions and the existence of a global attractor hold when $\varepsilon _a \in (0,2)$. Proofs are based on comparison principles that derive from the cooperative structure of the problem, and that provide invariant rectangles for smooth initial conditions, and on regularity properties.

We show that two non-isometric, smooth, globally hyperbolic Lorentzian metrics can have the same hyperbolic Dirichlet-to-Neumann map on an infinite cylinder with timelike boundary.

We focus on the classification of positive solutions to $(-\Delta)^s u=\frac{x_n^{\alpha}}{u^\gamma}$ in the half space with $\gamma>0$, subject to the Dirichlet condition. We show that when $-2s<\alpha<(\gamma-1)s$, all positive solutions exhibit one-dimensional symmetry and are monotone increasing in $x_n$. Moreover, we provide a complete classification of all such one-dimensional solutions via their “asymptotic $s$-order slope". When $\alpha$ lies outside this range, we demonstrate the nonexistence of global positive solutions.

In this note we consider a two-dimensional semi-discrete dislocation energy and propose a simple and physically motivated construction for the grain boundary between two crystal grains with a small orientation difference. In the case of a general Bravais lattice, the energy of this construction matches the logarithmic scaling predicted by Read and Shockley.

In this paper, we first derive the compressible Navier-Stokes/Landau-Lifshitz-Gilbert (NS-LLG) model for magnetoelastic materials via the energetic variational approach (EnVarA). It is important to emphasize that the manner in which the evolution of magnetoelastic materials is influenced by the fluid motion–specifically through the deformation gradient–determines the kinematics of the magnetization and consequently leads to distinct governing equations. Subsequently, we establish the local-in-time existence of solutions to the compressible NS-LLG system under finite initial energy. Finally, near the constant equilibrium for magnetoelasticity in the absence of an external magnetic field, we reformulate the evolutionary model, which allows an additional dissipative term to be identified from the elastic stress. Based on this reformulation, we justify the global well-posedness of the evolutionary magnetoelasticity system with zero external magnetic field, provided the initial data are sufficiently small. In particular, when the magnetic field $M$ vanishes, this model reduces to the viscoelastic model. Our results significantly relax the previous initial data requirements, only assume the most basic structural condition $\rho_{0} \operatorname{det} F_{0} = 1$.

In this paper, we first prove that the following generalized conservation principle holds on complete Riemannian manifolds: for every \(0<s<1\) and \(t>0\), \[ T_t^{(s)}\mathbf 1+\int_0^t T_\tau^{(s)}\mathcal R_s\,d\tau=1 \qquad\text{on }M, \] where \(\mathcal R_s\) is the intrinsic killing term measuring the loss of mass of the subordinate semigroup, and the condition \(\mathcal R_s\equiv0\) is equivalent to the stochastic completeness of \(M\). We then provide several new nonlocal characterizations of stochastic completeness. In particular, we show that stochastic completeness is equivalent to genuinely nonlocal conditions, including the zero-mean identity \[ \int_M (-\Delta)^s\varphi\,dV_g=0 \qquad\forall\,\varphi\in C_c^\infty(M), \] as well as the uniqueness of bounded distributional solutions to the associated fractional elliptic and parabolic equations. We also revisit the equivalent \(L^1\)-core characterization for the generator of the heat semigroup, which plays an important role in our approach. In addition, we prove \(L^p\)-contractivity and smoothing properties of the subordinate semigroup, establish both short-time and long-time asymptotic results for the fractional heat kernel, derive the short-time asymptotics of jump probabilities for the associated Markov process, and study the variational characterization and minimality properties of the fractional resolvent. Together, these results provide a unified analytic and probabilistic framework for the fractional Laplacian on complete Riemannian manifolds.

This paper presents a rigorous mathematical analysis of transverse electromagnetic (EM) field concentration between two adjacent obstacles within the framework of the quasi-static approximation. We investigate three degenerate conductivity models recently introduced in [22], two of these incorporating nonlocal boundary conditions to capture fundamental physical phenomena, such as surface nonlocality and thin-layer interactions. Our primary results establish sharp conditions for gradient blowup and derive the corresponding optimal blowup rates. These findings elucidate how nonlocal boundary conditions modify classical gradient estimates. Furthermore, we analyze the influence of wave frequency, demonstrating that it mitigates the severity of field concentration even in the limit of a vanishing gap distance. Consequently, this work extends the classical theory of field enhancement in plasmonic and metamaterial systems to incorporate nonlocal surface effects, yielding precise asymptotic formulas that are essential for the quantitative design of nanophotonic devices.

For a time dependent Schrödinger equation, the scattering map is the map sending the asymptotic profile of solution as $t\to-\infty$ to its asymptotic profile as $t\to+\infty$. In this paper we show that, for certain class of metrics, the scattering maps associated to two Schrödinger operators with two time dependent metrics only differ by a compact operator if and only if these two metrics are related by a pull-back of a diffeomorphism.

This paper investigates the collisionless quantum hydrodynamic, or quantum Euler, system in \(\mathbb{T}^3\) with the linear pressure law \(P(\rho)=\rho\). Since this pressure is associated with the logarithmic internal energy \(f(\rho)=\rho\log\rho\), the model admits a natural logarithmic Schrödinger approximation. By means of a regularized logarithmic Schrödinger equation, we rigorously construct global weak solutions to the quantum isothermal Euler system. The proof relies on the Madelung transform, the polar decomposition of the wave functions, and compactness arguments. In particular, an energy identity is used to recover the strong convergence of the hydrodynamic variables. More broadly, the analysis provides a robust Schrödinger approximation framework for QHD models whose internal energy contains an isothermal component.

We prove the monotone dependence of the heat kernel corresponding to the Vladimirov-Taibleson non-Archimedean fractional differentiation operator $D^\alpha$ on the order $\alpha$.

This work investigates the effectiveness of the Back-and-Forth Nudging (BFN) data assimilation algorithm, specifically its performance when employing the Azouani-Olson-Titi (AOT) continuous data assimilation downscaling nudging algorithm, for recovering initial conditions of dissipative dynamical systems. Contrary to previous reports in the literature, we show that, for several systems of interest, one can construct initial conditions that BFN cannot reliably recover. Our key finding is the construction of infinitely many distinct solutions for certain dissipative systems that share identical spatially sparse observational data. Since these observations are indistinguishable, no data assimilation method relying only on them can differentiate between these solutions or recover the correct initial condition. We illustrate these pathological initial conditions for the Lorenz 1963 model and several 1D partial differential equations: the heat equation, viscous linear transport, and viscous and inviscid Burgers equations. Our analytical results are supported by numerical simulations. To address the numerical ill-posedness of backward-in-time iterations, an essential step of BFN for dissipative models, we introduce a regularized backward step, the Voigt-regularized BFN (VBFN). We investigate its performance for the 2D Navier-Stokes equations and the viscous 1D KdV equation, comparing it with standard BFN and Diffusive BFN (dBFN). While VBFN improves numerical stability and reduces model bias relative to dBFN, it still cannot reconstruct the unobserved fine spatial scales of the reference solution. This reinforces our main conclusion: even with regularization, BFN-type algorithms are limited in recovering the full state, and in particular the initial data, from sparse spatial observations.

We study the periodic homogenization of the viscous Hamilton–Jacobi equation \[ u_t^\varepsilon + \frac{1}{2}|Du^\varepsilon|^2 + V\!\left(\frac{x}{\varepsilon}\right) = \frac{\varepsilon}{2}\Delta u^\varepsilon \qquad \text{in } \mathbb{R}^n \times (0,\infty), \] with initial datum $g \in W^{1,\infty}(\mathbb{R}^n)$, where $V$ is Lipschitz continuous and $\mathbb{Z}^n$-periodic. We prove the sharp global estimate \[ |u^\varepsilon(x,t)-u(x,t)| \leq \varepsilon\!\left(C+\frac{n}{2}\log\!\left(\frac{\max\{t,\varepsilon\}}{\varepsilon}\right)\right) \qquad \text{for all } (x,t)\in \mathbb{R}^n \times [0,\infty), \] where $\varepsilon \in (0,1]$, $u$ solves the limiting (homogenized) equation and $C>0$ is a constant depending only on $\|Dg\|_{L^\infty(\mathbb{R}^n)}$, $\|DV\|_{L^\infty(\mathbb{R}^n)}$, and $n$. We further show that if $g$ is locally semiconcave, then \[|u^\varepsilon(x,t)-u(x,t)| \leq C_{x,t}\varepsilon \qquad \text{for a.e. } (x,t)\in \mathbb{R}^n \times (0,\infty),\] where $C_{x,t}$ depends on $(x,t)$, $\|Dg\|_{L^\infty(\mathbb{R}^n)}$, and $\|DV\|_{L^\infty(\mathbb{R}^n)}$. More precisely, the above improved rate holds at every point $(x,t)$ where $u(\cdot,t)$ is twice differentiable at $x$. In particular, this occurs for a.e. $x\in \mathbb{R}^n$, since $u(\cdot,t)$ is locally semiconcave. We conclude by raising the open problem of whether the same $O(\varepsilon |\log \varepsilon|)$ rate remains valid for general strictly convex Hamiltonians or general periodic diffusions.

We consider the dissipative generalized Surface Quasi-Geostrophic equation with dissipation given by any fractional power of the Laplacian. In the inviscid limit, it is proved that anomalous dissipation of the Hamiltonian is prevented by the strong compactness of the solutions in the lowest norm that makes the nonlinearity well-defined. In fact, only the dynamics at certain frequencies matters. The argument is quite robust as it applies regardless of the criticality regime and of the presence of a, possibly noncompact, external forcing. This reveals a more general mechanism behind some recent results obtained for the Navier-Stokes and the critical dissipative Surface Quasi-Geostrophic equations. Because of nonuniqueness issues, in our broader context it is important to work with Leray solutions enjoying suitable higher-order bounds. The existence of such solutions is shown and it might be of independent interest. Finally, we prove that the strong compactness is guaranteed for any initial datum with critical integrability, from which global existence of conservative, although Onsager's supercritical, weak solutions of the inviscid problem is deduced. This offers the largest class of initial data for which global existence is known so far, matching with the one considered by Delort at the endpoint.

We study null controllability for the parabolic equation on $\mathbb{S}^{2}$ endowed with its canonical almost-Riemannian structure. For a spherical crown $\omega=\{\alpha<x_3<\beta\}$, where $0\le \alpha<\beta\le1$, we prove the sharp minimal time formula $T_{\min}(\omega)=\ln(1/\sqrt{1-\alpha^{2}})$ for null controllability in $\omega$. We also prove that, whenever the control region contains the equator, null controllability holds in every positive time. The proof combines two complementary tools. First, after Fourier decomposition with respect to the periodic variable, we establish observability estimates for a family of one-dimensional singular parabolic equations, with constants uniform with respect to the Fourier mode; the singularities at the poles are handled via a Hardy-Poincaré inequality. Second, for crowns away from the equator, we use the moment method to construct controls on the pole-touching crown $\alpha<x_3< 1$ from sharp weighted lower bounds on associated Legendre functions, and then pass to a general crown $\alpha<x_3<\beta$ by a cut-off argument on the full domain combined with the arbitrary-time controllability of crowns containing the equator. The result closes the large-time gap left in earlier work and gives the exact null-controllability threshold for the canonical almost-Riemannian heat equation on $\mathbb S^2$.

This paper develops a conceptual extension of the Kinetic Theory of Active Particles, building upon the framework introduced in [2]. Living systems cannot be adequately described within classical single-scale paradigms, even when refined. To overcome this limitation, we introduce a Multiscale Kinetic Theory of Active Particles (MS-KTAP), in which a sub-microscopic scale of interacting entities is incorporated into the description of collective dynamics. In this framework, the activity variable is interpreted as an emergent quantity arising from lower-scale regulatory mechanisms and influenced by interactions across higher scales. The proposed framework captures key features of living systems, including heterogeneity, adaptive decision-making, nonlinear and non-conservative interactions, spatial dynamics, and cross-scale feedback, within a unified mathematical structure. Competition and cooperation are thus described across multiple levels of organization. The first part of the paper derives the mathematical framework, while the second presents how specific models can be obtained. The paper concludes with perspectives on further developments, including possible integrations with scientific machine learning.

Graphs of solutions to the minimal surface equation over simply connected domains with boundary values 0 can have at most exponential growth.

We define a filter of time-frequency anisotropic global singularities of phase space for tempered distributions. The filter contains information from the corresponding anisotropic Gabor wave front set and admits propagation results for the Cauchy problem for certain linear evolution equations of Schrödinger type that generalize the harmonic oscillator.

We introduce and analyse a mathematical model describing the dynamics of particles generated by charge-exchange interactions. The model extends the well-established exchange-driven growth model, previously studied in several works, by allowing for particle densities defined on the entire integer lattice. Despite the many similarities between the two models, substantial differences arise both in their qualitative behaviour and in their mathematical analysis. Under suitable assumptions on the kernel in the collision operator, we establish global well-posedness in the class of nonnegative densities with finite first moment. Moreover, under a detailed balance condition, we investigate the structure of equilibria and analyse their stability by means of entropy methods.

We consider a reaction-diffusion equation in a one-dimensional space, where the diffusion coefficient changes sign from positive to negative and back to positive. The reaction term is bistable, with its interior zero located in the region where the diffusivity is negative. The model does not admit continuous wavefronts, i.e., continuous traveling waves that connect the steady states $0$ and $1$. We prove the existence of a family of shock wavefronts, that is, wavefronts with profiles exhibiting a jump discontinuity. We investigate the properties of these profiles and their propagation speeds. Finally, we apply the results to a recently proposed model describing the movement of a population composed of both isolated and grouped individuals.

In this paper, we theoretically and numerically study the minimizers for the Cahn-Hilliard energy with the Flory-Huggins potential under the strong anchoring condition, i.e., the Dirichlet boundary condition. We reveal bifurcation phenomena mediated by the boundary condition, the transition layer thickness, and the temperature of the system. Numerical simulations are also presented to approximate the minimizers of this energy by solving a gradient-flow equation, namely the Allen-Cahn equation constrained with strong anchoring conditions and random initial data. The effects of varying the transition layer thickness and temperature are presented to confirm the theoretical analysis.

We study solutions to the self-dual Abelian Yang–Mills–Higgs (YMH) equations in the singular limit $\e \to 0 $, where the associated self-dual Ginzburg–Landau type energy \begin{align*} E_\e\begin{pmatrix}u\\ A\end{pmatrix} = \int_M \left( |\nabla^A u|^2 + \e^2 |F_A|^2 + \frac{(1 - |u|^2)^2}{4\e^2} \right) \mathrm{dvol}_g \end{align*} exhibits concentration along codimension-two sets. Using techniques inspired by Allard's regularity theory, we construct approximate solutions concentrating near a minimal submanifold and analyse their perturbations via a linearised operator projected orthogonally to gauge and translational zero modes. By working in Fermi coordinates and enforcing Coulomb gauge conditions, we derive uniform Lipschitz and curvature estimates for the solutions and obtain Hölder regularity for the scalar and connection components. These results establish a geometric framework for understanding vortex sheet formation and provide a regularity theory for the limiting defect set in the context of Abelian gauge theories.

We consider the cubic nonlinear Schrödinger system with Rashba type Spin-Orbit Coupling (SOC) on $\mathbb{R}^2$. The system describes SO-coupled spinor BEC in physics. In the literature of physics, the small semi-vortex solutions, small ground state, and the so-called mixed mode, which are mixture of semi-vortex solutions, are investigated. The semi-vortex solutions cause from the resonance on the essential spectrum of the linear operator. In the present paper, we give mathematical proofs of the existence of the semi-vortex and the ground state by finding minimizers of the energy under small mass constraint based on concentration compactness argument. Moreover, we also discuss the mixed modes in the case where all the coefficients of the nonlinear terms are equal.

The nonlocal Cahn-Hilliard equation provides a natural extension of the classical model for phase separation by incorporating long-range interactions through a singular convolution kernel. While this formulation admits a rich existence and regularity theory, its numerical approximation remains challenging: discretisation of the nonlocal term leads to dense operators, and the singularity of the kernel requires special treatment in collocation-based schemes. In this work, we develop an efficient and error-controlled numerical framework for the nonlocal Cahn-Hilliard system with constant mobility, logarithmic potential, Newtonian interaction kernel, and no-flux boundary conditions. Our approach is based on a pseudospectral multishape method that accurately approximates the action of singular convolution operators. We present high-resolution numerical solutions for this nonlocal system of equations that can be achieved with limited computational resources.

We study the focusing semilinear heat equation with an additional defocusing Hénon-type nonlinearity, the coupling of which is measured by a constant $c >0$. For $c \in (0,c^*)$, the model admits a closed-form self-similar blowup solution in every space dimension $d \geq 1$. Restricting ourselves to the three-dimensional case, we study the stability of this solution under small non-radial perturbations. By working in intersection Sobolev spaces with additional angular regularity, we prove finite co-dimension stability for all admissible values of $c$. Furthermore, we analyze the spectrum of the underlying linearized operator and we prove stable blowup for the cubic-quintic case and $c$ sufficiently close to $c^*$. Finally, we discuss the situation for small values of $c$ and use a modified version of the classical GGMT criterion to give an upper bound on the number of unstable eigenvalues.

In this paper, we consider 2D incompressible Euler equations in an unbounded domain with a free surface and a fixed bottom at finite depth. The fluid motion is under the influence of gravity and surface tension. We construct initial data with a flat free surface and small velocity, such that the $L^\infty$ norm of the vorticity gradient has at least a double-exponential growth rate within the lifespan of the corresponding solution. This work generalizes the result of Zlatos to the free-surface setting and Hu–Luo–Yao to the case of an unbounded domain.

We prove that for solutions of the Euler equation on the sphere, the vorticity gradient can grow at most double-exponentially in time, and we show that this upper bound is sharp by constructing explicit solutions with odd symmetry that exhibit double-exponential growth in the hemisphere. We also extend the results to the case of a rotating sphere. This seems to be the first result on the growth of the vorticity gradient for ideal fluids on a compact manifold with non-trivial geometry.

We study the existence of negative eigenvalues for two-dimensional Schrödinger operators with real-valued potentials in the weak coupling regime. In his pioneering paper [Simon 1976] from half a century ago, Simon was the first to describe the unique negative eigenvalue emerging from the threshold of the essential spectrum of one- and two-dimensional Schrödinger operators. The aim of this paper is to extend Simon's results in two dimensions to a broader class of potentials, allowing for both stronger singularities and slower decay at infinity, at the cost of losing uniqueness of weakly coupled eigenvalues.

This paper aims to extend the concept of stochastic $\Sigma$-convergence to the framework of Orlicz-Sobolev spaces in order to deals with coupled stochastic and deterministic homogenization problems in this type of spaces. Thus, this concept is a combination of both well-known $\Sigma$-convergence [Acta Math. Sinica, English Series 30(9) 1621-1654] and stochastic two-scale convergence in the mean schemes [Asympt. Anal. (2025) 142, 291-320]. An application to the stochastic-deterministic homogenization (in the context of ergodic $H$-supralgebra) of a class of highly oscillatory minimizations problems involving integral functionals with convex and nonstandard growth integrands is also given, and some concrete homogenization problems following varied structure hypothesis are deduce from this latter.

We study isoperimetric problems modeled on the liquid drop model, with nonlocal interactions under a volume constraint. While balls are natural critical points, we show that, for an unbounded sequence of radii, non-spherical solutions bifurcate from the family of balls. These new solutions lie arbitrarily close to balls and can have arbitrarily large volume. Conversely, at radii outside this sequence, no bifurcation occurs, and nearby solutions are trivial, arising only from rigid motions.