We study a class of nonlocal pseudo-parabolic equations with Sonine kernels, where the nonlinearity involves time-varying delays. Our main goal is to address some qualitative questions including the solvability, stability and the Ho & uml;lder regularity for mild solutions. In addition, we show a result on the convergence of solutions to ones of corresponding subdiffusion equations. The main ingredients for our analysis are regularity estimates of resolvents, a nonlo cal Halanay inequality, the fixed point argument and the embedding of fractional Sobolev spaces.
The low-magnetic-Reynolds-number approximation has been widely used in mathematical modeling of conducting fluids under the influence of external magnetic fields. The aim of this paper is to justify this approximation using a rigorous analysis in the appropriate functional setting. We derive absolute and relative error estimates for both three-dimensional magnetohydrodynamic duct flow and two-dimensional shallow-channel flow. The derived relative error estimate vanishes as the magnetic Reynolds number tends to zero, thereby validating the approximation and also providing its order of accuracy and the range of applicability.
We present an asymptotic method for solving the linearized Euler equations for homentropic (entropy-constant) fluids within the geometrical acoustics limit, without resorting to any decomposition of the perturbed field a priori. The asymptotic solution obtained therefrom provides a preliminary answer to the long-standing question of whether disturbances that propagate at approximately the speed of sound relative to the unperturbed flow are purely irrotational. The answer is no. Likewise, convected disturbances with nearly solenoidal velocity fields have been shown to be capable of transporting pressure perturbations. Accordingly, the results suggest that performing the flow decomposition under an a priori uniqueness condition before the complete solution is known may yield misleading consequences.
This paper investigates a general linear parabolic partial differential equation driven by a multidimensional bounded stochastic process. Employing spectral analysis of the associated uniformly elliptic operator, specifically extensions of the Krein-Rutman theorem and maximum principles, we establish the existence and uniqueness (up to scaling) of a strictly positive eigenfunction. We prove that the corresponding simple positive eigenvalue strictly dominates the rest of the spectrum, implying that every strictly positive solution to the parabolic equation is uniquely of separated form. This result generalizes prior univariate Sturm-Liouville approaches to the correlated multivariate setting. On the applications side, we show that this uniqueness theorem yields a rigorous multi-factor version of the Ross recovery theory, uniquely identifying the market price of risk and the Girsanov kernel in bounded diffusion models. Furthermore, we propose a framework for valuing European call options on private equity where the volatility vector is uniquely determined. Finally, our analysis connects to large deviation principles for stochastic PDEs developed by Dupuis et al., offering a robustness perspective on the spectral characterization of long-term risk prices in bounded domains.
We introduce a discrete-space analogue of the No-U-Turn sampler on the symmetric group $S_n$, yielding a locally adaptive and reversible Markov chain Monte Carlo method for $\mathrm{Mallows}(d,σ_0)$. Here $d:S_n\times S_n\to[0,\infty)$ is any fixed distance on $S_n$, $σ_0\in S_n$ is a fixed reference permutation, and the target distribution on $S_n$ has mass function $π(σ)\propto e^{-βd(σ,σ_0)}$ where $β>0$ is the inverse temperature. The construction replaces Hamiltonian trajectories with measure-preserving group-orbit exploration. A randomized dyadic expansion is used to explore a one-dimensional orbit until a probabilistic \emph{no-underrun} criterion is met, after which the next state is sampled from the explored orbit with probability proportional to the target weights. On the theory side, embedding this transition within the Gibbs self-tuning (GIST) framework provides a concise proof of reversibility. Moreover, we construct a \emph{shift coupling} for orbit segments and prove an explicit edge-wise contraction in the Cayley distance under a mild Lipschitz condition on the energy $E(σ)=d(σ,σ_0)$. A path-coupling argument then yields an $O(n^2\log n)$ total-variation mixing-time bound.
In this paper, we investigate the existence, uniqueness, and exponential stability of almost periodic mild solutions for the chemotaxis-Navier-Stokes system with a matrix-valued sensitivity on a real hyperbolic space Hd(R) (d ?. 2) and on the wholeline time-axis RE. First, we prove the existence and uniqueness of almost periodic mild solutions for the corresponding linear systems by using the dispersive estimates of the scalar and vectorial heat semigroups and the boundedness of the matrix-valued sensitivity. Then, we use the results of linear systems and fixed-point arguments, and we establish the well-posedness of almost periodic mild solutions for the chemotaxis-Navier-Stokes systems. Finally, we employ the Gronwall inequality to prove the exponential stability of such solutions. Our obtained results are also valid for such systems on a bounded domain with smooth boundary in the Euclidean space Rd (d ?. 2).
The transport of subsurface water within a homogeneous porous unsaturated vadose zone is considered to be governed by Richards' conservation of mass equation with vegetation growing on its upper horizontal arid boundary surface and its horizontal lower surface being impermeable. It is assumed that Richards' equation and its lower boundary satisfy a linear and semi-infinite limit, respectively, while an existing vegetation-groundwater system is adopted on its upper one, groundwater being identified with subsurface water content on the upper boundary. A rhombic planform nonlinear stability analysis is applied to this system to investigate the formation of stationary twodimensional vegetative patterns consisting of periodic arrays of spots (leopard bush), gaps (pearled bush), or bicontinuous mazes (labyrinthine tiger bush) observed to occur on arid flat environments. The results of that analysis can be represented by plots in a root suction coefficient versus rainfall rate dimensionless parameter space. Then, a morphological sequence of bare ground and stable vegetative states is produced upon traversing an experimentally-determined root suction characteristic curve based on soil type as a function of rainfall rate through that parameter space. Finally, this predicted sequence along a rainfall gradient is used to motivate an aridity classification scheme and compared with both theoretical results from similar systems and observational evi-dence relevant to the occurrence of leopard, pearled, or labyrinthine tiger bush vegetative patterns.
Two-dimensional potential flow along a straight-walled duct with a sharp right-angled bend is considered. This problem is solved using two distinct methods, conformal mapping and matched modal expansions. The main purpose of the paper is to compare the two methods: each gives insight into the other. Applications to genuine waveguide problems are expected, when conformal mapping is no longer available.
The slow drift along a manifold of periodic orbits is a key mathematical structure underlying bursting dynamics in many scientific applications. While classical averaging theory, as formalised by the Pontryagin-Rodygin theorem, provides a leading-order approximation for this slow drift, the connection to the underlying geometry described by geometric singular perturbation theory (GSPT)-also known as Fenichel theory-is often not explicit, particularly at higher orders. This paper makes that connection rigorous and constructive using the parametrisation method. We provide a detailed, self-contained exposition of this functional analytic technique, showing how it synthesizes the geometric insight of invariant manifold theory with a systematic, perturbative algorithm. By treating the manifold's embedding and the reduced flow as coupled unknowns, the method constructs an averaged system that is guaranteed to govern the dynamics on the persisting normally hyperbolic manifold to any perturbative order. We translate the abstract theory into a concrete computational procedure using Floquet theory, spectral analysis, and the Fredholm alternative, yielding a practical guide for computing highaccuracy, higher-order averaged models, and we demonstrate its implementation, both analytically and numerically, through specific examples.
We consider the Markovian supermarket model with growing choices, where jobs arrive at rate n lambda n and each of n parallel servers processes jobs in its queue at rate 1. Each incoming job joins the shortest among dn E {1, ... , n} randomly selected queues. Under the assumption dn-+ oo and lambda n-+ lambda E (0, oo) as n-+ oo, a large deviation principle (LDP) for the occupancy process is established in a suitable infinitedimensional path space, and it is shown that the rate function is invariant with respect to the manner in which dn-+ oo. The LDP gives information on the rate of decay of probabilities of various types of rare events associated with the system. We illustrate this by establishing explicit exponential decay rates for probabilities of large total number of jobs in the system. As a corollary, we also show that probabilities of certain rare events can indeed depend on the rate of dn-+ oo.
We introduce the proximal optimal transport divergence, a novel discrepancy measure that interpolates between information divergences and optimal transport distances via an infimal convolution formulation. This divergence provides a principled foundation for optimal transport proximals and proximal optimization methods frequently used in generative modeling. We explore its mathematical properties, including smoothness, boundedness, and computational tractability, and establish connections to primal-dual formulations and adversarial learning. The proximal operator associated with the proximal optimal transport divergence can be interpreted as a transport map that pushes a reference distribution toward the optimal generative distribution, which approximates the target distribution that is only accessible through data samples. Building on the Benamou-Brenier dynamic formulation of classical optimal transport, we also establish a dynamic formulation for proximal OT divergences. The resulting dynamic formulation is a first order mean-field game whose optimality conditions are governed by a pair of nonlinear partial differential equations: a backward Hamilton-Jacobi equation and a forward continuity equation. Our framework generalizes existing approaches while offering new insights and computational tools for generative modeling, distributionally robust optimization, and gradient-based learning in probability spaces.
In this paper, we propose two approaches to derive the discrete Poincaré inequality for the B-schemes, a family of finite volume discretization schemes, for the one-dimensional Fokker-Planck equation in full space. We study the properties of the spatially discretized Fokker-Planck equation in the viewpoint of a continuous-time Markov chain. The first approach is based on Gamma-calculus, through which we show that the Bakry-Émery criterion still holds in the discrete setting. The second approach employs the Lyapunov function method, allowing us to extend a local discrete Poincaré inequality to the full space. The assumptions required for both approaches are roughly comparable with some minor differences. These methods have the potential to be extended to higher dimensions. As a result, we obtain exponential convergence to equilibrium for the discrete schemes by applying the discrete Poincaré inequality.
In this paper a reduced one-dimensional moving boundary model is studied that describes the evolution of a biofilm driven by the presence of a reaction limiting substrate. Global well-posedness is established for the resulting parabolic free boundary value problem in strong form in Sobolev spaces and for a quasi-stationary approximation in spaces of classical regularity. The general existence results are complemented by results about the qualitative properties of solutions including the existence, in general, and, additionally, the uniqueness and stability of non-trivial equilibria, in a special case.
In the present work, we revisit the topic of translational eigenmodes in discrete models. We focus on the prototypical example of the discrete nonlinear Schrödinger equation, although the methodology presented is quite general. We tackle the relevant discrete system based on exponential asymptotics and start by deducing the well-known (and fairly generic) feature of the existence of two types of fixed points, namely site-centered and inter-site-centered. Then, turning to the stability problem, we not only retrieve the exponential scaling (as e − π 2 / ( 2 ε ) e^{-\pi ^2/(2 \varepsilon )} , where ε \varepsilon denotes the spacing between nodes) and its corresponding prefactor power-law (as ε − 5 / 2 \varepsilon ^{-5/2} ), both of which had been previously obtained, but we also obtain a highly accurate leading-order prefactor and, importantly, the next-order correction, for the first time, to the best of our knowledge. This methodology paves the way for such an analysis in a wide range of lattice nonlinear dynamical equation models.
While ridges in the scalogram, determined by the squared modulus of analytic wavelet transform (AWT), are a widely accepted concept and utilized in nonstationary time series analysis, their behavior in noisy environments remains underexplored. Our objective is to provide a theoretical foundation for scalogram ridges by defining ridges as a potentially set-valued random process connecting local maxima of the scalogram along the scale axis and analyzing their properties when the signal fulfills the adaptive harmonic model and is contaminated by stationary Gaussian noise. In addition to establishing several key properties of the AWT for random processes, we investigate the probabilistic characteristics of the resulting random ridge points in the scalogram. Specifically, we establish the uniqueness property of the ridge point at individual time instances and prove the upper hemicontinuity of the ridge random process. Furthermore, we derive bounds on the probability that the deviation between the ridges of noisy and clean signals exceeds a specified threshold, and these bounds depend on the signal-to-noise ratio. To achieve these ridge deviation results, we derive maximal inequalities for the complex modulus of nonstationary Gaussian processes, leveraging classical tools such as the Borell-TIS inequality and Dudley's theorem, which might be of independent interest.
This paper asks if the following iterative procedure approximately orthogonalizes a set of n n linearly independent unit vectors while preserving their span: in each iteration, access a random pair of vectors and replace one with the component perpendicular to the other, renormalized to be a unit vector. We provide a positive answer: any given set of starting vectors converges almost surely to an orthonormal basis of their span. We specifically argue that the n n -volume of the parallelepiped generated by the vectors approaches 1 (i.e. the parallelepiped approaches a hypercube). If A A is the matrix formed by taking these vectors as columns, this volume is simply det ( | A | ) \det (\lvert A\rvert ) where | A | = ( A ∗ A ) 1 / 2 \lvert A\rvert =(A^*A)^{1/2} . We show that O ( n 2 log ( 1 / ( det ( | A | ) ε ) ) ) O(n^2\log (1/\left (\det (\lvert A\rvert )\varepsilon \right ))) iterations suffice to bring det ( | A | ) {\det (\lvert A\rvert )} above 1 − ε 1-\varepsilon with constant probability.
This paper is concerned with the existence of global-in-time weak solutions to the multicomponent reactive flows inside a moving domain whose shape in time is prescribed. The flow is governed by the 3D compressible Navier-Stokes-Fourier system coupled with the equations of species mass fractions. The fluid velocity is supposed to fulfill the complete slip boundary condition, whereas the heat flux and species diffusion fluxes satisfy the conservative boundary conditions. The existence of weak solutions is obtained by means of suitable approximation techniques. To this end, we need to rigorously analyze the penalization of the boundary behavior, viscosity and the pressure in the weak formulation.
We consider the question of global existence of smooth solutions to a multi-species aggregation-diffusion equation for a class of singular interaction kernels. We establish a smallness condition on the initial data which yields global existence of smooth solutions. We also give conditions on the species interaction which ensure that pointwise inequalities comparing species densities are preserved by the evolution.
