In this work a study of conformal Killing vector fields has been carried out with a new perspective of statistical connection. The study is organized into three main sections. In one section we listed the generators of conformal Killing algebra of locally rotationally symmetric (LRS) Bianchi type-I spacetimes which were obtained in a previous work [Mahmood et al., Mod. Phys. Lett. A 33, 1850063 (2018)]. In the second section we constructed statistical structure over the manifolds of the obtained LRS Bianchi type-I metrics and investigated the conditions under which the obtained vector fields for these spacetime metrics remain conformal Killing vector fields. We also discussed the vanishing Lie derivative for proper conformal Killing vectors under particular affine connections. Statistical structures of LRS Bianchi type-I spacetimes have been achieved by applying the Codazzi and torsion free conditions. In the last, we deduced some conditions under which an initial connection having torsion will produce a vanishing Lie derivative for proper conformal Killing vector fields.
A classification of soliton solutions is established for the modified Boussinesq (mBoussinesq) equation, which arises from the 3-reduction of the modified Kadomtsev-Petviashvili hierarchy. We prove that when all κ-parameters are nonzero, the soliton solutions can be expressed in terms of τ-functions associated with a single Grassmannian. When a zero κ-parameter is present, certain table solitons degenerate into kink solitons, and the solutions are described by τ-functions associated with two Grassmannians of different dimensions. On this basis, we present specific examples and apply combinatorial tools such as chord diagrams and pipe dreams to systematically analyze the asymptotic behavior and internal structures of the soliton solutions.
In this paper, we firstly obtain a Birkhoff normal form theorem for a non-autonomous Hamiltonian system. Based on this result, we investigate the subexponential long time stability of equilibrium for quasi-periodic forced nonlinear Schrödinger equation in a Gevrey space.
This paper is concerned with the following L2-norm constrained Kirchhoff equation −1+b∫R3|∇u|2dxΔu+V(x)u+λu=|u|83u,x∈R3,u(x)∈H1(R3),u(x)≥0,∫R3|u|2dx=m2 in the mass-critical setting. The parameters b > 0 and m > 0 are given in advance, and the unknown parameter λ appears as a Lagrange multiplier. Here, the potential V(x) is positive, vanishing at infinity, and may have singular points. Under some explicit smallness assumptions on V(x), we show the existence of a bound state solution (v,λ)∈H1(R3)×R+ and the nonexistence of ground state solutions. To overcome the lack of compactness resulting from the L2-norm constraint, we introduce some new and subtle energy estimates that fully utilize the features of the mass critical exponent.
We investigate equilibration-on-average and effective equilibration for quantum systems evolving on infinite-dimensional Hilbert spaces under Hamiltonians with purely continuous spectrum. In this regime, the standard infinite-time dephasing picture underlying finite-dimensional equilibration theory ceases to be informative: the corresponding ergodic average becomes trivial in the relevant sense and therefore cannot serve as a physically meaningful equilibrium state. This makes equilibration intrinsically a finite-time phenomenon. Motivated by this obstruction, we introduce a framework based on finite-time ergodic averages and spectral coarse graining at finite resolution. Within this setting, we derive explicit upper bounds for equilibration-on-average of bounded observables under purely continuous-spectrum dynamics. Unlike their finite-dimensional counterparts, these bounds depend not only on the Hamiltonian and the initial state, but also on the observation time horizon and on the resolution scale of the spectral partition. We also obtain corresponding bounds for effective equilibration with respect to finite families of physically realistic POVM (Positive Operator-Valued Measure) measurements. Our results provide a quantitative formulation of equilibration for continuous-spectrum quantum systems in a regime where the usual infinite-time dephasing paradigm breaks down. Within the coarse-grained framework developed here, the resulting bounds depend explicitly on both the finite observation time and the chosen spectral resolution. An explicit example based on a free-particle Hamiltonian illustrates the mechanism and interpretation of the general bounds.
We study a class of discontinuous-flux Riemann problems with mass concentration in gas dynamics. Specifically, we consider the singular Riemann problem for the one-dimensional unsteady isentropic compressible Euler equations, where the flux has a jump across a discontinuous curve x = x(t) that separates two different polytropic gases. The main result establishes the existence of Radon measure-valued solutions to this problem. We also show that for certain initial data, the problem admits multiple solutions consisting of piecewise constant states connected by delta shocks satisfying the over-compressing condition. The essential difficulty stems from the flux discontinuity along the discontinuity curve combined with the distinct polytropic gas properties on either side, which leads to more complex nonlinearity and makes the construction of solutions challenging. This work extends the existing theory of Radon measure-valued solutions for compressible Euler systems with a discontinuous flux to a general case. We conclude that for conservation laws featuring discontinuous flux functions with mass concentration, genuine nonlinearity does not guarantee the uniqueness of piecewise constant solutions containing delta shocks under the over-compressing condition.
In this paper, we consider the parabolic–elliptic–elliptic Keller–Segel–Stokes system with nonlinear diffusion and indirect signal production nt+u⋅∇n=Δnm−∇⋅(nS(n)∇v);u⋅∇v=Δv−v+w;u⋅∇w=Δw−w+n;ut=Δu−∇P+n∇ϕ;∇⋅u=0 in a bounded domain Ω⊂R3 with smooth boundary, where ϕ ∈ W2,∞(Ω) and m > 0, and the chemotactic sensitivity function S∈C2(0,∞) satisfies 0 ≤ S(ξ) ≤ CS(ξ + 1)−α for all ξ ≥ 0 with some CS > 0 and α∈R. It is proved that under the structural assumption m+α>1 and proper regularity assumptions on the initial data, the associated initial-boundary value problem possesses at least one global bounded weak solution. Our result not only accommodates both porous-medium-type diffusion and fast (singular) diffusion, but also covers saturated sensitivity (α > 0) and stronger chemotactic sensitivity (α ≤ 0). To the best of our knowledge, this is the first result on the existence of global bounded weak solutions in a parabolic–elliptic–elliptic Keller–Segel–Stokes system of type (⋆) and covers the result of linear diffusion under the condition that α > 0.
This paper investigates the representation theory of the affine Lie superalgebra sl(2,1)̂ by introducing and studying a new class of modules termed quasi-Whittaker modules. While Whittaker modules have been extensively studied for various Lie algebras and quantum groups, their exploration for affine Lie superalgebras remains largely undeveloped. Drawing inspiration from methods used for the quasihighest weight sl(2,1)̂-modules and the structure of irreducible Whittaker modules on the affine Lie algebra sl2̂ and Heisenberg Lie algebra, we construct these quasi-Whittaker modules Ŵψ,k,l for sl(2,1)̂ by inducing from irreducible Whittaker modules of its even subalgebra. Our main result establishes the irreducibility of this quasi-Whittaker module Ŵψ,k,l. This work represents an initial effort to investigate Whittaker-type representations for affine Lie superalgebras.
We study a new approach to generally covariant quantum mechanics applied in the case of a Friedmann–Lemaïtre–Robertson–Walker cosmological background. For positive spatial curvature we find a discrete series of solutions of the Klein–Gordon equation that can reasonably be called gravitationally bound “cosmological atom” states. For all cases of curvature, these modes, as well as more conventional atomic spatial modes bound by an external potential, extend to solutions of the Klein–Gordon equations viewed as stationary modes of Klein–Gordon quantum mechanics where wavefunctions are over spacetime and evolution is with respect to an external “geodesic time” parameter s. For general nonstationary states with fixed spatial eigenvector, the theory reduces to a novel 1-dimensional quantum system on the time t axis with potential 1/a(t)2, where a(t) is the Friedmann expansion factor. Its behaviour, and hence the evolution of spatial states, changes critically when the Hubble constant exceeds 2/3 of the particle mass, as typically occurs during inflation. We also find washout of the evolution of spatial observables at late times and a backward-traveling reflected mode generated when the value of H transitions to a larger value.
This paper is concerned with the Cauchy problem for the three-dimensional isentropic compressible Navier–Stokes equations, considering both far-field vacuum and non-vacuum cases. Our main novelties are twofold: First, we establish a new and refined scaling-invariant initial condition, which is simpler and less restrictive than those used in recent works {e.g., Wen [Adv. Math. 482, 110628 (2025)]}. Specifically, global existence and uniqueness of strong solutions are proven under the smallness of the quantity: ρ̄3E0∇u0L22+ρ̄γE01+ρ̄1+γ/3E02/3. This framework dispenses with the previously required ‖ρ0γ‖L22 term, thereby relaxing the initial data assumptions. A key challenge addressed is controlling the supercritical nonlinear convective terms in the momentum equation without relying on the L2 norm of the pressure. This is achieved via a combination of energy estimates, interpolation trick, and a refined analysis of the density’s upper bound, showing that the L2 pressure estimate is unnecessary. Second, the result is extended to the physically relevant case of non-zero far-field density (ρ∞ > 0), a scenario with additional complexity due to the non-trivial equilibrium state. This generalization is achieved by adapting our a priori estimates to a modified energy functional, further demonstrating the robustness and generality of our new framework.
In this paper, we study the global well-posedness and decaying rate of the three dimensional magnetohydrodynamic equations, either with horizontal viscosity and vertical magnetic diffusion or with vertical viscosity and horizontal magnetic diffusion around any constant magnetic field background ω=(ω1,ω2,ω3)∈R3 with ω3 ≠ 0 and (ω1, ω2) satisfying the Diophantine condition.
This paper presents the Riemann problem and wave interactions for the one-dimensional generalized Chaplygin gas equations with a nonlinear time-dependent source term. We rigorously construct the Riemann solutions for this nonhomogeneous hyperbolic system and show that, for a class of initial data, the solutions develop singular measures in the form of δ-shock waves. For such solutions, we derive the generalized Rankine–Hugoniot conditions and obtain explicit formulas for the propagation speed, trajectory, and strength of the δ-shock through a suitable δ-entropy framework. We further provide a detailed analysis of wave interactions involving δ-shocks by introducing three-state piecewise constant perturbations, which enables the construction of global-in-time solutions. By performing a vanishing perturbation limit as ɛ → 0, we establish the stability of the Riemann solutions and rigorously characterize their asymptotic behavior. These results extend the theory of δ-shocks to nonhomogeneous systems with time-dependent source terms and offer new insights into the interplay between source effects and singular wave dynamics.
We consider the quantum Gibbs state of an interacting Bose gas on the 2D torus T-2. We set temperature, chemical potential and coupling constant in a regime where classical field theory gives leading order asymptotics. In the same limit, the repulsive interaction potential is set to be short-range: it converges to a Dirac delta function with a rate depending polynomially on the other scaling parameters. We prove that the free-energy of the interacting Bose gas (counted relatively to the non-interacting one) converges to the free energy of the Phi(4 )(2)non-linear Schrodinger-Gibbs measure, thereby revisiting recent results and streamlining proofs thereof. We combine the variational method of Lewin-Nam-Rougerie to connect, with controlled error, the quantum free energy to a classical Hartree-Gibbs one with smeared non-linearity. The convergence of the latter to the Phi(4)(2) free energy then follows from arguments of Frohlich-Knowles-Schlein-Sohinger. This derivation parallels recent results of Nam-Zhu-Zhu.
We constructed a new exactly solvable model of the non-relativistic quantum linear harmonic oscillator with the position-dependent mass M(x) =2m(0)/e(alpha x)+1. We solved the Schr & ouml;dinger equation for this model in the absence and presence of an external homogeneous field. We showed that in both cases the wave functions of the bound states are expressed through Jacobi polynomials, and the corresponding energy spectra are non-equidistant. We also showed that in the limit alpha -> 0 the wave functions and energy spectra coincide with the corresponding expressions for a linear harmonic oscillator with the constant mass m(0) in the absence and presence of an external homogeneous field. We present also two new limit relations that reduce the Jacobi polynomials directly to the Hermite polynomials with shifted and non-shifted arguments. The proofs of these limit relations are based on the method of mathematical induction.
This paper explores the b-weighted upper metric mean dimension and the average b-weighted upper metric mean dimension with potential for any subset. Specifically, we establish a variational principle for these dimensions.
This paper investigates the existence and regularity of pullback random attractors for a class of non-autonomous stochastic g-Navier-Stokes equations driven by nonlinear colored noise. We first establish the existence and uniqueness of pullback random attractors in the g-weighted space H-g. We further prove the regularity of the attractor in the strong topology of V-g, showing that it is a bi-spatial attractor which is compact in H-g while attracting bounded sets in V-g. The main difficulty in proving pullback asymptotic compactness in H-g is overcome by deriving uniform estimates in both H-g and V-g and leveraging the compact embedding V-g hooked right arrow H-g. To establish compactness in the stronger space V-g, we employ a spectral decomposition method. This yields the flattening property of solutions and thereby verifies the asymptotic compactness in V-g.
In this paper, we investigate the global existence and long-time behavior of solutions to the massive Dirac-Klein-Gordon system in R(1+2 )for a class of large initial data. We establish global existence, sharp pointwise decay estimates and linear scattering under the assumption that only the initial data for the Dirac field are small. This assumption permits the initial data for the Klein-Gordon field to have a finite weighted Sobolev norm, without imposing any restriction on its size. Our analysis is carried out under a natural mass condition, consistent with that appearing in the small-data theory. The main difficulties arise from the presence of large data, the lack of symmetry and the weak decay properties of solutions in two spatial dimensions. To overcome these challenges, we exploit a vanishing structure hidden in the Dirac energy and introduce new dynamical variables that reveal a null structure inherent in the massive Dirac-Klein-Gordon system.
In this paper, we consider the asymptotic behavior of solutions of an evolution equation containing the non-autonomous p-Laplacian equation with polynomial growth nonlinearity of arbitrary order and dynamic boundary conditions driven by nonlinear colored noise. We first prove the existence of weak solutions by the Faedo-Galerkin method, but the uniqueness of solutions cannot be guaranteed due to the lack of Lipschitz continuity of diffusion and nonlinear terms. Then we establish the asymptotic compactness of the corresponding cocycle by Sobolev compactness embedding theorem and the measurability of the random attractor by proving the weak upper semi-continuity of the multi-valued non-autonomous cocycle. Finally, we prove the existence and uniqueness of pullback random attractor for the multi-valued non-autonomous cocycle generated by the solution operator.
In this paper, we are concerned with the uniqueness and nonlinear stability of steady flow for the lake equation in a bounded domain Omega of R-2. We prove the uniqueness of a family of solutions zeta(epsilon) obtained in Cao et al. [IMA J. Appl. Math. 87(1), 50-79 (2022)] over the admissible set A(epsilon )provided supp (zeta(epsilon)) -> x(0) as epsilon -> 0 and the vorticity function f(s)=s(+)(p) for some p > 1, where x(0) is an element of Omega is a non-degenerate maximum point of the depth function b(x). Furthermore, as a by-product, we obtain the nonlinear stability of zeta(epsilon) for the lake equation.
This paper focuses on shock solutions to the one-dimensional piston problem for nonisentropic compressible Euler equations. Given that the initial data and piston velocity are small perturbations of the background solution, by leveraging the local existence of solutions and obtaining uniform a priori estimates, we prove the global well-posedness of the shock solution in C1 space. The dissipation mechanism inherent in shock waves is important in the proof.
