Class Of Solutions Research Articles

Analysis of fracton-gravity stability

We study the stability of fracton gravity, a variant of linearized gravity where the gauge symmetry is restricted to longitudinal diffeomorphisms. These transformations can be connected to a spacetime generalization of dipole symmetry, hence the tag fracton. We find that fracton gravity features an instability in the spin-1 sector corresponding to solutions with growing amplitude. This dynamical instability can be removed by tuning the couplings of the theory. Nonetheless, the Hamiltonian for the spin-1 modes remains always unbounded from below when evaluated on the classical solutions. We find no other sources of instability in the spin-2 or spin-0 sectors. We analyze in detail the canonical formulation and the constraints arguing that neither auxiliary fields nor gauge-fixing conditions can be employed to remove the problematic vector modes or stabilize them. Published by the American Physical Society 2024

Evolution of Tidal Disruption Event Disks with Magnetically Driven Winds

We present a time-dependent, one-dimensional, magnetically driven disk wind model based on magnetohydrodynamic (MHD) equations, in the context of tidal disruption events (TDEs). We assume that the disk is geometrically thin, is gas pressure dominated, and explicitly accounts for magnetic braking and turbulent viscosity through an extended α-viscosity prescription. We find a particular wind solution for a set of basic equations that satisfies the necessary and sufficient conditions for vertically unbound MHD flows. The solution shows that the disk evolves with mass loss due to wind and accretion from the initial Gaussian density distribution. We confirm that the mass accretion rate follows the power law of time t −19/16 at late times in the absence of wind, which matches the classical solution of J. K. Cannizzo et al. We find that the mass accretion rate is steeper than the t −19/16 curve when the wind is present. Mass accretion is also induced by magnetic braking, known as the wind-driven accretion mechanism, which results in a faster decay with time of both the mass accretion and mass-loss rates. In the disk emission, the ultraviolet (UV) luminosity is the highest among the optical, UV, and X-ray luminosities. While the optical and X-ray emission is observationally insignificant without magnetic braking, the X-ray emission is brighter at late times, especially in the presence of magnetic braking. This provides a possible explanation for observed delayed X-ray flares. Our model predicts that late-time bolometric light curves steeper than t −19/16 in UV-bright TDEs are potentially compelling indicators of magnetically driven winds.

On the Large-x Asymptotic of the Classical Solutions to the Non-Linear Benjamin Equation in Fractional Sobolev Spaces

In this work, we study the large-x asymptotic of classical solutions to the non-linear Benjamin equation modeling propagation of small amplitude internal waves in a two fluid system. In our analysis, we extend known Hs-well-posedness results to the case of the variable-weight Sobolev spaces. The spaces provide a direct control over the asymptotics of classical solutions and their weak derivatives, and permit us to compute the bulk large-x asymptotic of classical solutions explicitly in terms of input data. The asymptotic formula provides a precise description of the qualitative behaviour of classical solutions in weighted spaces and yields a number of weighted persistence and continuation results automatically.

Weak compactons of nonlinearly dispersive KdV and KP equations

AbstractA weak formulation is devised for the equation, which is a nonlinearly dispersive generalization of the gKdV equation having compacton solutions. With this formulation, explicit weak compacton solutions are derived, including ones that do not exist as classical (strong) solutions. Similar results are obtained for a nonlinearly dispersive generalization of the gKP equation in two dimensions, which possesses line compacton solutions.

A polar diatomic molecule under a high-frequency laser field: classical analytical solution

AbstractDynamics of a polar diatomic molecule (represented as the oscillating rotator model) under external fields is the most fundamental problem and the testbed in molecular physics—analogously to the dynamics of hydrogenic atom (represented as the model of an electron in the Coulomb field) under external fields being the most fundamental problem and the testbed in atomic physics. We present the classical analytical study of the dynamical Stark effect in the oscillating rotator under a high-frequency laser field. We obtain the analytical results by using the method of effective potentials. We demonstrate that under the laser field, the rotation frequency increases, while the oscillation frequency decreases, and the amplitude of the oscillations increases. More significantly, the laser field causes the generation of the second harmonic in the oscillations of this system. This is an important, counterintuitive result in studying this fundamental physical system. We also drew attention to some flaws in the literature on this subject.

Logarithmically refined Gagliardo–Nirenberg interpolation and application to blow-up exclusion in a singular chemotaxis–consumption system

A family of interpolation inequalities is derived, which differ from estimates of classical Gagliardo–Nirenberg type through the appearance of certain logarithmic deviations from standard Lebesgue norms in zero-order expressions. Optimality of the obtained inequalities is shown. A subsequent application reveals that when posed under homogeneous Neumann boundary conditions in smoothly bounded planar domains and with suitably regular initial data, for any choice of \alpha>0 the Keller–Segel-type migration–consumption system u_{t} = \Delta (uv^{-\alpha}) , v_{t} = \Delta v-uv , admits a global classical solution.

A lower bound estimate of solutions to the Cauchy problems for a hyperbolic Monge–Ampère equation

This paper investigates the Cauchy problems for a hyperbolic Monge–Ampère equation which can be reduced to a quasilinear hyperbolic system via Riemannian invariants. Based on the first a priori estimate of the solutions, the second a priori estimate of the derivation of the solutions, and the third a priori estimate of the continuous mould of the first‐order partial derivatives of the solutions to quasilinear hyperbolic system, a lower bound estimate of classical solutions to the Cauchy problems for a hyperbolic Monge–Ampère equation is derived.

Global existence and boundedness in a two‐dimensional parabolic chemotaxis system with competing attraction and repulsion effects

In this paper, we investigate a chemotaxis system under homogeneous Neumann boundary conditions within a bounded domain with a smooth boundary. The system describes the movement of cells in response to two chemical signal substances: one acts as a chemoattractant, while the other serves as a chemorepellent, both produced by the cells. The system takes into account chemotactic sensitivity in the reaction movement when detecting these chemicals. Under certain assumptions, we demonstrate the existence of a unique global bounded classical solution for the proposed problem. To further understand the time evolution of the system's solutions, we conduct numerical experiments and analyze the dynamic properties of the norm of the solutions with respect to variations in chemical production rates.

Classical solutions to a pursuit‐evasion model with prey‐taxis and indirect predator‐taxis

Classical solutions to a pursuit‐evasion model with prey‐taxis and indirect predator‐taxis

Embedding space approach to Jackiw-Teitelboim gravity

We present a coordinate-free background space construction of Euclidean Jackiw-Teitelboim gravity. It is written as a gauge theory obtained from the Killing vectors and conformal Killing vectors of a hyperboloid embedded in a three-dimensional background. A novel feature of the gauge theory is that vanishing field strength does not necessarily imply that the gauge potentials (written in the embedding space) are pure gauges, not even locally. On the other hand, the projection of the potentials along the hyperboloid are pure gauges, as is the case in the standard approach. As is usual, metric tensors are dynamically generated from the classical solutions of the theory, which here do not rely on coordinate charts on the two-dimensional surface. We find a special class of solutions whereby the derived metric tensor on the surface is the induced metric from the background space. The gauge theory construction given here has a natural generalization to a noncommutative space, which does not require the use of coordinates, symbols, or a star product. Published by the American Physical Society 2024

Rectifiability, finite Hausdorff measure, and compactness for non‐minimizing Bernoulli free boundaries

AbstractWhile there are numerous results on minimizers or stable solutions of the Bernoulli problem proving regularity of the free boundary and analyzing singularities, much less is known about critical points of the corresponding energy. Saddle points of the energy (or of closely related energies) and solutions of the corresponding time‐dependent problem occur naturally in applied problems such as water waves and combustion theory. For such critical points —which can be obtained as limits of classical solutions or limits of a singular perturbation problem—it has been open since (Weiss, 2003) whether the singular set can be large and what equation the measure satisfies, except for the case of two dimensions. In the present result we use recent techniques such as a frequency formula for the Bernoulli problem as well as the celebrated Naber–Valtorta procedure to answer this more than 20 year old question in an affirmative way: For a closed class we call variational solutions of the Bernoulli problem, we show that the topological free boundary (including degenerate singular points , at which as ) is countably ‐rectifiable and has locally finite ‐measure, and we identify the measure completely. This gives a more precise characterization of the free boundary of in arbitrary dimension than was previously available even in dimension two. We also show that limits of (not necessarily minimizing) classical solutions as well as limits of critical points of a singularly perturbed energy are variational solutions, so that the result above applies directly to all of them.

Dynamic behavior in a pursuit-evasion system with signaling mechanism

We mainly focus on the predator-prey problem associated with indirect predator taxis{ut=Δu+χ1∇⋅(u∇w)+u(1−μ1u−αv),vt=Δv−χ2∇⋅(ψ(v)∇u)+v(−1−μ2v+βu),0=Δw+v−w in a smooth bounded domain Ω⊂Rn(n≤3) with homogeneous Neumann boundary conditions, where the parameters χ1, χ2, μ1, μ2, α and β are positive. The chemotaxis sensitivity function with/without “volume-filling” effect ψ(v)=v1+σv, where σ≥0 is a constant. Compared with the well-known predator-prey system, the discussion of signaling mechanism w forms a novel feature of our system. For the case σ=0, we identify a positive constant χ0 such that the above system possesses global classical solutions with relative bound if χ2<χ0; For the case σ>0, we also show that the classical solutions of the system are globally bounded without any restrictions on system parameters. Furthermore, utilizing the Lyapunov functionals, we establish stability for the prey-only situation, the steady state of coexistence is also investigated.

High-frequency vibration analysis of laminated composite plates using energy flow and shear deformation theories

Energy Flow Analysis (EFA) is an efficient approach for characterizing high-frequency vibrations through time-averaged energy density distributions. This study employs EFA to investigate vibroacoustic behavior in laminated composite plates subjected to high-frequency excitation. Governing equations of motion are derived using Classical Plate Theory (CPT), First Order Shear Deformation Theory (FSDT), and Higher Order Shear Deformation Theories (HSDT). Wave propagation parameters like wave number and group velocity are obtained from each theory and compared to assess accuracy. Energy density and intensity formulations are then developed based on classical solutions of the governing equations. EFA analysis indicates that HSDT provides more accurate predictions of wave parameters, especially at very high frequencies where it accounts better for shear deformation effects. Validation against classical energy density solutions shows acceptable accuracy with less than 0.5dB difference in the far-field. Comparisons further demonstrate the superiority of HSDT over FSDT for thicker plates in both classical and EFA analyses. This research establishes the effectiveness of EFA for high-frequency vibration analysis of composite laminates. The main novelty of this research lies in the integration of HSDT into the application of EFA for composite laminates, providing a more precise consideration of shear deformation effects at high frequencies. Specifically, HSDT enhances modeling of critical shear deformation effects at elevated frequencies. EFA delivers computationally efficient solutions while maintaining acceptable accuracy, improving characterization and design of composite structures under high-frequency loading.

On the expansion of a flow into vacuum for spherically symmetric relativistic hydrodynamic equations

Fluid expansion into a vacuum is an interesting and important problem in continuum physics as well as in mathematics. This paper studies the relativistic Euler equations with an initial value that is a combination of a radial flow in a spherical domain and vacuum in the remaining domain. We prove that this expansion into vacuum problem admits a global-in-time classical solution, provided that the initial mass-energy density in the spherical domain is sufficiently small.

Acoustic black hole combined with microperforated plate for a rectangular waveguide

The concept of an acoustic black hole (ABH) for anechoic termination of waveguides has been well-established and extensively studied. The fundamental principle underlying an ABH is the deceleration of acoustic waves, leading to the accumulation and subsequent absorption of acoustic energy within the thermoviscous boundary layer formed along the ribs forming its internal structure. The primary advantage of ABHs, in contrast to classical solutions, lies in their avoidance of porous or fibrous materials, rendering them suitable for deployment in challenging environments. While the majority of published works have focused on ABHs with circular cross-sections, recent advancements have demonstrated the effectiveness of the ABH effect also in waveguides with rectangular cross-sections. However, it has been revealed that these ABHs require very fine internal structure for optimal performance, posing challenges in manufacturing. This study addresses this issue by utilizing a microperforated plate to provide the necessary resistance. To model these ABHs, a simple mathematical model is proposed, enabling the optimization of their parameters. Subsequently, their performance is investigated through numerical experiments. The validity of the results is confirmed by comparing them with a detailed mathematical model, ensuring the accuracy and reliability of the proposed approach.

A singular system involving mixed local and non-local operators

This article sets forth results on the existence, non-existence, uniqueness, and regularities properties, as well as boundary behavior of solutions for singular systems involving mixed local and non-local elliptic operators (see System (S) below). More precisely, we first establish a new weak comparison principle for a singular equation. Afterward, we discuss the non-existence of positive classical solutions, as well as construct suitable ordered pairs of sub-solutions and super-solutions. This allows us to obtain the existence of a pair of positive weak solutions for System (S) by employing Schauder’s fixed-point theorem in the associated conical shell. Finally, we adapt a method of Krasnoselsky to establish the uniqueness of such a positive pair of solutions.

Quasinormal Modes and Stability Analysis of Rotating Hairy Black Holes in Asymptotically Flat Spacetime

Abstract Kerr black holes, characterized solely by their mass and angular momentum, are considered the predominant type in the universe. Recent studies, however, suggest that black holes may possess additional parameters, known as “hair”. This paper investigates the quasinormal modes (QNMs) of rotating hairy black holes in asymptotically flat spacetimes using the continued fraction method to analyze their stability. Our results demonstrate that the introduction of hair parameters modifies the QNM frequencies, indicating a nuanced stability profile that reduces to the classical Kerr solution under specific conditions. These findings provide new insights into the dynamic properties of hairy black holes and their potential astrophysical implications.

Global boundedness and asymptotic stability for a food chain model with nonlinear diffusion

This paper is concerned with a food chain model with nonlinear diffusion ut = Δu + u(1 − u − b1v), vt=∇⋅((v+1)m∇v)−∇⋅(ξv∇u)+vu−b2w1+v+w−θ1−α1v,wt=∇⋅((w+1)l∇w)−∇⋅(χw∇v)+wv1+v+w−θ2−α2w in a smooth bounded domain Ω ⊂ Rn(n ≥ 2) with homogeneous Neumann boundary conditions, where the parameters ξ, χ, α1, bi, θi (i = 1, 2) &amp;gt; 0 and α2 ≥ 0 as well as m, l∈R. We study the global boundedness of classical solutions to the problem if either n = 2 and m ≥ 0, l &amp;gt; − 1 or n ≥ 3 and m&amp;gt;1−2n, l &amp;gt; − 1. Moreover, we prove the global stability of the prey-only steady state and semi-coexistence steady as well as coexistence steady states under certain conditions on parameters.

Reduced-dimensional modelling for nonlinear convection-dominated flow in cylindric domains

The aim of the paper is to construct and justify asymptotic approximations for solutions to quasilinear convection–diffusion problems with a predominance of nonlinear convective flow in a thin cylinder, where an inhomogeneous nonlinear Robin-type boundary condition involving convective and diffusive fluxes is imposed on the lateral surface. The limit problem for vanishing diffusion and the cylinder shrinking to an interval is a nonlinear first-order conservation law. For a time span that allows for a classical solution of this limit problem corresponding uniform pointwise and energy estimates are proven. They provide precise model error estimates with respect to the small parameter that controls the double viscosity-geometric limit. In addition, other problems with more higher Péclet numbers are also considered.

A diffusive predator-prey system with hunting cooperation in predators and prey-taxis: II stationary pattern formation

This paper is a continuation of the study conducted by Ko and Ryu (2024) [5], which introduces and analyzes a generalized predator-prey reaction-diffusion system incorporating (repulsive) prey-taxis and a hunting cooperation effect in predators, under homogeneous Neumann boundary conditions. In the study, the existence and uniqueness of global and classical solutions for the time- and space-dependent system are analytically examined. Furthermore, the study examines the local and global stability and convergence rate at the constant predator-extinction and coexistence states. In our paper, we analyze the stationary system corresponding to the system in [5], with a specific focus on examining the existence and nonexistence of positive and nonconstant solutions. The nonexistence occurs when the diffusion rate of prey is sufficiently high. On the other hand, the existence occurs when the prey-tactic rate is sufficiently high, indicating a strong repulsive prey-taxis, and the diffusion rate of prey is sufficiently low. For this investigation, we separately employ the energy method and the Leray-Schauder degree theory.