Class Of Solutions Research Articles

Iterative methods with new topological properties to show a new classical solutions of the Burgers equation

Iterative methods with new topological properties to show a new classical solutions of the Burgers equation

A statistical study of energetic particle events associated with interplanetary shocks observed by Solar Orbiter in solar cycle 25

We studied energetic particle intensity profiles observed by Solar Orbiter during the time period from April 2020 to April 2023, associated with the passage of interplanetary (IP) shocks. For our study we considered 58 IP forward shocks and analysed the possible correlations between some IP shock parameters and the electron and proton responses to the passage of the IP shocks. We investigated which shock signatures are more likely related to the efficiency of the IP shocks with respect to particle acceleration. We introduced a variable that characterises the contamination induced by protons in the electron channels of the Electron Proton Telescope (EPT) part of the Energetic Particle Detector (EPD) suite of instruments on board Solar Orbiter, which allowed us to identify the cases in which the intensity time profiles of electrons at energies le240 keV showed a real response at the passage of IP shocks. In the case of protons, we searched for the response in seven energy ranges from 52 keV to 15 MeV, and based on the shape of the proton response at low energies (∼100 keV), we divided the profiles into weak responses, peaks (regular or irregular), plateaus, and unclear responses. For the regular peak and plateau types we constructed an average time profile by applying superposed epoch analysis. For the response in electrons and protons, and for the different types of proton responses at different energies, we analysed the corresponding IP shock parameters, aiming to understand which ones are important to form a certain type of time profile or to achieve a certain energy. We also included a comparison between the proton intensity time profiles in the upstream region and, assuming the predictions of the diffusive shock acceleration (DSA) theory, identified the values of the mean free path in several cases. We found that the IP shock efficiency in the energisation of both electrons and protons is strongly energy dependent. Cases of electron acceleration are rare. Only in about ∼8$%$ of the events for energies le100 keV and in ∼2$%$ for energies le250 keV did the electron intensities show an unambiguous response at the passage of IP shocks (with those accompanied by a response being mainly oblique or quasi-perpendicular). The shocks for which we identified a response in ∼100 keV proton intensity time profiles come to ∼ 83% of the IP shocks under study, and are parallel or quasi-parallel. The ability to accelerate protons to higher energies and to form a particular shape of the particle response to the IP shock passage mostly depends on the IP shock speed. Based on the analysis of time profiles and the occurrence of unambiguous electron acceleration at shocks, the acceleration mechanism behind the electron energisation is unlikely to be DSA, but shock drift acceleration (SDA) remains a candidate for the acceleration mechanism. Proton time profiles of the plateau type around the IP shock front can be achieved with an IP shock speed above 800 km s^-1 and an ambient mean free path ≤ 0.015 au, reproducing the asymptotic steady-state ion distribution reached in the classical DSA solution.

Kolmogorov Equation for a Stochastic Reaction–Diffusion Equation with Multiplicative Noise

Reaction–diffusion equations can model complex systems where randomness plays a role, capturing the interaction between diffusion processes and random fluctuations. The Kolmogorov equations associated with these systems play an important role in understanding the long-term behavior, stability, and control of such complex systems. In this paper, we investigate the existence of a classical solution for the Kolmogorov equation associated with a stochastic reaction–diffusion equation driven by nonlinear multiplicative trace-class noise. We also establish the existence of an invariant measure ν for the corresponding transition semigroup Pt, where the infinitesimal generator in L2(H,ν) is identified as the closure of the Kolmogorov operator K0.

Non-existence of classical solutions to a two-phase flow model with vacuum

Abstract In this paper, we study the well-posedness of classical solutions to a two-phase flow model consisting of the pressureless Euler equations coupled with the isentropic compressible Navier–Stokes equations via a drag forcing term. We consider the case that the fluid densities may contain a vacuum, and the viscosities are density-dependent functions. Under suitable assumptions on the initial data, we show that the finite-energy (i.e. in the inhomogeneous Sobolev space) classical solutions to the Cauchy problem of this coupled system do not exist for any small time.

Small-mass solutions in a two-dimensional logarithmic Chemotaxis–Navier–Stokes system with indirect nutrient consumption

This paper is concerned with a two-dimensional singular chemotaxis–Navier–Stokes system under the no-flux/Neumann/Neumann/Dirichlet boundary condition which models the movement of a bacteria population with indirect nutrient consumption in a fluid environment. In this model, we consider a general tensor-valued chemotactic sensitivity, where the bacteria movement orients toward higher concentration of nutrients under the Weber–Fechner law of stimulus perception, resulting in a logarithmic singularity in the chemotactic sensitivity tensor when the nutrient concentration drops to 0. We demonstrate that, when the initial bacteria population is suitably small, the system possesses a globally bounded classical solution, which, inter alia, exponentially stabilizes toward the spatially homogeneous state where the bacteria concentration settles on an even distribution of the initial population. This rigorously confirms that, at least in the two-dimensional setting, in comparison with the direct mechanism of nutrient consumption, an indirect mechanism can induce much more regularity in the solutions to the chemotaxis–fluid system even with a singular tensor-valued sensitivity.

Boundedness in a one-dimensional quasilinear attraction–repulsion chemotaxis system with flux limitation

This paper is concerned with a quasilinear attraction–repulsion chemotaxis system with flux limitation in the one-dimensional setting. The main results assert global existence and boundedness of classical solutions to the system under smallness conditions on initial data and certain assumptions on parameters.

Global Classical Solutions to the Two-phase Flow Model with Slip Boundary Condition in 3D Bounded Domains

Global Classical Solutions to the Two-phase Flow Model with Slip Boundary Condition in 3D Bounded Domains

Solution of the Boussinesq Problem in a Depth‐Dependent Nonhomogeneous Half‐Space

ABSTRACTIn this study, the Boussinesq problem is modeled using the half‐space model of the soil foundation to determine the collapse of the ground surface under a point load applied to the surface when the elastic modulus of the ground changes as a continuous function of depth. Analytical solutions are developed for the cases where the elastic modulus increases depending on the depth coordinate. The solutions are presented as the product of the classical Boussinesq solution and a dimensionless function that considers the heterogeneity with depth. Finally, a comparative analysis of the subsidence at dimensionless distances from the point load for different heterogeneity functions is performed compared to the classical solution.

A comprehensive study of two-phase flow in deformable porous media

This study explored the initial-boundary value problem of an immiscible, incompressible two-phase flow within a deformable porous medium system, modeled in an N-dimensional (N≥3) Euclidean space. The primary objective was to devise a novel model that effectively captured previously overlooked physical phenomena and established the global existence of a weak solution to the system. After proving the existence of a classical solution to the approximate problem using Schauder's fixed point theorem under a series of reasonable assumptions, we obtained the convergence to a weak solution of the original problem in two different cases by applying the Fréchet–Kolmogorov theorem. Subsequently, we verified the applicability of this model to the commonly used van Genuchten model in the dynamic simulation of oil storage, providing theoretical support for engineering applications.

Inverse Boundary Value Problem for Fourth-Order Pseudo-Hyperbolic Equation with Nonlocal Integral Conditions of the Second Kind

In this paper, we consider a nonlinear inverse boundary value problem for a fourthorder pseudo hyperbolic equation with nonlocal conditions of the integral type. First, we introduce the definition of a classical solution to the problem. The purpose of this paper is to determine the unknown coefficient of the right-hand side and to solve the problem of interest. The problem is considered in a rectangular domain. To study the solvability of the inverse problem, we perform a transformation from the original problem to some auxiliary inverse problem with trivial boundary conditions. Using the principle of contraction mappings, we prove the existence and uniqueness of solutions to the auxiliary problem. Then we again perform a transformation to the problem and as a result obtain the solvability of the inverse problem.

Existence of a positive solution for a second order Sturm-Liouville boundary value problem on the half-line with a nonlinear derivative dependence via variational methods

In this paper, a class of second order Sturm-Liouville boundary value problem on the half-line with dependence derivative is considered. The existence of a positive classical solution is proved by using variational methods and iterative methods.

Stability and global analysis of the Vlasov–Maxwell–Einstein system in astrophysical plasmas

This paper provides an analysis of the relativistic Vlasov–Maxwell–Einstein (RVME) system, which models the dynamics of relativistic plasmas in the presence of strong gravitational fields. We establish the existence and uniqueness of global classical solutions using a fixed-point iteration method combined with a priori estimates. The stability of these solutions is analyzed with respect to small perturbations in the initial data to derive explicit stability bounds. The implications of our results are briefly discussed in the context of astrophysical scenarios, particularly focusing on the dynamics of plasmas near compact objects such as black holes and neutron stars.

Global Boundedness and Mass Persistence of Solutions to A Chemotaxis-Competition System with Logistic Source

This article examines the population dynamics of solutions such as global existence, global boundedness, and mass persistence, to a parabolic elliptic type of chemotaxis-competition system including logistics kinetics in a smooth bounded domain. Tello and Winkler were the first to investigate the global existence and global boundedness of the system mentioned above. Then Tao and Winkler examined qualitative properties of the given system such as the mass persistence of solutions. This study improves some known results and reveals that under some suitable conditions, there exists a classical solution to the system described above that is globally bounded. In addition, it is shown that the population as a whole is never extinct.

Global existence and decay estimates of classical solutions to a chemotaxis-consumption system

Global existence and decay estimates of classical solutions to a chemotaxis-consumption system

On classical solutions and canonical transformations for Hamilton–Jacobi–Bellman equations

AbstractIn this note, we show how canonical transformations reveal hidden convexity properties for deterministic optimal control problems, which in turn result in global existence of solutions to first‐order Hamilton–Jacobi–Bellman equations.

Global boundedness of solutions for a quasilinear chemotaxis‐Stokes system with nonlinear production

AbstractIn this paper we study the global boundedness of solutions to the quasilinear chemotaxis‐Stokes system with nonlinear production subject to no‐flux/no‐flux/Dirichlet boundary conditions in a smoothly bounded domain with , where , , and generalize the prototypes and for all with and . It is shown that if for , or for , then for any reasonably regular initial datum, the corresponding initial‐boundary value problem of () possesses a unique globally bounded classical solution. Compared to the global boundedness condition for in the relevant fluid‐free system [J. Differ. Equ. 268 6729–6777 (2020)], the present result indicates that the slow fluid motion described by the Stokes equation might possibly be adverse to the global solvability in the case that with . What is more, this paper provides for the quasilinear problems with general signal production a universal approach capable of deriving the boundedness of solutions in the optimal range of parameters.

Global dynamics of an SIS epidemic model with cross-diffusion: applications to quarantine measures

Abstract This paper considers an SIS model with a cross-diffusion dispersal strategy for the infected individuals describing the public health intervention measures (like quarantine) during the outbreak of infectious diseases. The model adopts the frequency-dependent transmission mechanism and includes demographic changes (i.e. population recruitment and death) subject to homogeneous Neumann boundary conditions. We first establish the existence of global classical solutions with the uniform-in-time bound. Then, we define the basic reproduction number R 0 by a weighted variational form. Due to the presence of the cross-diffusion on infected individuals, we employ a change of variable and apply the index theory along with the principal eigenvalue theory to establish the threshold dynamics in terms of R 0 based on the fact that the sign of the principal eigenvalue of the weighted eigenvalue problem is the same as that of the corresponding unweighted eigenvalue problem. Furthermore, we obtain the global stability of the unique disease-free equilibrium and constant endemic equilibrium under some conditions. Finally, we discuss some open questions and use numerical simulation to demonstrate the applications of our analytical results, showing that the cross-diffusion dispersal strategy can reduce the value of R 0 and help eradicate the diseases even if the habitat is high-risk in contrast to the situation without cross-diffusion.

Finite upper bounds of existence time of classical solutions to relativistic Euler equations with finite energy

AbstractIn this paper, we consider the relativistic Euler equations with vacuum state and finite energy such that the existence of local classical solutions is guaranteed by smooth enough initial data. We first show that the propagation speed of support of classical solutions is at most by considering the non‐vacuum case with a perturbation, where represents the derivative of the pressure function with respect to the density. It follows that the support does not expand in most cases where one has . Subsequently, by introducing a weighted second moment of inertia and a weighted radial component of momentum, we prove that the classical solutions of the relativistic Euler equations with vacuum state and finite energy can only exist up to a finite time. Moreover, a class of concrete finite upper bounds of existence time that depends only on initial data and given parameters is established. Last, we provide specific examples of our test functions.

(2+1) Lorentzian quantum cosmology from spin-foams: opportunities and obstacles for semi-classicality

Abstract We construct an effective cosmological spin-foam model for a ( 2 + 1 ) dimensional spatially flat Universe, discretized on a cubical lattice, containing both space- and time-like regions. Our starting point is the recently proposed coherent state spin-foam model for (2+1) Lorentzian quantum gravity. The full amplitude is assumed to factorize into single vertex amplitudes with boundary data corresponding to Lorentzian 3-frusta. A stationary phase approximation is performed at each vertex individually, where the inverse square root of the Hessian determinant serves as a measure for the effective path integral. Additionally, a massive scalar field is coupled to the geometry, and we show that its mass renders the partition function convergent. For a single 3-frustum with time-like struts, we compute the expectation value of the bulk strut length and show that it generically agrees with the classical solutions and that it is a discontinuous function of the scalar field mass. Allowing the struts to be space-like introduces causality violations, which drive the expectation values away from the classical solutions due to the lack of an exponential suppression of these configurations. This is a direct consequence of the semi-classical amplitude only containing the real part of deficit angles, in contrast with the Lorentzian Regge action used in effective spin-foams. We give an outlook on how to evaluate the partition function on an extended discretization including a bulk spatial slice. This serves as a foundation for future investigations of physically interesting scenarios such as a quantum bounce or the viability of massive scalar field clocks. Our results demonstrate that the effective path integral in the causally regular sector serves as a viable quantum cosmology model, but that the agreement of expectation values with classical solutions is tightly bound to the path integral measure.

Approximating the solution of a linear heat transfer problem for concrete subjected to one-sided heating under standard fire conditions

To estimate the fire resistance limit of reinforced concrete structures, it is essential to understand the temperature distribution within the concrete cross-section under standard fire conditions. Existing approximate analytical methods rely on the classical solution of the heat transfer equation assuming a constant surface temperature. The authors developed a degree approximation of the standard fire temperature curve. This approximation enables an approximate analytical solution to the heat transfer problem with a varying surface temperature corresponding to standard fire conditions. The aim of this work was to derive a convenient formula for heat transfer engineering calculations applicable to concrete with arbitrary thermophysical properties. The derived formula accurately predicts the temperature at any point within the concrete at a given time. The authors’ solution was compared with the high-precision numerical simulations (ANSYS, MATLAB) for various concrete types. Because the proposed approximation does not involve special functions, its implementation does not require any specialized software. The accuracy, simplicity, and versatility of this formula make it suitable for use in fire resistance engineering calculations to determine the time-dependent temperature distribution within concrete under standard fire conditions.