Global Uniqueness Research Articles

The inverse problem of identifying complex hyperbolic equation source terms in electromagnetic propagation

Abstract This article investigates the inverse problem of determining the source term of the hyperbolic equation for electromagnetic propagation using terminal data. This study is an important method for identifying propagation sources in electromagnetics. Unlike wave equations, the complexity of the underlying equations can make theoretical analysis quite difficult. Firstly, the uniqueness of the inverse problem was proved using the energy method. Then, based on the optimal control framework, the inverse problem was transformed into an optimal control problem, and the existence of the optimal solution and its necessary conditions were established. Secondly, the global uniqueness and stability of the optimal solution have been proven, which is a completely new conclusion. This has laid a solid theoretical foundation for numerical algorithms. Finally, it is proposed to apply the Landweber iteration method and conjugate gradient method to this problem, and some numerical examples are provided to demonstrate the effectiveness and convergence speed of these two algorithms.

Local and Global Uniqueness Theorems of the N-th Order Partial Differential Equations

In this paper, we consider inequalities in which the function is an element of n-th partially order space. Local and Global uniqueness theorem of solutions of the n-the order Partial differential equation Obtained which are applications of Gronwall's inequalities.

설화의 국가무형유산 종목 지정을 위한 제언

As Korea joined the “UNESCO Convention” in 2005, Korea implemented the “Intangible Cultural Property Act” in 2016. As a result, a field of “Ra(라). Oral Tradition and Expression”, was newly established, making it possible to designate a national cultural heritage item for folktales. Nevertheless, until August 2024, not a single event has been realized in the designation of national intangible heritage of folktales. This is due to the view that since the 1980s, the traditional Iyagipan(이야 기판) has disappeared, and the transmission of oral stories has also been cut off. However, this recognition is a misunderstanding caused by denying that the current transmission of folktales continues actively through the “Beautiful Story Grandma” project and numerous old story telling channels on YouTube etc. Another position against the designation of folktales as national intangible heritage is the opinion that Gureongdeongdeongsinseonbi(구 렁덩덩신선비), Kongji Patzzi(콩쥐팥쥐) are folktales with global universality and lack uniqueness, and can lead to conflict with other countries. However, the value of folktales can be found in various realization patterns such as national identity and cultural uniqueness. Therefore, the designation of a national intangible heritage item of folktales is an important task to protect our identity and uniqueness. Finally, I would like to suggest some opinions to revitalize the designation of folktales as national intangible heritage items. It is urgent to clarify the value of intangible heritage of the current transmission of folktales and to redefine the definition, category, and classification system of the “Ra(라)” field. In addition, systematic policy establishment and long-term research projects should be promoted, and the discovery of local intangible heritage should be actively supported. Aside from that, the construction of a digital archive of folktales should protect and promote the current transmission of folktales.

Strong Solutions of Brusselator System

The study involves a mathematical analysis of the Brusselator system on a convex bounded three-dimensional open domain, considering Neumann boundary conditions. We establish the global existence and uniqueness of the strong solution for this system. Achieving high regularity for the strong solution requires stringent conditions on the initial data. The study demonstrates the continuous dependence of the solution on the initial conditions.

The Future of Referencing Specimens Is Near: Cite the Digital Specimen DOI

Specimens are often mentioned in scholarly publications or data infrastructures by referencing the local identifiers attached to the objects held in specimen collections. However, these are often only unique to the issuing institution and not resolvable. Transforming these into globally unique identifiers, such as the 'Darwin Core Triplet' constructed from codes that specify the institution, the collection and the accession or catalog number, is an imperfect solution. These cannot be unambiguously validated nor be dereferenced (Guralnick et al. 2015, Groom et al. 2019), are vulnerable to human error, and rely on accession numbers that sometimes change over time. Earlier approaches to give specimens globally unique and resolvable identifiers had limited success (Page 2009). Attaching labels with new identifiers is too costly and time consuming, apart from digital infrastructure challenges. The Digital Extended Specimen is a new object on the internet, containing all known information about the specimen, thus acting as a surrogate for the physical object. To distinguish it from its physical counterpart, it needs a new identifier, which also provides a solution to unambiguously and persistently refer to specimens. Digital specimens get Digital Object Identifiers (DOIs) and Persistent Identifiers, in the infrastructure developed in the BiCIKL project (see deliverable D7.1 and this blog post). Ties to these accession or catalog numbers of physical specimens, allow these local identifiers to change over time, while still persistently resolving to the digital object. DOIs guarantee global uniqueness, persistence and reliable resolution through the oversight of the DOI Foundation and its registration agencies (RAs). DiSSCo, Distributed System of Scientific Collections, is creating a partnership with DataCite RA, where both partners invest to provide Digital Specimen DOIs and enhance their findability, leveraging from their metadata services. This aids the potential of Digital Specimen DOIs to reliably create links with data in other infrastructures, cite individual specimens, and improve the FAIR-ness (Findable, Accessible, Interoperable, Reusable) of specimen data worldwide. We will present how to use these new DOIs in practice. The use of Digital Specimen DOIs in a publication to reference a specimen is already piloted (Deeleman-Reinhold et al. 2024). All specimen records cited in the publication have been assigned a DOI that points to a digital representation of the specimen, the Digital Specimen, which can evolve over time to become a digital extended specimen by digitally linking it to relevant ecological, environmental, and related data such as the publication itself. The DOIs redirect both to a machine readable and a human readable version and include metadata in the DOI record for machine actionability that is also findable in DataCite Commons. The Crosscite citation formatter offers guidance to users how to cite a Digital Specimen DOI, similar to DOIs that refer to other works.

The global uniqueness of a dissipative fractional Helmholtz equation

In this paper, we consider the Dirichlet problem for the fractional Helmholtz equation with dissipation, and study the inverse problem of determining the source function, potential function and dissipation of the equation using the Dirichlet-to-Neumann map and Runge approximation. We prove the global uniqueness of these three functions of the equation under low-frequency conditions. As the main result, this implies that one can use the external data to uniquely recover the unknown functions of the equation in the low-frequency case, which will be of great significance in photoacoustic tomography and thermoacoustic tomography.

The global existence and uniqueness of smooth solutions to a fluid-particle interaction model in the flowing regime

The global existence and uniqueness of smooth solutions to a fluid-particle interaction model in the flowing regime

On an Oberbeck-Boussinesq model relating to the motion of a viscous fluid subject to heating

Abstract This article surveys some results in the study of Iannelli [Su un modello di Oberbeck-Boussinesq relativo al moto di un fluido viscoso soggetto a riscaldamento, Fisica Matematica, Istituto Lombardo (rend. Sc.) A 121 (1987), 145–191], in which the motion of a viscous, compressible fluid in a two-dimensional domain, subject to heating at the walls, is studied. A global existence and uniqueness theorem for the time-dependent problem is given, and also, under more stringent assumptions, an existence and uniqueness theorem in the stationary case is given. A theorem on the asymptotic behavior for t → ∞ t\to \infty of the time-dependent solutions is proved.

Global Existence, Uniqueness, and Non-negativity of the Solutions to Stochastic Partial Differential Equations of Infectious Diseases

Global Existence, Uniqueness, and Non-negativity of the Solutions to Stochastic Partial Differential Equations of Infectious Diseases

Long time decay of incompressible convective Brinkman-Forchheimer in L 2(ℝ3)

Abstract In this article, we study the global existence, uniqueness, and continuity for the solution of incompressible convective Brinkman-Forchheimer on the whole space R 3 {{\mathbb{R}}}^{3} when 4 μ β ≥ 1 4\mu \beta \ge 1 . Additionally, we give an asymptotic type of convergence of the global solution.

Determining coefficients for a fractional p-Laplace equation from exterior measurements

We consider an inverse problem of determining the coefficients of a fractional p -Laplace equation in the exterior domain. Assuming suitable local regularity of the coefficients in the exterior domain, we offer an explicit reconstruction formula in the region where the exterior measurements are performed. This formula is then used to establish a global uniqueness result for real-analytic coefficients. In addition, we also derive a stability estimate for the unique determination of the coefficients in the exterior measurement set.

Global well-posedness of the 3D damped micropolar Bénard system with horizontal dissipation

In this paper, we consider the global well-posedness of the 3D damped micropolar Bénard system with horizontal dissipation. Global existence and uniqueness of the solution of system (1.1) are proved for β≥4 and α>0.

Global well-posedness and large time behavior for the 3D magneto-micropolar equations

This paper focuses on the global well-posedness and optimal decay estimates to the 3D magneto-micropolar equations in Besov spaces. We obtain the global existence and uniqueness of solutions in Besov spaces. Moreover, the decay estimates of the global solutions are established provided that the initial data belongs to the negative Besov space.

Stability and exponential decay for the compressible viscous non-resistive MHD system

How to construct global solutions of the compressible viscous magnetohydrodynamic (MHD) equations without magnetic diffusion even with small initial data in R3 or T3 is still an extremely challenging open problem. The difficulty comes from the lack of magnetic diffusion and the fact that solutions to inviscid equations generally grow in time. Motivated by this open problem, the present paper focuses on a special case of this MHD system in T3 when the magnetic field is vertical. We establish the global existence and uniqueness of smooth solutions to this system near a steady-state solution. In addition, the solution is shown to be stable and decay exponentially in time. The proof discovers and makes use of the smoothing and stabilizing effect of the steady magnetic field on the perturbations.

Well-posedness of stochastic multi-weighted complex networks with regime-switching diffusions

In this paper, a stochastic multi-weighted complex network with regime-switching diffusions (SMCNRSD) is considered. Here, the jump and the diffusion processes are interdependent, which generalizes the most existing network model. In the study of stochastic multi-weighted complex networks, stability analysis is an important topic. But so far, some work on stability requires stochastic multi-weighted complex networks to satisfy the global linear growth condition, which cannot be applied to some high-order or nonlinear networks. Therefore, one of the research contents of this study is to obtain the global existence and uniqueness of SMCNRSD under local conditions. Besides, the SMCNRSD usually does not stabilize spontaneously for reasons such as stochastic disturbance, multiple weights and nonlinear coupling. To stabilize SMCNRSD, periodic intermittent control is introduced to obtain the stability criterion. In application, stochastic multi-weighted Chua’s circuits with regime-switching diffusions are taken into account. The corresponding existence and uniqueness as well as stability criteria are given under local conditions. To further illustrate the validity of theoretical results, numerical simulations are given.

The Burgers-type equation driven by a stochastic measure

We study the one-dimensional equation driven by a stochastic measure μ \mu . For μ \mu we assume only σ \sigma -additivity in probability. Our results imply the global existence and uniqueness of the solution to the heat equation and the local existence and uniqueness of the solution to the Burgers equation. The averaging principle for such equation is studied.

Existence and uniqueness of solutions for forward and backward nonlocal Fokker-Planck equations with time-dependent coefficients

Based on the Feynman-Kac formula, we develop a new approach to show the global existence and uniqueness of the solutions for the forward and backward time-dependent multidimensional nonlocal Fokker-Planck equations, which are associated with a class of time-dependent stochastic differential equations driven by symmetric (or asymmetric) non-Gaussian Lévy process. First, after deriving the forward time-dependent multidimensional nonlocal Fokker-Planck equation, the Feynman-Kac formula for the forward nonlocal Fokker-Planck equation is established by using Itô's formula and techniques for the backward nonlocal Fokker-Planck equation corresponding to the backward stochastic differential equation driven by Lévy process. Then, from the Feynman-Kac formula, we prove the global existence and uniqueness of the solutions for the forward and backward time-dependent multidimensional nonlocal Fokker-Planck equations by using techniques from stochastic analysis. Finally, the dynamic evolution of the probability density function corresponding to the stochastic model for the MeKS network (reflecting the interactions among the MecA complex, comK, comS) is investigated over an extended period by Feynman-Kac formula and Monte Carlo simulations. In particular, we reveal the influence of symmetric and asymmetric stable noises as well as tempered stable noise on the probability density function corresponding to the stochastic MeKS network model.

Global existence and attractivity for Riemann-Liouville fractional semilinear evolution equations involving weakly singular integral inequalities

In this paper, we obtain several results on the global existence, uniqueness and attractivity for fractional evolution equations involving the Riemann-Liouville type by exploiting some results on weakly singular integral inequalities in Banach spaces. Some boundedness conditions of the nonlinear term are considered to obtain the main results that generalize and improve some well-known works.

Global existence and partial regularity for the Landau-Lifshitz-Gilbert equation with helicity

In this paper, the Landau-Lifshitz-Gilbert equation with helicity is considered. In R3 or a bounded regular domain Ω of R3, we establish the global existence of a weak solution. In Rn, a global existence criterion and uniqueness of the smooth solution are given. In R1, the local smooth solution is indeed global with large initial data. In R2, we prove the existence of a global weak solution, which is smooth with the exception of at most finite singular points.

Weak Solutions to an Initial-Boundary Value Problem of a Consistent Three-phase-field Model

Global warming has caused the melting of glaciers, especially the Greenland sea ice which is a great threat to human beings. The phase-field method is used to describe the evolution of multi-phase microstructure of Greenland sea ice. This paper shall investigate the global existence, uniqueness and regularity of weak solutions to an initial-boundary value problem for a three-phase-field model based on criteria that lead to both physical and mathematical consistency. For global solutions, it is calculated by applying the method of continuation of local solutions. This model allows for the formulation of spurious phases, which is completely consistent with physical reality.