Local Uniqueness Research Articles

Stochastic Quasi-Geostrophic Equation with Jump Noise in Lp Spaces

In this paper, we consider a 2D stochastic quasi-geostrophic equation driven by jump noise in a smooth bounded domain. We prove the local existence and uniqueness of mild Lp(D)-solutions for the dissipative quasi-geostrophic equation with a full range of subcritical powers α∈(12,1] by using the semigroup theory and fixed point theorem. Our approach, based on the Yosida approximation argument and Itô formula for the Banach space valued processes, allows for establishing some uniform bounds for the mild solutions and we prove the global existence of mild solutions in L∞(0,T;Lp(D)) space for all p>22α−1, which is consistent with the deterministic case.

Approximation of solutions to integro-differential time fractional order parabolic equations in L^{p}-spaces

In this paper we study the initial boundary value problem for a class of integro-differential time fractional order parabolic equations with a small positive parameter ε. Using the Laplace transform, Mittag-Leffler operator family, C_{0}-semigroup, resolvent operator, and weighted function space, we get the existence of a mild solution. For suitable indices pin [1,+infty ) and sin (1,+infty ), we first prove that the mild solution of the approximating problem converges to that of the corresponding limit problem in L^{p}((0,T), L^{s}(Omega )) as varepsilon rightarrow 0^{+}. Then for the linear approximating problem with ε and the corresponding limit problem, we give the continuous dependence of the solutions. Finally, for a class of semilinear approximating problems and the corresponding limit problems with initial data in L^{s}(Omega ), we prove the local existence and uniqueness of the mild solution and then give the continuous dependence on the initial data.

A priori and a posteriori error analysis for semilinear problems in liquid crystals

In this paper, we develop a unified framework for the a priori and a posteriori error control of different lowest-order finite element methods for approximating the regular solutions of systems of partial differential equations under a set of hypotheses. The systems involve cubic nonlinearities in lower order terms, non-homogeneous Dirichlet boundary conditions, and the results are established under minimal regularity assumptions on the exact solution. The key contributions include (i) results for existence and local uniqueness of the discrete solutions using Newton–Kantorovich theorem, (ii) a priori error estimates in the energy norm, and (iii) a posteriori error estimates that steer the adaptive refinement process. The results are applied to conforming, Nitsche, discontinuous Galerkin, and weakly over penalized symmetric interior penalty schemes for variational models of ferronematics and nematic liquid crystals. The theoretical estimates are corroborated by substantive numerical results.

The Role of Boundary Spanner in Maintaining the Sustainability of Padang City Branding

The lack focus on brand positioning and brand identity in Indonesia, especially in creative industries based on local uniqueness, reflects a lack of development and competitiveness. This research critically evaluates the sustainable development of Padang City city branding with a focus on the role of leaders as boundary spanners in maintaining the sustainability of city branding and building brand awareness. Although required to create uniqueness, Padang City Branding is still influenced by the previous city branding development discourse, causing a lack of specifications and similarities with other cities in West Sumatra. Although the majority of the population is Muslim, the concept of halal tourism has not been created well. This study aims to understand the role of leaders in each institution as boundary spanners in maintaining the sustainability of city branding. As boundary spanners, leaders in each institution are considered central in maintaining the existence of city branding and building efficient policy networks. Observation-participating research methods using the Internet assist researchers in interacting with boundary spanners and understanding culture from the perspectives of its members. This research yields in-depth insights into the leader's key role as a boundary spanner in the context of Padang City city branding.

The solvability for a nonlinear degenerate hyperbolic–parabolic coupled system arising from nematic liquid crystals

Abstract This paper focuses on the Cauchy problem for a one-dimensional quasilinear hyperbolic–parabolic coupled system with initial data given on a line of parabolicity. The coupled system is derived from the Poiseuille flow of full Ericksen–Leslie model in the theory of nematic liquid crystals, which incorporates the crystal and liquid properties of the materials. The main difficulty comes from the degeneracy of the hyperbolic equation, which makes that the system is not continuously differentiable and then the classical methods for the strictly hyperbolic–parabolic coupled systems are invalid. With a choice of a suitable space for the unknown variable of the parabolic equation, we first solve the degenerate hyperbolic problem in a partial hodograph plane and express the smooth solution in terms of the original variables. Based on the smooth solution of the hyperbolic equation, we then construct an iterative sequence for the unknown variable of the parabolic equation by the fundamental solution of the heat equation. Finally, we verify the uniform convergence of the iterative sequence in the selected function space and establish the local existence and uniqueness of classical solutions to the degenerate coupled problem.

Unique local weak solutions of the non-resistive MHD equations in homogeneous Besov space

ABSTRACT In this paper, the local existence and uniqueness of weak solutions to a d-dimensional non-resistive MHD equations in homogeneous Besov spaces are studied. Specifically we obtain the local existence of a weak solution ( u , b ) of the non-resistive MHD equations for the initial data u 0 ∈ B ˙ p , 1 d p − 1 ( R d ) and b 0 ∈ B ˙ p , 1 d p ( R d ) with 1 ≤ p ≤ ∞ , and the uniqueness of the weak solution when 1 ≤ p ≤ 2 d . Compared with the previous results for the non-resistive MHD equations, in the local existence part, the range of p extends to 1 ≤ p ≤ ∞ from 1 ≤ p ≤ 2 d , but the uniqueness of the solution requires 1 ≤ p ≤ 2 d yet.

Approximating local solution of IVPs of nonlinear first order ordinary hybrid integrodifferential equations

In this paper, we prove a couple of approximation results for local existence and uniqueness of the solution of a IVP of nonlinear first order ordinary hybrid integrodifferential equations by using the Dhage monotone iteration method based on a hybrid fixed point theorem of Dhage (2022) and Dhage {et al.} (2022). An approximation result for the Ulam-Hyers stability of the local solution of the considered hybrid differential equation is also established. Finally, our main abstract results are also illustrated with a couple of numerical examples.

Local wellposedness of the relativistic Vlasov–Poisson equation in Besov space

We obtain the local existence and uniqueness of solution to the relativistic Vlasov–Poisson system in Besov space, extending the non-relativistic result to the relativistic case. Weaker regularity is required for the initial datum by optimizing the commutator estimates related to the field. Commutator estimates associated with relativistic velocity are proved by dyadic decomposition of the frequency space and the decreasing properties of Bessel potential.

New general decay results for a fully dynamic piezoelectric beam system with fractional delays under memory boundary controls

In this paper, we investigate the general stability results of a fully dynamic piezoelectric beam system with internal fractional delays under memory boundary controls. Firstly, by introducing two new equations and using two Volterra's inverse operators to deal with fractional delay terms and boundary memory terms, respectively, a new equivalent system is obtained. Secondly, by combining Faedo–Galerkin approximations with the compactness argument, we prove the local existence and uniqueness of weak solution. Finally, under a broader class of kernel functions, by constructing Lyapunov functionals and new equations and applying some technical lemmas, we establish the optimal and explicit energy decay results for two cases, namely, initial values and without imposing initial values equal to 0, that is, . It is worth pointing out that we focus more on the analysis of , because the stability results of are special cases of . This is the first time to analyze the stability of strong coupling problem by combining internal fractional delay and boundary viscoelastic damping. The obtained general decay results are not necessarily exponential or polynomial, which generalize the relevant results in the existing literature.

Global solutions and blow-up for Klein–Gordon equation with damping and logarithmic terms

In this paper, the initial boundary value problem for Klein–Gordon equation with weak and strong damping terms and nonlinear logarithmic term is investigated, which is known as one of the nonlinear wave equations in relativistic quantum mechanics and quantum field theory. Firstly, we prove the local existence and uniqueness of weak solution by using the Galerkin method and Contraction mapping principle. The global existence, energy decay and finite time blow-up of the solution with subcritical initial energy are established. Then these conclusions are extended to the critical initial energy. Besides, the finite time blow-up result with supercritical initial energy is shown.

Optimal control problems for the reaction–diffusion–convection equation with variable coefficients

The solvability of optimal control problems is proved on both weak and strong solutions of a boundary value problem for the nonlinear reaction–diffusion–convection equation with variable coefficients. In the second case, the requirements for smoothness of the multiplicative control are reduced. The study of extremal problems is based on the proof of the solvability of the corresponding boundary value problems and on the qualitative analysis of their solutions properties. The large data existence results for weak solutions, the maximum principle as well as the local existence and uniqueness of a strong solution are established. Moreover, an optimal feedback control problem is considered. Using methods of the theory of topological degree for set-valued perturbations (with aspheric image sets) of generalized monotone operators, we obtain sufficient conditions for the solvability of this problem in the class of weak solutions.

Existence of multi‐dimensional contact discontinuities for the ideal compressible magnetohydrodynamics

AbstractWe establish the local existence and uniqueness of multi‐dimensional contact discontinuities for the ideal compressible magnetohydrodynamics (MHD) in Sobolev spaces, which are most typical interfacial waves for astrophysical plasmas and prototypical fundamental waves for hyperbolic systems of conservation laws. Such waves are characteristic discontinuities for which there is no flow across the discontinuity surface while the magnetic field crosses transversely, which lead to a two‐phase free boundary problem where the pressure, velocity and magnetic field are continuous across the interface whereas the entropy and density may have jumps. To overcome the difficulties of possible nonlinear Rayleigh–Taylor instability and loss of derivatives, here we use crucially the Lagrangian formulation and Cauchy's celebrated integral (1815) for the magnetic field. These motivate us to define two special good unknowns; one enables us to capture the boundary regularizing effect of the transversal magnetic field on the flow map, and the other one allows us to get around the troublesome boundary integrals due to the transversality of the magnetic field. In particular, our result removes the additional assumption of the Rayleigh–Taylor sign condition required by Morando, Trakhinin and Trebeschi (J. Differ. Equ. 258 (2015), no. 7, 2531–2571; Arch. Ration. Mech. Anal. 228 (2018), no. 2, 697–742) and holds for both 2D and 3D and hence gives a complete answer to the two open questions raised therein. Moreover, there is no loss of derivatives in our well‐posedness theory. The solution is constructed as the inviscid limit of solutions to some suitably‐chosen nonlinear approximate problems for the two‐phase compressible viscous non‐resistive MHD.

Inverse Problems for the Diffusion–Drift Model of Charging of an Inhomogeneous Polar Dielectric

The problems of reconstructing the unknown parameters of the model of electron-induced charging of an inhomogeneous polar dielectric from additional information about the volume charge density distribution and the electric field strength are studied. Within the optimization approach, these inverse problems are reduced to control problems and their solvability is proved. For extremum problems, optimality systems are derived and, based on their analysis, local uniqueness of the solution of one of the considered problems is proved. Taking into account the introduced characteristic of the inhomogeneity of the dielectric, auxiliary results on the solvability and properties of solutions of the boundary value problem, obtained earlier for the model of charging of a homogeneous dielectric, are corrected.

Local uniqueness of blow-up solutions for critical Hartree equations in bounded domain

In this paper we are interested in the following critical Hartree equation -Δu=(∫Ωu2μ∗(ξ)|x-ξ|μdξ)u2μ∗-1+εu,inΩ,u=0,on∂Ω,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} {\\left\\{ \\begin{array}{ll} -\\Delta u =\\displaystyle {\\Big (\\int _{\\Omega }\\frac{u^{2_{\\mu }^*} (\\xi )}{|x-\\xi |^{\\mu }}d\\xi \\Big )u^{2_{\\mu }^*-1}}+\\varepsilon u,~~~\ ext {in}~\\Omega ,\\\\ u=0,~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\ ext {on}~\\partial \\Omega , \\end{array}\\right. } \\end{aligned}$$\\end{document}where Nge 4, 0<mu le 4, varepsilon >0 is a small parameter, Omega is a bounded domain in mathbb {R}^N, and 2_{mu }^*=frac{2N-mu }{N-2} is the critical exponent in the sense of the Hardy–Littlewood–Sobolev inequality. By establishing various versions of local Pohozaev identities and applying blow-up analysis, we first investigate the location of the blow-up points for single bubbling solutions to above the Hartree equation. Next we prove the local uniqueness of the blow-up solutions that concentrates at the non-degenerate critical point of the Robin function for varepsilon small.

STRATEGI PENGEMBANGAN PEREKONOMIAN BADAN USAHA MILIK DESA (BUMDES) MELALUI KEARIFAN LOKAL PUJASERA DI DESA KETERUNGAN KECAMATAN KRIYAN – SIDOARJO

The purpose of this research is to formulate a strategy for developing BUMDes Pujasera based on local wisdom in alleviating poverty in the village of Ketungan, Kriyan sub-district - Sidoarjo. The research respondents were 18 heads of traditional markets in the village of Keterungan, Kriyan sub-district - Sidoarjo. This study uses qualitative research with a case study approach with SWOT analysis. The results of the study show that opportunities for traditional markets in the future are people's income levels, inflation growth rates, traditional market revitalization policies, security stability, social concern for traditional markets, community respect, changes in people's tastes, and adoption of information technology, while the elements that pose a threat are the existence of modern markets as competitors and population growth. The strengths in the future are product diversity, local product uniqueness, product quality, merchandise arrangement, flexible pricing, and strategic location. Weaknesses are the availability of parking areas, transaction processing, promotions, cooperation with the tourism industry, orderly site zoning, market cleanliness, availability of public facilities, and customer service. Based on the results of the SWOT analysis, it is necessary to maintain the position of the traditional market business in this digital era and the basis for its development strategy is the "Grow and Develop Strategy".

An inverse problem of reconstructing the refractive index interrain parabolic equations ∗ ∗ Supported by National Natural Science Foundation of China (Nos. 61663018, 11961042), and NSF of Gansu Province of China (No. 22JR5RA341).

This work deals with an inverse problem of identifying the refractive index in terrain parabolic equations, which has important applications in radio systems and other applied scientific fields. Unlike other inverse coefficient problems arising in the real field, the underlying mathematical model is discussed in the complex field which may lead to many mathematical tools not available, and the corresponding theoretical analysis rather difficult. Based on the optimal control framework, the recognition problem is transformed into an optimization problem, and the existence and necessary condition of the minimizer for the cost functional are deduced. The local uniqueness and stability of the minimizer are derived from the necessary condition. Finally, a gradient descent iterative scheme is designed to solve the numerical solution of the inverse problem. The validity of the proposed algorithm is illustrated by some typical numerical examples.

Stochastic quantization of laser propagation models

This paper identifies certain interesting mathematical problems of stochastic quantization type in the modeling of Laser propagation through turbulent media. In some of the typical physical contexts, the problem reduces to stochastic Schrödinger equation with space–time white noise of Gaussian or Poisson or Lévy type. We identify their mathematical resolution via stochastic quantization. Nonlinear phenomena such as Kerr effect can be modeled by a stochastic nonlinear Schrödinger equation in the focusing case with space–time white noise. A treatment of stochastic transport equation, the Korteweg–De Vries equation as well as a number of other nonlinear wave equations with space–time white noise is also given. The main technique is the S-transform (we will actually use the closely related Hermite transform) which converts the stochastic partial differential equation (PDE) with space–time white noise to a deterministic PDE defined on the Hida–Kondratiev white noise distribution space. We then utilize the inverse S-transform/Hermite transform known as the characterization theorem combined with the infinite-dimensional implicit function theorem for analytic maps to establish local existence and uniqueness theorems for path-wise solutions of this class of problems. The particular focus of this paper on singular white noise distributions is motivated by practical situations where the refractive index fluctuations in the propagation medium in space and time are intense due to turbulence, ionospheric plasma turbulence, marine-layer fluctuations, etc. Since a large class of PDEs, that arise in nonlinear wave propagation, have polynomial-type nonlinearities, white noise distribution theory is an effective tool in studying these problems subject to different types of white noises.

REVISITED CULTURAL HERITAGE &amp; ARCHITECTURAL HERITAGE PREFERENCES TO LEVERAGE SUSTAINABLE TOURISM PERFORMANCE: EMPIRICAL RESEARCH ON INDONESIA WORLD HERITAGE SITES

Present study investigated the local benevolence uniqueness (LBU) and Aesthetical Environmental Look (AEL) on Sustainable Tourism Performance (STP) mediated by Revisited Cultural Heritage Value (RCHV) &amp; Architectural Heritage Preferences (AHP. Data was gathering through survey with questionnaires based from January to June 2022, also questionnaires were distributed to email as well as other social media platform with google forms. Data collection is gained from the domestics’ tourists who have made traveling more than twice and visiting cultural heritage sites. 500 questionnaires were sent and 400 were returned and 383 could be used for further analysis. with non-purposive sampling and purposive sampling technique, respondents should have made traveling). Structural Equation Modeling (SEM) with AMOS 22 was deployed to analyze data. Statistical outputs demonstrated the research findings showing all the proposed hypotheses have significant impact on sustain tourism performance.

Local Existence and Uniqueness of Strong Solution to the Inhomogeneous Primitive Equations with Vacuum

Local Existence and Uniqueness of Strong Solution to the Inhomogeneous Primitive Equations with Vacuum

Existence and local uniqueness of normalized peak solutions for a Schrödinger-Newton system

In this paper, we investigate the existence and local uniqueness of normalized peak solutions for a Schr\"odinger-Newton system under the assumption that the trapping potential is degenerate and has non-isolated critical points. First we investigate the existence and local uniqueness of normalized single-peak solutions for the Schr\"odinger-Newton system. Precisely, we give the precise description of the chemical potential $\mu$ and the attractive interaction $a$. Then we apply the finite dimensional reduction method to obtain the existence of single-peak solutions. Furthermore, using various local Pohozaev identities, blow-up analysis and the maximum principle, we prove the local uniqueness of single-peak solutions by precise analysis of the concentrated points and the Lagrange multiplier. Finally, we also prove the nonexistence of multi-peak solutions for the Schr\"odinger-Newton system, which is markedly different from the corresponding Schr\"odinger equation. The nonlocal term results in this difference. The main difficulties come from the estimates on Lagrange multiplier, the different degenerate rates along different directions at the critical point of $P(x)$ and some complicated estimates involved by the nonlocal term. To our best knowledge, it may be the first time to study the existence and local uniqueness of solutions with prescribed $L^{2}$-norm for the Schr\"odinger-Newton system.