Existence Theory Research Articles

Spatial Dynamics and Solitary Hydroelastic Surface Waves

This paper presents an existence theory for solitary waves at the interface between a thin ice sheet (modelled using the Cosserat theory of hyperelastic shells) and an ideal fluid (of finite depth and in irrotational motion). The theory takes the form of a review of the Kirchgässner reduction to a finite-dimensional Hamiltonian system, highlighting the refinements in the theory over the years and presenting some novel aspects including the use of a higher-order Legendre transformation to formulate the problem as a spatial Hamiltonian system, and a Riesz basis for the phase space to complete the analogy with a dynamical system. The reduced system is to leading order given by the focussing cubic nonlinear Schrödinger equation, agreeing with the result of formal weakly nonlinear theory (which is included for completeness). We give a precise proof of the persistence of two of its homoclinic solutions as solutions to the unapproximated reduced system which correspond to symmetric hydroeleastic solitary waves.

Existence Theorems for Hybrid Fractional Differential Equations with ψ -Weighted Caputo–Fabrizio Derivatives

In this study, two classes of hybrid boundary value problems involving ψ -weighted Caputo–Fabrizio fractional derivatives are considered. Based on the properties of the given operator, we construct the hybrid fractional integral equations corresponding to the hybrid fractional differential equations. Then, we establish and extend the existence theory for given problems in the class of continuous functions by Dhage’s fixed point theory. Furthermore, as special cases, we offer further analogous and comparable conclusions. Finally, we give two examples as applications to illustrate and validate the results.

Regularity of viscosity solutions to fully nonlinear elliptic transmission problems

We develop the regularity theory of viscosity solutions to transmission problems for fully nonlinear second order uniformly elliptic equations. Our results give a complete theory of existence, uniqueness, comparison principle, and regularity of solutions to flat interface transmission problems; and the C0,α, C1,α and C2,α regularity of viscosity solutions up to the transmission surface for the case of curved interfaces.

The existence theory of solution in Sobolev space for fractional differential equations

We study the solution theory in Sobolev space for fractional differential equations of Caputo type: CDxαu(x)=f(x,u),0<α<1, with given initial value. Under proper conditions, we establish the existence theorem of solution in space Wp1(0,b). We emphasize that our existence theorem has no any continuity requirement on f(x,u) with respect to x∈[0,b]. The uniqueness condition is also given. Several examples are provided to support our theoretical analysis.

On a multivalued prescribed mean curvature problem and inclusions defined on dual spaces

This article addresses two main objectives. First, it establishes a functional analytic framework and presents existence results for a quasilinear inclusion describing a prescribed mean curvature problem with homogeneous Dirichlet boundary conditions, involving a multivalued lower order term. The formulation of the problem is done in the space of functions with bounded variation. The second objective is to introduce a general existence theory for inclusions defined on nonreflexive Banach spaces, which is specifically applicable to the aforementioned prescribed mean curvature problem. This problem can be formulated as a multivalued variational inequality in the space of functions with bounded variation, which, under suitable conditions, is equivalent to an inclusion involving a maximal monotone mapping of type (D) and a generalized pseudomonotone mapping. We prove an abstract existence theorem for inclusions of this form, under some coercivity conditions involving both the maximal monotone and the generalized pseudomonotone mappings.

Global existence of weak solutions to a BGK model relaxing to the barotropic Euler equations

We establish the global-in-time existence of weak solutions to a variant of the BGK model proposed by Bouchut (1999) which leads to the barotropic Euler equations, where the pressure is given by p(ρ)=ργ with γ∈(1,1+2/(d+2)], in the hydrodynamic limit. For the existence of solutions, we deal with γ∈(1,3] for the one-dimensional case, and γ∈(1,1+2/(d+2)]∪{1+2/d} for the multi-dimensional case. In particular, our existence theory makes the quantified estimates of hydrodynamic limit from the BGK-type equations to the multi-dimensional barotropic Euler system discussed by Berthelin and Vasseur (2005) completely rigorous.

The compressible Navier–Stokes equations with slip boundary conditions of friction type

We study a mathematical model of a viscous compressible fluid obeying the slip boundary condition of friction type. We present a notion of weak solutions to this model, in which the momentum equation and the associated energy inequality are combined into a single relation. Moreover, the slip boundary condition of friction type is incorporated into this relation by the use of a boundary integral. Our main result proves the existence of such weak solutions. The proof of this result combines the classical existence theory for the compressible Navier–Stokes equations with an approximation of the aforementioned boundary integral via a convex regularization of the absolute value function.

Analysis of compressible bubbly flows. Part I: construction of a microscopic model

In this note, we introduce a microscopic model for the motion of gas bubbles in a viscous fluid. By interpreting a bubble as a compressible fluid with infinite shear viscosity, we derive a pde/ode system coupling the density/velocity/pressure in the surrounding fluid with the linear/angular velocities and radii of the bubbles. We provide a 1D analogue of the system and construct an existence theory for this simplified system in a natural regularity framework. The second part of the paper is a preparatory work for the derivation of an averaged or macroscopic model.

A variational theory for integral functionals involving finite-horizon fractional gradients

The center of interest in this work are variational problems with integral functionals depending on nonlocal gradients with finite horizon that correspond to truncated versions of the Riesz fractional gradient. We contribute several new aspects to both the existence theory of these problems and the study of their asymptotic behavior. Our overall proof strategy builds on finding suitable translation operators that allow to switch between the three types of gradients: classical, fractional, and nonlocal. These provide useful technical tools for transferring results from one setting to the other. Based on this approach, we show that quasiconvexity, which is the natural convexity notion in the classical calculus of variations, gives a necessary and sufficient condition for the weak lower semicontinuity of the nonlocal functionals as well. As a consequence of a general varGamma -convergence statement, we obtain relaxation and homogenization results. The analysis of the limiting behavior for varying fractional parameters yields, in particular, a rigorous localization with a classical local limit model.

Differential inclusions for the Schouten tensor and nonlinear eigenvalue problems in conformal geometry

Let g0 be a smooth Riemannian metric on a closed manifold Mn of dimension n≥3. We study the existence of a smooth metric g conformal to g0 whose Schouten tensor Ag satisfies the differential inclusion λ(g−1Ag)∈Γ on Mn, where Γ⊂Rn is a cone satisfying standard assumptions. Inclusions of this type are often assumed in the existence theory for fully nonlinear elliptic equations in conformal geometry. We assume the existence of a continuous metric g1 conformal to g0 satisfying λ(g1−1Ag1)∈Γ′‾ in the viscosity sense on Mn, together with a nondegenerate ellipticity condition, where Γ′=Γ or Γ′ is a cone slightly smaller than Γ. In fact, we prove not only the existence of metrics satisfying such differential inclusions, but also existence and uniqueness results for fully nonlinear eigenvalue problems for the Schouten tensor. We also give a number of geometric applications of our results. We show that the sign of a nonlinear eigenvalue for the σ2 operator is a conformal invariant in three dimensions. We also give a generalisation of a theorem of Aubin & Ehrlick on pinching of the Ricci curvature, and an application in the study of Green's functions for fully nonlinear Yamabe problems.

Job satisfaction and influencing factors among young medical staff based on ERG theory

Objective: On the basis of the existence, relatedness, and growth (ERG) theory, this study examines job satisfaction and its influencing factors among young medical staff at a public grade A tertiary general hospital in a prefecture level city, and explores corresponding management strategies according to the factual evidence. Methods: An electronic questionnaire survey was conducted on a random sample of 500 young medical staff at the hospital. Descriptive statistical analysis and single-factor variance analysis were performed. Results: The satisfaction of survival needs among young medical staff is reflected in their work itself and remuneration; relatedness needs are reflected by relationships with colleagues,organizational management, hospital culture, and professional identity; and growth needs are reflected in professional development. Conclusion: To address the ERG needs of young medical staff and improve their satisfaction, hospitals may consider optimizing the practice environment, boosting remuneration, improving internal communications, deepening hospital culture construction, and strengthening employee career management.

Existence theory and numerical simulations of variable order model of infectious disease

This study uses variable order differentiation and integration to investigate the disease dynamical model of COVID-19. Here, we update the results of the qualitative and quantitative analysis. We obtain necessary conclusions for the existence theory of the solution to the suggested model in order to satisfy the aforementioned criteria using fixed point theories of Banach and Schauder. Additionally, we simulate the outcomes mathematically and graphically using the Euler modified technique for numerical purposes. There are several graphs provided that relate to various variable ordering. In addition, we compare our simulated results with the real data results also in case of infected class.

Well-posedness and qualitative analysis of a SEIR model with spatial diffusion for COVID-19 spreading

In this paper, we study the well-posedness and the qualitative behavior of equilibria of a SEIR epidemic models with spatial diffusion for the spreading of COVID-19. The well-posedness of the model is proved using both the Semigroup Theory of sectorial operators and existence results for abstract parabolic differential equations. The asymptotical local stability of both disease-free and endemic equilibria are established using standard linearization theory, and confirmed by illustrative numerical simulations. The asymptotical global stability of both disease-free and endemic equilibria are established using a Lyapunov function.

P-harmonic functions with nonlinear Neumann conditions on the sphere and measure data

We study renormalized solutions to the problem 1\\right\\rbrace \\\\ \\left\\lvert {\ abla u} \\right\\rvert^{p-2}u_\ u + g(u) = \\mu & \\mbox{ on } \\left\\lbrace x \\in \\mathbb{R}^N : \\left\\lvert {x} \\right\\rvert=1\\right\\rbrace \\end{cases} \\end{align*} $$]]> { − Δ p u = 0 in { x ∈ R N : | x | > 1 } | ∇ u | p − 2 u ν + g ( u ) = μ on { x ∈ R N : | x | = 1 } where 1<p<N, N ≥ 2 , u ν is the normal derivative of u, μ is a bounded Radon measure, and g : R → R is a continuous function. We consider first the case when g is trivial where, by making use of the symmetries of the problem and the stability of renormalized solutions, we obtain an existence theory. Afterwards, we show existence of solutions for different types of nonlinearities, including power functions. We also provide a characterization of removable sets for the problem with source.

Existence theory for nabla fractional three-point boundary value problems via continuation methods for contractive maps

In this article, we analyse an $\alpha$-th order, $1 &lt; \alpha \leq 2$, nabla fractional three-point boundary value problem (BVP). We construct the Green's function associated to this problem and derive a few of its important properties. We then establish sufficient conditions on existence and uniqueness of solutions for the corresponding nonlinear BVP using the modern ideas of continuation methods for contractive maps. Our results extend recent results on nabla fractional BVPs. Finally, we provide an example to illustrate the applicability of main results.

Existence theory and stability analysis to a coupled nonlinear fractional mixed boundary value problem

In this manuscript, we conclude a comprehensive approach to a class of nonlinear coupled system of fractional differential equations with mixed type boundary value conditions. Subsequently, the solution of coupled system exists and unique under mixed type boundary value conditions with the reference of Schaefer and Banach fixed-point theorems. Further, we developed the Hyers- Ulam stability for the considered problem. Finally, we set an example for the support of our results.

Minimizers of nonlocal polyconvex energies in nonlocal hyperelasticity

Abstract We develop a theory of existence of minimizers of energy functionals in vectorial problems based on a nonlocal gradient under Dirichlet boundary conditions. The model shares many features with the peridynamics model and is also applicable to nonlocal solid mechanics, especially nonlinear elasticity. This nonlocal gradient was introduced in an earlier work, inspired by Riesz’ fractional gradient, but suitable for bounded domains. The main assumption on the integrand of the energy is polyconvexity. Thus, we adapt the corresponding results of the classical case to this nonlocal context, notably, Piola’s identity, the integration by parts of the determinant and the weak continuity of the determinant. The proof exploits the fact that every nonlocal gradient is a classical gradient.

Solutions of spinorial Yamabe-type problems on 𝑆^{𝑚}: Perturbations and applications

This paper is part of a program to establish the existence theory for the conformally invariant Dirac equation \[ D g ψ = f ( x ) | ψ | g 2 m − 1 ψ D_g\psi =f(x)|\psi |_g^{\frac 2{m-1}}\psi \] on a closed spin manifold ( M , g ) (M,g) of dimension m ≥ 2 m\geq 2 with a fixed spin structure, where f : M → R f:M\to \mathbb {R} is a given function. The study on such nonlinear equation is motivated by its important applications in Spin Geometry: when m = 2 m=2 , a solution corresponds to an isometric immersion of the universal covering M ~ \widetilde M into R 3 \mathbb {R}^3 with prescribed mean curvature f f ; meanwhile, for general dimensions and f ≡ c o n s t a n t f\equiv constant , a solution provides an upper bound estimate for the Bär-Hijazi-Lott invariant. Comparing with the existing issues, the aim of this paper is to establish multiple existence results in a new geometric context, which have not been considered in the previous literature. Precisely, in order to examine the dependence of solutions of the aforementioned nonlinear Dirac equations on geometrical data, concrete analyses are made for two specific models on the manifold ( S m , g ) (S^m,g) : the geometric potential f f is a perturbation from constant with g = g S m g=g_{S^m} being the canonical round metric; and f ≡ 1 f\equiv 1 with the metric g g being a perturbation of g S m g_{S^m} that is not conformally flat somewhere on S m S^m . The proof is variational: solutions of these problems are found as critical points of their corresponding energy functionals. The emphasis is that the solutions are always degenerate: they appear as critical manifolds of positive dimension. This is very different from most situations in elliptic PDEs and classical critical point theory. As corollaries of the existence results, multiple distinct embedded spheres in R 3 \mathbb {R}^3 with a common mean curvature are constructed, and furthermore, a strict inequality estimate for the Bär-Hijazi-Lott invariant on S m S^m , m ≥ 4 m\geq 4 , is derived, which is the first result of this kind in the non-locally-conformally-flat setting.

Solving existence results in multi-term fractional differential equations via fixed points

Many researchers are interested in the existence theory of solutions to fractional differential equations. In the literature, existence results have been obtained by using a variety of fixed point problems, including the fixed-point problems of Lefschetz, Kleene, Tychonoff, and Banach. In this article, we propose a generalized version of the contraction principle in the context of controlled rectangular metric space. With this result, we address the existence and uniqueness results for the following fractional-order differential equations.1. The nonlinear multi-term fractional delay differential equation L(D)ζ(ϖ)=σ(ϖ,ζ(ϖ),ζ(ϖ−τ)),ϖ∈J=[0,T],T>0;ζ(ϖ)=σ̄(ϖ),ϖ∈[−τ,0]. where, L(D)=γwcDδw+γw−1cDδw−1+...+γ1cDδ1+γ0cDδ0,γ♭∈R(♭=0,1,2,3....w),γw≠0,0≤δ0<δ1<δ2....δw−1<δw<1, and cDδ denotes the Caputo fractional derivative of order δ.2. The Caputo type fractional differential equation cDδζ(ϖ)=σ(ϖ,ζ(ϖ)),5<δ≤6,ϖ∈[0,T], with ζ(0)=−ζ(T),ζ′(0)=−ζ′(T),ζ′′(0)=−ζ′′(T),ζ′′′(0)=−ζ′′′(T),ζ4(0)=−ζ4(T),ζ5(0)=−ζ5(T)

Solvability of nonlinear diffusion models with flux at the boundary driven by nonmonotone local and memory interactions

We investigate local and global solvability for nonlinear diffusion models with boundary flux driven by competing local and memory interactions. New local existence and comparison theories are developed for such models which are not amenable to a-priori arguments for nonnegative solutions, as well as not permitting standard monotonicity techniques of analysis. A transformation used to address nonnegativity subsequently results in classical solutions for strictly positive initial states. An alternating construction introduced to obtain solutions via monotone limits from above and below is somewhat reminiscent of the method of upper and lower solutions. Subsolution/supersolution comparison results that follow are applied to analyze boundedness, positivity, and blow up in finite time of solutions for power law cases of localized absorption and memory driven dissipation at the boundary. This connects the present work with our previous analyses of models having memory driven absorption at the boundary, shown to exactly parallel known results in cases of localized absorption, and begins to reveal comparative strengths of each.