Nonlocal Term Research Articles

Crank-Nicolson finite element methods for nonlocal problems with p -Laplace-type operator

A theoretical analysis of a Crank-Nicolson Galerkin finite element method for a class of nonlinear nonlocal diffusion problems associated with p -Laplace-type operator is presented here. It is shown, by a rigorous analysis that the unconditionally optimal error estimates for the fully discrete scheme are established. The presence of the nonlocal term in the models destroys the sparsity of the Jacobian matrices when solving the problem numerically using finite element method and Newton-Raphson method. As a consequence, computations consume more time and space in contrast to local problems. To overcome this difficulty, a new algorithm is proposed to avoid the full Jacobian matrix. Finally, some numerical simulations are presented to illustrate our theoretical analysis.

Stationary Pattern and Global Bifurcation for a Predator–Prey Model with Prey-Taxis and General Class of Functional Responses

This paper will explore a predator–prey model that incorporates prey-taxis and a general functional response in a bounded domain. Firstly, we will examine the stability and pattern formation of both local and nonlocal models. Our main finding is that the inclusion of nonlocal terms enhances linear stability, and the system can generate patterns due to the effects of prey-taxis. Secondly, we consider the nonlinear prey-taxis as the bifurcation parameter in order to analyze the global bifurcation of this model. Specifically, we identify a branch of nonconstant solutions that emerges from the positive constant solution when the prey-tactic sensitivity is repulsive. Finally, we will validate the effectiveness of the theoretical conclusions using numerical simulation methods.

Mass concentration for Ergodic Choquard Mean-Field Games

We study concentration phenomena in the vanishing viscosity limit for second-order stationary Mean-Field Games systems defined in the whole space RN with Riesz-type aggregating coupling and external confining potential. In this setting, every player of the game is attracted toward congested areas and the external potential discourages agents to be far away from the origin. Assuming some restrictions on the strength of the attractive nonlocal term depending on the growth of the Hamiltonian, we study the asymptotic behavior of solutions in the vanishing viscosity limit. First, we obtain existence of classical solutions to potential-free MFG systems with Riesz-type coupling. Secondly, we prove concentration of mass around minima of the potential.

Reconciling absence of vDVZ discontinuity with absence of ghosts in nonlocal linearized gravity

The modern massive gravity theories resolve a historical tension between the absence of the so called vDVZ mass discontinuity and the absence of ghosts via a fine tuned gravitational potential and a sophisticated screening mechanism. Those theories have originated the modern covariant bimetric models which are local, ghost free and cosmologically viable apparently, they contain a massive plus a massless graviton in the spectrum. It seems hard to solve the mentioned tension if we do insist in a model with a minimal number of degrees of freedom, with only one massive spin-2 particle in the spectrum, even if we allow nonlocal theories. Here we show that this problem can be circumvented in linearized nonlocal theories by the introduction of exponential terms with infinite derivatives. The model admits non linear completions via nonlocal quadratic terms in curvatures. We also investigate the role of the exponential factors in linearized models where the graviton remains massless and a mass scale is introduced via nonlocal terms, they are also ghost free and approach the Einstein–Hilbert theory as we go much above the introduced mass scale.

Asymptotic profiles for a nonlinear Kirchhoff equation with combined powers nonlinearity

We study asymptotic behavior of positive ground state solutions of the nonlinear Kirchhoff equation −(a+b∫RN|∇u|2)Δu+λu=uq−1+up−1inRN,(Pλ)as λ→0 and λ→+∞, where N=3 or N=4, 2<q≤p≤2∗, 2∗=2NN−2 is the Sobolev critical exponent, a>0, b≥0 are constants and λ>0 is a parameter. In particular, we prove that in the case 2<q<p=2∗, as λ→0, after a suitable rescaling the ground state solutions of (Pλ) converge to the unique positive solution of the equation −Δu+u=uq−1 and as λ→+∞, after another rescaling the ground state solutions of (Pλ) converge to a particular solution of the critical Emden–Fowler equation −Δu=u2∗−1. We establish a sharp asymptotic characterization of such rescalings, which depends in a non-trivial way on the space dimension N=3 and N=4. We also discuss a connection of our results with a mass constrained problem associated to (Pλ) with normalization constraint ∫RN|u|2=c2. As a consequence of the main results, we obtain the existence, non-existence and asymptotic behavior of positive normalized solutions of such a problem. In particular, we obtain the exact number and their precise asymptotic expressions of normalized solutions if c>0 is sufficiently large or sufficiently small. Our results also show that in the space dimension N=3, there is a striking difference between the cases b=0 and b≠0. More precisely, if b≠0, then both p0≔10/3 and pb≔14/3 play a role in the existence, non-existence, the exact number and asymptotic behavior of the normalized solutions of the mass constrained problem, which is completely different from those for the corresponding nonlinear Schrödinger equation and which reveals the special influence of the nonlocal term.

Stable solutions to double phase problems involving a nonlocal term – erratum

Stable solutions to double phase problems involving a nonlocal term – erratum

The numerical solution of the free-boundary cell motility problem

The cell motility problem has been investigated for a long time. Today, many biologists, physicists, and mathematicians are looking for new research instruments for this process. A simple 2D model of a free-boundary cell moving on a homogeneous isotropic surface is presented in the paper. It describes the dynamics of the complex actomyosin liquid, whose special properties influence the boundary dynamics and cell motility. The model consists of a system of equations with the free boundary domain and contains a non-local term. In this work, we present a numerical solution of this problem.

Stable solutions to double phase problems involving a nonlocal term

AbstractIn this paper, we study weak solutions, possibly unbounded and sign-changing, to the double phase problem \begin{equation*} -\text{div} (|\nabla u|^{p-2} \nabla u + w(x)|\nabla u|^{q-2} \nabla u) = \left(\frac{1}{|x|^{N-\mu}}*f|u|^r\right) f(x)|u|^{r-2}u \quad\text{in}\ \mathbb{R}^N, \end{equation*} where $q\ge p\ge2$, r &gt; q, $0 \lt \mu \lt N$ and $w,f \in L^1_{\rm loc}(\mathbb{R}^N)$ are two non-negative functions such that $w(x) \le C_1|x|^a$ and $f(x) \ge C_2|x|^b$ for all $|x| \gt R_0$, where $R_0,C_1,C_2 \gt 0$ and $a,b\in\mathbb{R}$. Under some appropriate assumptions on p, q, r, µ, a, b and N, we prove various Liouville-type theorems for weak solutions which are stable or stable outside a compact set of $\mathbb{R}^N$. First, we establish the standard integral estimates via stability property to derive the non-existence results for stable weak solutions. Then, by means of the Pohožaev identity, we deduce the Liouville-type theorem for weak solutions which are stable outside a compact set.

A lattice Boltzmann model for the interface tracking of immiscible ternary fluids based on the conservative Allen-Cahn equation

A lattice Boltzmann (LB) model for the interface tracking of immiscible ternary fluids is proposed based on the conservative Allen-Cahn (A-C) equation. A nonlocal equilibrium distribution function source term is adopted in the LB equation to eliminate the unwanted numerical items. With the new LB equation and the carefully designed equilibrium distribution function, Chapman-Enskog analysis shows that the present model can correctly recover the conservative A-C equation with second-order accuracy. A series of benchmark cases have been used to validate the accuracy and stability of the present model, including the classical diagonal translation of two concentric circular interfaces, the rigid-body rotation of Zalesak's disk, and the periodic shear deformation of two tangent circular interfaces. By coupling the dynamic LB model for Navier-Stokes equations, the stationary droplets, the spreading of a liquid lens, the spinodal decomposition, and the Raleigh-Taylor instability have also been simulated in the framework of ternary fluids. Utilizing the geometric interpolation method, the wettability effect of the horizontal boundary is considered and numerical tests with different wall contact angles have been performed to verify the applicability of the present model. All the numerical tests show that the present model can capture the interfaces among the immiscible ternary fluids accurately. Furthermore, using fewer discrete velocity directions, the present model shows higher numerical accuracy and stability compared with other existing numerical models. Therefore, it is promising to improve computational efficiency.

Data-driven modelling and dynamic analysis of the multistable energy harvester with non-Gaussian Lévy noise

In engineering, due to the complex structural characteristics of system and the non-Gaussian properties of random excitation, it is difficult to establish an accurate stochastic dynamic model for the strongly nonlinear multistable vibration energy harvester (VEH), especially for these driven by non-Gaussian Lévy noise. From the view of machine learning, a data-driven model identification method is devised to extract the non-Gaussian governing laws of the multistable VEH with the aid of the observed sample trajectory data. Based on the Nonlocal Kramers-Moyal formulas, the Lévy, drift and diffusion terms can be approximately expressed by the sample trajectories of the system. By implementing the least square method and the stepwise sparse regressor algorithm, the optimal drift and diffusion coefficients can be identified, and then the non-Gaussian stochastic differential equation of VEH is extracted. Two examples are utilized to verify the feasibility and effectiveness of the data-driven modelling method in VEH, which indicates that the identified results agree well with the original system. Finally, the stochastic dynamic behaviors induced by non-Gaussian Lévy noise are explored based on the data-driven penta-stable VEH. The proposed method can provide the theoretical guidance for the modelling and dynamics research of VEH in engineering.

Optimal Control of Nonlocal Continuity Equations: Numerical Solution

The paper addresses an optimal ensemble control problem for nonlocal continuity equations on the space of probability measures. We admit the general nonlinear cost functional, and an option to directly control the nonlocal terms of the driving vector field. For this problem, we design a descent method based on Pontryagin’s maximum principle (PMP). To this end, we derive a new form of PMP with a decoupled Hamiltonian system. Specifically, we extract the adjoint system of linear nonlocal balance laws on the space of signed measures and prove its well-posedness. As an implementation of the designed descent method, we propose an indirect deterministic numeric algorithm with backtracking. We prove the convergence of the algorithm and illustrate its modus operandi by treating a simple case involving a Kuramoto-type model of a population of interacting oscillators.

Modeling plant water deficit by a non-local root water uptake term in the unsaturated flow equation

In this paper we present a novel way to mathematically frame the concept of ecological memory of plant water stress in the context of root water uptake in unsaturated flow equations. Inspired by recent eco-hydrological papers, we model the water absorption by roots with a non-local sink term, accounting also for a memory effect. In order to model such a memory effect, an integral equation is defined; the main purpose of this work is to provide sufficient conditions on the functions at play for ensuring existence and uniqueness of its solution. Finally, tailored numerical methods are implemented, and numerical simulations are also provided.

Nonlinear elasticity with vanishing nonlocal self-repulsion

We prove that for nonlinear elastic energies with strong enough energetic control of the outer distortion of admissible deformations, almost everywhere global invertibility as constraint can be obtained in the $\Gamma$ -limit of the elastic energy with an added nonlocal self-repulsion term with asymptocially vanishing coefficient. The self-repulsion term considered here formally coincides with a Sobolev–Slobodeckiĭ seminorm of the inverse deformation. Variants near the boundary or on the surface of the domain are also studied.

Orbital-free QM/MM simulation combined with a theory of solutions.

In a recent study, we developed a kinetic-energy density functional that can be utilized in orbital-free quantum mechanical/molecular mechanical (OF-QM/MM) simulations. The functional includes the nonlocal term constructed from the response function of the reference system of the QM solute. The present work provides a method to combine the OF-QM/MM with a theory of solutions based on the energy representation to compute the solvation free energy of the QM solute in solution. The method is applied to the calculation of the solvation free energy Δμ of a QM water solute in an MM water solvent. It is demonstrated that Δμ is computed as -7.7 kcal/mol, in good agreement with an experimental value of -6.3 kcal/mol. We also develop a theory to map the free energy δμ due to electron density polarization onto the coordinate space of electrons. The free energy density obtained by the free-energy mapping for the QM water clarifies that each hydrogen atom makes a positive contribution (+34.7 kcal/mol) to δμ, and the oxygen atom gives the negative free energy (-71.7 kcal/mol). It is shown that the small polarization free energy -2.4 kcal/mol is generated as a result of the cancellation of these counteracting energies. These analyses are made possible by the OF-QM/MM approach combined with a statistical theory of solutions.

Nonlocal Mechanistic Models in Ecology: Numerical Methods and Parameter Inferences

Animals utilize their surroundings to make decisions on how to navigate and establish their territories. Some species gather information about competing groups by observing them from a distance, detecting scent markings, or relying on memories of encounters with rival populations. Gathering such information involves a nonlocal process, prompting the development of mechanistic models that incorporate nonlocal terms to explore species movement. These models, however, pose analytical and computational challenges. In this study, we focus on a multi-species advection–diffusion model that incorporates nonlocal advection. To efficiently compute solutions for this system involving a large number of interacting species, we introduce a numerical scheme using spectral methods. Additionally, we examine the influence of various parameters and interaction potentials on population densities. Our investigation aims to provide a method to identify the primary factors driving species movements, and we validate our approach using synthetic data.

Normalized solutions for nonlinear Kirchhoff type equations with low-order fractional Laplacian and critical exponent

In this paper, we study the following fractional Kirchhoff equation with critical growth a+b∫R3|(−Δ)s2u|2dx(−Δ)su+λu=μ|u|q−2u+|u|2s∗−2uinR3,under the constraint ∫R3|u|2dx=c2, where s∈(0,1), a,b,c>0, 2<q<2s∗, μ>0, and λ∈R appears as a Lagrange multiplier. The equation has been extensively studied in the case s∈(34,1). In contrast, no existence result of normalized solutions is available for the case s∈(0,34]. Because the complicated competition between the Kirchhoff nonlocal term and the Sobolev critical term, such problem cannot be studied by applying standard variational methods, even by restricting its underlying energy functional on the Pohožaev manifold. In this paper, by using appropriate transform, we first get the equivalent system of the above problem. With the equivalence result, we obtain the nonexistence, existence and multiplicity of normalized solutions in the case s∈(0,34].

Solving stochastic equations with unbounded nonlinear perturbations

This paper is interested in semilinear stochastic equations having unbounded nonlinear perturbations in the deterministic part and/or in the random part. Moreover, the linear part of these equations is governed by a not necessarily analytic semigroup. The main difficulty with these equations is how to define the concept of mild solutions due to the chosen type of unbounded perturbations. To overcome this problem, we first proved a regularity property of the stochastic convolution with respect to the domain of ‘admissible’ unbounded linear operators (not necessarily closed or closable). This is done using Yosida extensions of such unbounded linear operators. After proving the well-posedness of these equations, we also establish the Feller property for the corresponding transition semigroups. Several examples like heat equations and Schrödinger equations with nonlocal perturbations terms are given. Finally, we give an application to a general class of semilinear neutral stochastic equations.

Critical mass capacity for two-dimensional Keller–Segel model with nonlocal reaction terms

This paper deals with the parabolic–elliptic Keller–Segel system on , involving a source term of logistic type defined in terms of the mass capacity M 0 and the total mass of the individuals. We exhibit that the qualitative behaviour of solutions is decided by the mass capacity M 0 and the initial mass m 0. For general solutions, the existence of a global weak solution is proved under the assumption that both M 0 and m 0 are less than 8π, whereas there exist solutions blowing up in finite time under the hypotheses of either with any integrable initial data or accompanied with large initial data. Moreover, gives rise to a compromise that solutions exist globally and blow up as time goes to infinity. For radially symmetric solutions, we introduce a strategy of relegating the lack of mass conservation via a transformation to the density and then obtain that there are stationary solutions given by with λ > 0. Subsequently we prove that if the initial data is strictly below for some λ > 0, then the solution vanishes in as . If the initial data is strictly above for some λ > 0, then the solution either blows up in finite time or has a mass concentration at the origin as time goes to infinity. Finally, our results are complemented by numerical simulations that demonstrate the asymptotic behaviour of solutions.

Dynamics analysis of a nonlocal diffusion dengue model

Due to the unrestricted movement of humans over a wide area, it is important to understand how individuals move between non-adjacent locations in space. In this research, we introduce a nonlocal diffusion introduce for dengue, which is driven by integral operators. First, we use the semigroup theory and continuously Fréchet differentiable to demonstrate the existence, uniqueness, positivity and boundedness of the solution. Next, the global stability and uniform persistence of the system are proved by analyzing the eigenvalue problem of the nonlocal diffusion term. To achieve this, the Lyapunov function is derived and the comparison principle is applied. Finally, numerical simulations are carried out to validate the results of the theorem, and it is revealed that controlling the disease’s spread can be achieved by implementing measures to reduce the transmission of the virus through infected humans and mosquitoes.

Multiplicity of positive solutions for the fractional Schrödinger–Poisson system with critical nonlocal term

This paper deals with the following fractional Schrödinger–Poisson system: [Formula: see text] with multiple competing potentials and a critical nonlocal term, where [Formula: see text], [Formula: see text] or [Formula: see text], and [Formula: see text] is the fractional critical exponent. By combining the Nehari manifold analysis and the Ljusternik–Schnirelmann category theory, we establish how the coefficient [Formula: see text] of the nonlocal critical nonlinearity affects the number of positive solutions. We propose a new relation between the number of positive solutions and the category of the global maximal set of [Formula: see text].