Nonlocal Evolution Research Articles

Some remarks on lattice and nonlocal dispersal evolution systems

<p style='text-indent:20px;'>In this paper, we first characterize the continuity of a map in the space <inline-formula><tex-math id="M1">\begin{document}$ \mathcal{C} = BC(\mathcal{H},\mathbb{R}^m) $\end{document}</tex-math></inline-formula> equipped with the compact open topology. Then we show that linear lattice and nonlocal dispersal equations generate uniformly continuous semigroups in the Banach space <inline-formula><tex-math id="M2">\begin{document}$ \mathcal{B} = BC(\mathcal{H},\mathbb{R}^m) $\end{document}</tex-math></inline-formula> equipped with the supremum norm. Finally, we illustrate how to prove nonlinear lattice and nonlocal dispersal equations generate monotone semiflows with respect to the compact open topology.</p>

The efficient ADI Galerkin finite element methods for the three-dimensional nonlocal evolution problem arising in viscoelastic mechanics

In this article, we investigate and analyze new methods for the numerical solution of the three-dimensional nonlocal evolution problem arising in viscoelastic mechanics. Then these methods combine Galerkin finite element methods (FEMs) for the spatial discretization with corresponding alternating direction implicit (ADI) algorithms, based on the backward Euler (BE) method and Crank-Nicolson (CN) method, respectively, from which, the Riemann-Liouville (R-L) integral term is approximated by relevant convolution quadrature rules. The $ L^2 $-norm stability and convergence of two ADI Galerkin schemes are proved by the energy argument. Numerical results confirm the predicted space-time convergence orders.

Existence of variational solutions to nonlocal evolution equations via convex minimization

We prove existence of variational solutions for a class of nonlocal evolution equations whose prototype is the double phase equation [see formula in PDF] The approach of minimization of parameter-dependent convex functionals over space-time trajectories requires only appropriate convexity and coercivity assumptions on the nonlocal operator. As the parameter tends to zero, we recover variational solutions. Under further growth conditions, these variational solutions are global weak solutions. Further, this provides a direct minimization approach to approximation of nonlocal evolution equations.

Stability analysis for nonlocal evolution equations involving infinite delays

Stability analysis for nonlocal evolution equations involving infinite delays

Nonlocal in-time telegraph equation and telegraph processes with random time

In this paper we study the properties of a non-markovian version of the telegraph process, whose non-markovian character comes from a nonlocal in-time evolution equation that is satisfied by its probability density function. In the first part of the paper, using the theory of Volterra integral equations, we obtain an explicit formula for its moments, and we prove that the Carleman condition is satisfied. This shows that the distribution of the process is uniquely determined by its moments. We also obtain an explicit formula for the moment generating function. In the second part of the paper, we prove that the distribution of this process coincides with the distribution of a process of the form T(|W(t)|) where T(t) is the classical telegraph process, and |W(t)| is a random time whose distribution is related to a nonlocal in-time version of the wave equation. To this end, we construct the probability density function via subordination from the distribution of the classic telegraph process. Our results exhibit a strong interplay between this type of processes and subdiffusion theory.

Machine learning of nonlocal micro-structural defect evolutions in crystalline materials

The presence and evolution of defects that appear in the manufacturing process play a vital role in the failure mechanisms of engineering materials. In particular, the collective behavior of dislocation dynamics at the mesoscale leads to avalanche, strain bursts, intermittent energy spikes, and nonlocal interactions producing anomalous features across different time- and length-scales, directly affecting plasticity, void and crack nucleation. Discrete Dislocation Dynamics (DDD) simulations are often used at the meso-level, but the cost and complexity increase dramatically with simulation time. To further understand how the anomalous features propagate to the continuum, we develop a probabilistic model for dislocation motion constructed from the position statistics obtained from DDD simulations. We obtain the continuous limit of discrete dislocation dynamics through a Probability Density Function for the dislocation motion, and propose a nonlocal transport model for the PDF. We develop a machine-learning framework to learn the parameters of the nonlocal operator with a power-law kernel, connecting the anomalous nature of DDD to the origin of its corresponding nonlocal operator at the continuum, facilitating the integration of dislocation dynamics into multi-scale, long-time material failure simulations.

The compact difference scheme for the fourth‐order nonlocal evolution equation with a weakly singular kernel

In this paper, we main discuss an efficient numerical algorithm for the fourth‐order nonlocal evolution equation with a weakly singular kernel. The second‐order fractional convolution quadrature rule and L1 method are proposed to approximate the Riemann–Liouville (R‐L) fractional integral term and the temporal Caputo derivative, respectively. In order to obtain a fully discrete method, the compact difference scheme is used to discretize the second‐order and fourth‐order spatial derivative. Further, two new approaches are adopted for stability analysis, and then the optimal error estimates in the discrete ‐norm and ‐norm are obtained. At last, we give three test problems to illustrate the validity of the methods.

Nonlocal semilinear evolution inclusions with time lag

In this paper, nonautonomous semilinear evolution inclusions with time lag and nonlocal initial conditions are studied. We define the so-called limit and weak solutions. We prove the existence of solutions when the multivalued part of the right-hand side and the initial functions are Lipschitz under natural conditions on the Lipschitz constants. Afterwards, under the same conditions, a variant of the important in optimal control relaxation theorem is proved.

Solutions to nonlocal evolution equations governed by non-autonomous forms and demicontinuous nonlinearities

We deal with the existence of solutions having L^2-regularity for a class of non-autonomous evolution equations. Associated with the equation, a general non-local condition is studied. The technique we used combines a finite dimensional reduction together with the Leray–Schauder continuation principle. This approach permits to consider a wide class of nonlinear terms by allowing demicontinuity assumptions on the nonlinearity.

Topological properties of solution sets for Riemann-Liouville fractional nonlocal delay control systems with noncompact semigroups and applications to approximate controllability

This paper investigates the topological properties of the mild solution set for a control system monitored by Riemann-Liouville fractional nonlocal delay evolution equations with noncompact semigroups. We first obtain the Rδ-property of the mild solution set by employing the method of the Hausdorff measure of noncompactness. Then, we apply the obtained result to show that the presented control problem admits a reachable invariant set under nonlinear perturbations. Furthermore, we apply the obtained result to characterize the approximate controllability of the presented control system. Finally, we provide an example to illustrate the application of the abstract results.

Existence of variational solutions to doubly nonlinear nonlocal evolution equations via minimizing movements

We prove existence of variational solutions for a class of doubly nonlinear nonlocal evolution equations whose prototype is the double phase equation $$\begin{aligned}&\partial _t u^m + P.V.\int _{\mathbb {R}^N} \frac{|u(x,t)-u(y,t)|^{p-2}(u(x,t)-u(y,t))}{|x-y|^{N+ps}}\\&\qquad \quad +a(x,y)\frac{|u(x,t)-u(y,t)|^{q-2}(u(x,t)-u(y,t))}{|x-y|^{N+qr}} \,\mathrm{d}y = 0,\,m>0,\,p>1,\,s,r\in (0,1). \end{aligned}$$We make use of the approach of minimizing movements pioneered by De Giorgi (in Boundary value problems for partial differential equations and applications, volume 29 of RMA Res. Notes Appl. Math., Masson, Paris, pp 81–98, 1993. https://mathscinet.ams.org/mathscinet-getitem?mr=1260440) and Ambrosio (Accademia Nazionale delle Scienze detta dei XL. Rendiconti. Serie V. Memorie di Matematica e Applicazioni. Parte I, 19:191–246) and refined by Bögelein, Duzaar, Marcellini and coauthors to study nonlinear parabolic equations with nonstandard growth.

Investigating existence results for fractional evolution inclusions with order r ∈ (1, 2) in Banach space

Abstract This manuscript investigates the issue of existence results for fractional differential evolution inclusions of order r ∈ (1, 2) in the Banach space. In the beginning, we analyze the existence results by referring to the fractional calculations, cosine families, multivalued function, and Martelli’s fixed point theorem. The result is also used to investigate the existence of nonlocal fractional evolution inclusions of order r ∈ (1, 2). Finally, a concrete application is given to illustrate our main results.

A numerical method for a nonlocal diffusion equation with additive noise

We consider a nonlocal evolution equation representing the continuum limit of a large ensemble of interacting particles on graphs forced by noise. The two principle ingredients of the continuum model are a nonlocal term and a Q-Wiener process describing the interactions among the particles in the network and stochastic forcing, respectively. The network connectivity is given by a square integrable function called a graphon. We prove that the initial value problem for the continuum model is well-posed. Further, we construct semidiscrete (discrete in space and continuous in time) and fully discrete schemes for the nonlocal model. The former is obtained by a discontinuous Galerkin method and the latter is based on further discretizing time using the Euler–Maruyama method. We prove convergence and estimate the rate of convergence in each case. For the semidiscrete scheme, the rate of convergence is expressed in terms of the regularity of the graphon, the Q-Wiener process, and the initial data. We work in generalized Lipschitz spaces, which allows us to treat models with data of lower regularity. This is important for applications as many interesting types of connectivity, including small-world and power-law, are expressed by graphons that are not smooth. The error analysis of the fully discrete scheme, on the other hand, reveals that for some models common in applied science, one has a higher speed of convergence than that predicted by the standard estimates for the Euler–Maruyama method. The rate of convergence analysis is supplemented with detailed numerical experiments, which are consistent with our analytical results. As a by-product, this work presents a rigorous justification for taking continuum limit for a large class of interacting dynamical systems on graphs subject to noise.

Three-Point Boundary Value Problems of Coupled Nonlocal Laplacian Equations

This paper discusses a kind of coupled nonlocal Laplacian evolution equation with Caputo time-fractional derivatives and proportional delays. Green function and mild solution are firstly established by employing the approach of eigenvalues’ expansions and Fourier analysis technique. By the properties of eigenvalues and Mittag–Leffler functions, several vital estimations of Green functions are presented. In view of these estimations and some appropriate assumptions, the existence and uniqueness of the mild solution are studied by utilizing the Leray–Schauder fixed-point theorem and the Banach fixed-point theorem. Finally, an example is provided to illustrate the effectiveness of our main results.

An efficient ADI difference scheme for the nonlocal evolution problem in three-dimensional space

An efficient ADI difference scheme for the nonlocal evolution problem in three-dimensional space

Structural Stability of Autonomous Semilinear Nonlocal Evolution Equations and the Related Semi-dynamical Systems

Structural Stability of Autonomous Semilinear Nonlocal Evolution Equations and the Related Semi-dynamical Systems

Existence, persistence, and continuation of solutions for a generalized 0-Holm-Staley equation

We consider a family of non-local evolution equations including the 0−Holm-Staley equation. We show that the family considered does not possess compactly supported solutions as long as the initial data is non-trivial. Also, we prove different unique continuation results for the solutions of the family studied. Persistence properties of the solutions are also investigated as well as we describe the scenario for the global existence of solutions of the 0−Holm-Staley equation. In particular, the proof of global existence of solutions as well as our demonstrations for unique continuation results of solutions partially answers some questions pointed out in Himonas and Thompson (2014) [19].

Damage-augmented nonlocal lattice particle method for fracture simulation of solids

Fracture of solids, particularly for ductile metallic materials, typically involves elastoplastic deformation and associated damaging processes. This paper proposes a general damage-augmented nonlocal lattice particle method (LPM) to model this coupled behavior. The concept of interchangeability between particle-wise and bond-wise properties in LPM is first introduced and validated. It is shown that tensors can naturally represent material state variables, which is rarely seen in most lattice methods. A tensor-based return-mapping algorithm based on implicit integration is thus implemented to simulate J2 plasticity. Next, the damage-augmented LPM is proposed to properly simulate the material deterioration by combining LPM with a nonlocal damage evolution rule. The proposed method can handle the brittle fracture and pure elastoplastic deformation and simulate ductile fracture phenomena with moderately large time steps. The particle-size/lattice dependency issues of macroscopic mechanical responses are reduced under the proposed framework. Numerical examples of predicting the elastoplastic behavior of engineering structures with/without damage and fracture are provided. Several conclusions and limitations of the proposed method are also discussed.

A discussion on boundary controllability of nonlocal impulsive neutral integrodifferential evolution equations

Sufficient conditions for boundary controllability of nonlocal impulsive neutral integrodifferential evolution equations are explored. In the first problem, the outcomes are proven with the help of Sadovskii's fixed point theorem collaborated with strongly continuous semigroup theory. In the second problem, we considered nonautonomous evolution equations, and the result is proven by employing Sadovskii's fixed point theorem in collaboration with the evolution system. Examples are included to demonstrate the theoretical results for both types of equations.

A nonlinear time-domain BEM for the performance of 3D flapping-wing thrusters in directional waves

Recent studies indicate that flapping-foil biomimetic systems can significantly improve ship propulsion in waves. In most cases horizontal wing(s) are used performing vertical oscillatory motion induced by ship heaving and pitching responses in waves, while foil's rotational motion is actively controlled. Aiming to the detailed investigation of the free-surface effects, a GPU-accelerated Boundary Element Method (GPU-BEM) has been developed for the hydrodynamic analysis of 3D lifting bodies in non-linear waves with an angle of incidence and motions. The nonlinear free-surface conditions are satisfied at the exact location of the free-surface boundary. A nonlinear pressure-type Kutta condition is applied at the trailing edge and the dynamic evolution of the free vortex wake is obtained by a time-stepping method. The free-surface boundary conditions and Kutta condition are used to set up the nonlinear and nonlocal evolution equations. For the treatment of the dynamical system, the present BEM is coupled to the dynamical equations and it is integrated in time domain with an iterative scheme. Results are presented for a few subproblems of special interest, such as the wave resistance and the enforced radiation problems of smooth immersed bodies and compared with other methods for verification. Subsequently, the unsteady biomimetic thruster is examined in unbounded domain, beneath the free surface in the absence of incident waves, and in the presence of obliquely incident waves, illustrating the performance of the system. The present method can be exploited for the systematic investigation and the design of flapping-foil thrusters for augmenting ship propulsion in waves.