Nonlocal Term Research Articles

Existence and asymptotic behavior for L2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^2$$\\end{document}-norm preserving nonlinear heat equations

We consider a nonlinear parabolic equation with a nonlocal term which preserves the L2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^2$$\\end{document}-norm of the solution. We study the local and global well-posedness on a bounded domain, as well as the whole Euclidean space, in H1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$H^1$$\\end{document}. Then we study the asymptotic behavior of solutions. In general, we obtain weak convergence in H1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$H^1$$\\end{document} to a stationary state. For a ball, we prove strong convergence to the ground state when the initial condition is positive.

PT-symmetric PINN for integrable nonlocal equations: Forward and inverse problems.

Since the PT-symmetric nonlocal equations contain the physical information of the PT-symmetric, it is very appropriate to embed the physical information of the PT-symmetric into the loss function of PINN, named PTS-PINN. For general PT-symmetric nonlocal equations, especially those equations involving the derivation of nonlocal terms due to the existence of nonlocal terms, directly using the original PINN method to solve such nonlocal equations will face certain challenges. This problem can be solved by the PTS-PINN method, which can be illustrated in two aspects. First, we treat the nonlocal term of the equation as a new local component so that the equation is coupled at this time. In this way, we successfully avoid differentiating nonlocal terms in neural networks. On the other hand, in order to improve the accuracy, we make a second improvement, which is to embed the physical information of the PT-symmetric into the loss function. Through a series of independent numerical experiments, we evaluate the efficacy of PTS-PINN in tackling the forward and inverse problems for the nonlocal NLS equation, the nonlocal derivative NLS equation, the nonlocal (2+1)-dimensional NLS equation, and the nonlocal three-wave interaction systems. The numerical experiments demonstrate that PTS-PINN has good performance. In particular, PTS-PINN has also demonstrated an extraordinary ability to learn large space-time scale rogue waves for nonlocal equations.

On the singular limit problem in nonlocal balance laws: Applications to nonlocal lane-changing traffic flow models

We present a convergence result from nonlocal to local behavior for a system of nonlocal balance laws. The velocity field of the underlying conservation laws is diagonal. In contrast, the coupling to the remaining balance laws involves a nonlinear right-hand side that depends on the solution, nonlocal term, and other factors. The nonlocal operator integrates the density around a specific spatial point, which introduces nonlocality into the problem. Inspired by multi-lane traffic flow modeling and lane-changing, the nonlocal kernel is discontinuous and only looks downstream. In this paper, we prove the convergence of the system to the local entropy solutions when the nonlocal operator (chosen to be of an exponential type for simplicity) converges to a Dirac distribution. Numerical illustrations that support the main results are also presented.

A planar Kirchhoff equation with exponential growth and double nonlocal term

We investigate the existence of a positive solution for an elliptic problem of the Kirchhoff type involving a nonlocal operator. The nonlinearity considered in the equation combined a nonlocal term with an exponential growth governed by the Trudinger-Moser inequality.

Cauchy Problem with Summable Initial-Value Functions for Parabolic Equations with Translated Potentials

We study the Cauchy problem for differential–difference parabolic equations with potentials undergoing translations with respect to the spatial-independent variable. Such equations are used for the modeling of various phenomena not covered by the classical theory of differential equations (such as nonlinear optics, nonclassical diffusion, multilayer plates and envelopes, and others). From the viewpoint of the pure theory, they are important due to crucially new effects not arising in the case of differential equations and due to the fact that a number of classical methods, tools, and approaches turn out to be inapplicable in the nonlocal theory. The qualitative novelty of our investigation is that the initial-value function is assumed to be summable. Earlier, only the case of bounded (essentially bounded) initial-value functions was investigated. For the prototype problem (the spatial variable is single and the nonlocal term of the equation is single), we construct the integral representation of a solution and show its smoothness in the open half-plane. Further, we find a condition binding the coefficient at the nonlocal potential and the length of its translation such that this condition guarantees the uniform decay (weighted decay) of the constructed solution under the unbounded growth of time. The rate of this decay (weighted decay) is estimated as well.

Logarithmic or algebraic: Roughening of an active Kardar-Parisi-Zhang surface.

The Kardar-Parisi-Zhang (KPZ) equationsets the universality class for growing and roughening of nonequilibrium surfaces without any conservation law and nonlocal effects. We argue here that the KPZ equationcan be generalized by including a symmetry-permitted nonlocal nonlinear term of active origin that is of the same order as the one included in the KPZ equation. Including this term, the 2D active KPZ equationis stable in some parameter regimes, in which the interface conformation fluctuations exhibit sublogarithmic or superlogarithmic roughness, with nonuniversal exponents, giving positional generalized quasi-long-ranged order. For other parameter choices, the model is unstable, suggesting a perturbatively inaccessible algebraically rough interface or positional short-ranged order. Our model should serve as a paradigmatic nonlocal growth equation.

A convergent finite-volume scheme for nonlocal cross-diffusion systems for multi-species populations

An implicit Euler finite-volume scheme for a nonlocal cross-diffusion system on the one-dimensional torus, arising in population dynamics, is proposed and analyzed. The kernels are assumed to be in detailed balance and satisfy a weak cross-diffusion condition. The latter condition allows for negative off-diagonal coefficients and for kernels defined by an indicator function. The scheme preserves the nonnegativity of the densities, conservation of mass, and production of the Boltzmann and Rao entropies. The key idea is to “translate” the entropy calculations for the continuous equations to the finite-volume scheme, in particular to design discretizations of the mobilities, which guarantee a discrete chain rule even in the presence of nonlocal terms. Based on this idea, the existence of finite-volume solutions and the convergence of the scheme are proven. As a by-product, we deduce the existence of weak solutions to the continuous cross-diffusion system. Finally, we present some numerical experiments illustrating the behavior of the solutions to the nonlocal and associated local models.

A theorem of existence of solution for the nonlocal formulation of the problem of microwave heating

Microwave electromagnetic heating are widely used in many industrial processes. The mathematics involved is based on the Maxwell’s equations coupled with the heat equation. The thermal conductivity is strongly dependent on the temperature, itself an unknown of the system of P.D.E. We propose here a model which simplifies this coupling using a nonlocal term as the source of heating. We prove that the corresponding mathematical initial-boundary value problem has solutions using the Schauder’s fixed point theorem.

Weakly nonlinear analysis of a two-species non-local advection–diffusion system

Nonlocal interactions are ubiquitous in nature and play a central role in many biological systems. In this paper, we perform a bifurcation analysis of a widely-applicable advection–diffusion model with nonlocal advection terms describing the species movements generated by inter-species interactions. We use linear analysis to assess the stability of the constant steady state, then weakly nonlinear analysis to recover the shape and stability of non-homogeneous solutions. Since the system arises from a conservation law, the resulting amplitude equations consist of a Ginzburg–Landau equation coupled with an equation for the zero mode. In particular, this means that supercritical branches from the Ginzburg–Landau equation need not be stable. Indeed, we find that, depending on the parameters, bifurcations can be subcritical (always unstable), stable supercritical, or unstable supercritical. We show numerically that, when small amplitude patterns are unstable, the system exhibits large amplitude patterns and hysteresis, even in supercritical regimes. Finally, we construct bifurcation diagrams by combining our analysis with a previous study of the minimizers of the associated energy functional. Through this approach we reveal parameter regions in which stable small amplitude patterns coexist with strongly modulated solutions.

Existence and Limit Behavior of Constraint Minimizers for a Varying Non-Local Kirchhoff-Type Energy Functional

In this paper, we study the constrained minimization problem for an energy functional which is related to a Kirchhoff-type equation. For s=1, there many articles have analyzed the limit behavior of minimizers when η>0 as b→0+ or b>0 as η→0+. When the equation involves a varying non-local term ∫R3|∇u|2dxs, we give a detailed limit behavior analysis of constrained minimizers for any positive sequence {ηk} with ηk→0+. The present paper obtains an interesting result on this topic and enriches the conclusions of previous works.

Existence of transonic shocks in a cylindrical nozzle with given exit normal momentums

In this paper, we study the existence of transonic shocks in a cylindrical nozzle. More precisely, given the compatible outer normal momentum at the exit of a cylindrical nozzle, there exists a transonic shock solution such that the incoming supersonic flow is compressed to some subsonic flow. The shock position is uniquely determined by the exit normal momentum when we fix one point of the shock. This problem can be reduced to solve a nonlinear free boundary value problem for a hyperbolic–elliptic coupled system and the key ingredient is to solve a second‐order quasilinear elliptic equation with a nonlocal term by the method of separation of variables.

Modeling the interplay of oscillatory synchronization and aggregation via cell–cell adhesion

We present a model of systems of cells with intracellular oscillators (‘clocks’). This is motivated by examples from developmental biology and from the behavior of organisms on the threshold to multicellularity. Cells undergo random motion and adhere to each other. The adhesion strength between neighbors depends on their clock phases in addition to a constant baseline strength. The oscillators are linked via Kuramoto-type local interactions. The model is an advection-diffusion partial differential equation with nonlocal advection terms. We demonstrate that synchronized states correspond to Dirac-delta measure solutions of a weak version of the equation. To analyze the complex interplay of aggregation and synchronization, we then perform a linear stability analysis of the incoherent, spatially uniform state. This lets us classify possibly emerging patterns depending on model parameters. Combining these results with numerical simulations, we determine a range of possible far-from equilibrium patterns when baseline adhesion strength is zero: There is aggregation into separate synchronized clusters with or without global synchrony; global synchronization without aggregation; or unexpectedly a ‘phase wave’ pattern characterized by spatial gradients of clock phases. A 2D Lattice-Gas Cellular Automaton model confirms and illustrates these results.

On Global Solutions of Hyperbolic Equations with Positive Coefficients at Nonlocal Potentials

We study hyperbolic equations with positive coefficients at potentials undergoing translations with respect to the spatial independent variable. The qualitative novelty of the investigation is that the real part of the symbol of the differential-difference operator contained in the equation is allowed to change its sign. Earlier, only the case where the said sign is constant was investigated. We find a condition relating the coefficient at the nonlocal term of the investigated equation and the length of the translation, guaranteeing the global solvability of the investigated equation. Under this condition, we explicitly construct a three-parametric family of smooth global solutions of the investigated equation.

Blowup analysis for a quasi-exact 1D model of 3D Euler and Navier–Stokes

We study the singularity formation of a quasi-exact 1D model proposed by Hou and Li (2008 Commun. Pure Appl. Math. 61 661–97). This model is based on an approximation of the axisymmetric Navier–Stokes equations in the r direction. The solution of the 1D model can be used to construct an exact solution of the original 3D Euler and Navier–Stokes equations if the initial angular velocity, angular vorticity, and angular stream function are linear in r. This model shares many intrinsic properties similar to those of the 3D Euler and Navier–Stokes equations. It captures the competition between advection and vortex stretching as in the 1D De Gregorio (De Gregorio 1990 J. Stat. Phys. 59 1251–63; De Gregorio 1996 Math. Methods Appl. Sci. 19 1233–55) model. We show that the inviscid model with weakened advection and smooth initial data or the original 1D model with Hölder continuous data develops a self-similar blowup. We also show that the viscous model with weakened advection and smooth initial data develops a finite time blowup. To obtain sharp estimates for the nonlocal terms, we perform an exact computation for the low-frequency Fourier modes and extract damping in leading order estimates for the high-frequency modes using singularly weighted norms in the energy estimates. The analysis for the viscous case is more subtle since the viscous terms produce some instability if we just use singular weights. We establish the blowup analysis for the viscous model by carefully designing an energy norm that combines a singularly weighted energy norm and a sum of high-order Sobolev norms.

Phase-field modeling and effective simulation of non-isothermal reactive transport

We consider single-phase flow with solute transport where ions in the fluid can precipitate and form a mineral, and where the mineral can dissolve and release solute into the fluid. Such a setting includes an evolving interface between fluid and mineral. We approximate the evolving interface with a diffuse interface, which is modeled with an Allen–Cahn equation. We also include effects from temperature such that the reaction rate can depend on temperature, and allow heat conduction through fluid and mineral. As Allen–Cahn is generally not conservative due to curvature-driven motion, we include a reformulation that is conservative. This reformulation includes a non-local term that makes the use of standard Newton iterations for solving the resulting non-linear system of equations very slow. We instead apply L-scheme iterations, which can be proven to converge for any starting guess, although giving only linear convergence. The three coupled equations for diffuse interface, solute transport and heat transport are solved via an iterative coupling scheme. This allows the three equations to be solved more efficiently compared to a monolithic scheme, and only few iterations are needed for high accuracy. Through numerical experiments we highlight the usefulness and efficiency of the suggested numerical scheme and the applicability of the resulting model.

Boundedness of solutions to a chemotaxis–haptotaxis model with nonlocal terms

Boundedness of solutions to a chemotaxis–haptotaxis model with nonlocal terms

Null controllability for degenerate parabolic equations with a nonlocal space term

<p style='text-indent:20px;'>We consider two degenerate heat equations with a nonlocal space term, studying, in particular, their null controllability property. To this aim, we first consider the associated nonhomogeneous degenerate heat equations: we study their well posedness, the Carleman estimates for the associated adjoint problems and, finally, the null controllability. Then, as a consequence, using the Kakutani's fixed point Theorem, we deduce the null controllability property for the initial nonlocal problems.</p>

Positivity and dynamics preserving discretization schemes for nonlinear evolution equations

Discretization of a continuous-time system of dierential equations becomes inevitable due to the lack of analytical solutions. Standard discretization techniques, however, have many things that could be improved, e.g., the positivity of the solution and dynamic consistency may be lost, and stability and convergence may depend on the step length. A nonstandard nite dierence (NSFD) scheme is sometimes used to avoid inconsistencies. There are two fundamental issues regarding the construction of NSFD models. First, how to construct the denominator function of the discrete rst-order derivative? Second, how to discretize the nonlinear terms of a given dierential equation with nonlocal terms? We dene here a uniform technique for nonlocal discretization and construction of denominator function for NSFD models. We have discretized a couple of highly nonlinear continuous-time population models using these consistent rules. We give analytical proof in each case to show that the proposed NSFD model has identical dynamic properties to the continuous-time model. It is also shown that each NSFD system is positively invariant, and its dynamics do not depend on the step size. Numerical experiments have also been performed in favour of such claims.

On the transport of tracer particles in two-dimensional plasma edge turbulence

Shear flows in turbulent fluids have been known to act as transport barriers for some time. An example of a shear flow generating mechanism is the E×B shear in plasma, which has a substantial impact on the dynamics of magnetic confinement fusion devices. The influence of this may be seen in the scrape-off layer where blobs or filaments may be sheared and velocity impacted, and in the edge and core of the plasma, where the formation of transport barriers and suppression of turbulence is strongly associated with such shearing effects. A dynamical picture of transport through these effects has been elusive—the development of a reduced model would be beneficial. We consider the application of an “observational” random walk to such transport, in order to determine whether it is a suitable approach upon which to base the development of reduced models. The observational random walk is modification of the random walk approach, introducing an intrinsic time separating observations, which reproduces the basic results of previous random walk models given a Gaussian jump function, assuming spatially homogenous jump function. We demonstrate that the jump function can be inferred from the statistics of passive particles propagated by E×B drift on a synthetic turbulence field and that the transport equation found from the jump function matches the expected diffusive transport very well. We, then, consider passive particles on simulations of the classic and modified Hasagawa–Wakatani equations in a statistical steady state for a variety of adiabaticity values and find normal transport in the near-hydrodynamic limit. When zonal flows appear, we find jump functions with non-Gaussian features, which result in transport equations with fractional differential terms in addition to, or in place of, diffusion terms. We surmise that the non-local fractional terms are related to the zonal flows acting as transport barriers. Overall, we find that the approach developed is a suitable starting point for the development of reduced models.

On a parabolic-elliptic Keller–Segel system with nonlinear signal production and nonlocal growth term

On a parabolic-elliptic Keller–Segel system with nonlinear signal production and nonlocal growth term