Neumann Boundary Conditions Research Articles

On a $p(x,.)$-integrodifferential problem with Neumann boundary conditions

This study examines a class of a bi-nonlocal problem involving a generalized integro-differential operator of elliptic type, characterized by a singular kernel, with nonlocal nonlinear Neumann boundary conditions. We establish the existence of infinitely many solutions in a general fractional Sobolev space with variable exponent.

Ergodic and Chaotic Properties of the Heat Equation

AbstractWe consider a semiflow generated by the heat equation on the half-line with zero Neumann boundary condition. If the initial functions are from some weighted space X, then we prove that there exists an invariant mixing measure and X is the topological support of this measure. This result implies chaotic properties of the semiflow.

Logarithmically refined Gagliardo–Nirenberg interpolation and application to blow-up exclusion in a singular chemotaxis–consumption system

A family of interpolation inequalities is derived, which differ from estimates of classical Gagliardo–Nirenberg type through the appearance of certain logarithmic deviations from standard Lebesgue norms in zero-order expressions. Optimality of the obtained inequalities is shown. A subsequent application reveals that when posed under homogeneous Neumann boundary conditions in smoothly bounded planar domains and with suitably regular initial data, for any choice of \alpha>0 the Keller–Segel-type migration–consumption system u_{t} = \Delta (uv^{-\alpha}) , v_{t} = \Delta v-uv , admits a global classical solution.

Global existence and boundedness in a two‐dimensional parabolic chemotaxis system with competing attraction and repulsion effects

In this paper, we investigate a chemotaxis system under homogeneous Neumann boundary conditions within a bounded domain with a smooth boundary. The system describes the movement of cells in response to two chemical signal substances: one acts as a chemoattractant, while the other serves as a chemorepellent, both produced by the cells. The system takes into account chemotactic sensitivity in the reaction movement when detecting these chemicals. Under certain assumptions, we demonstrate the existence of a unique global bounded classical solution for the proposed problem. To further understand the time evolution of the system's solutions, we conduct numerical experiments and analyze the dynamic properties of the norm of the solutions with respect to variations in chemical production rates.

Transfer Learning-Based Physics-Informed Convolutional Neural Network for Simulating Flow in Porous Media with Time-Varying Controls

A physics-informed convolutional neural network (PICNN) is proposed to simulate two-phase flow in porous media with time-varying well controls. While most PICNNs in the existing literature worked on parameter-to-state mapping, our proposed network parameterizes the solutions with time-varying controls to establish a control-to-state regression. Firstly, a finite volume scheme is adopted to discretize flow equations and formulate a loss function that respects mass conservation laws. Neumann boundary conditions are seamlessly incorporated into the semi-discretized equations so no additional loss term is needed. The network architecture comprises two parallel U-Net structures, with network inputs being well controls and outputs being the system states (e.g., oil pressure and water saturation). To capture the time-dependent relationship between inputs and outputs, the network is well designed to mimic discretized state-space equations. We train the network progressively for every time step, enabling it to simultaneously predict oil pressure and water saturation at each timestep. After training the network for one timestep, we leverage transfer learning techniques to expedite the training process for a subsequent time step. The proposed model is used to simulate oil–water porous flow scenarios with varying reservoir model dimensionality, and aspects including computation efficiency and accuracy are compared against corresponding numerical approaches. The comparison with numerical methods demonstrates that a PICNN is highly efficient yet preserves decent accuracy.

Stabilisation of linear waves with inhomogeneous Neumann boundary conditions

ABSTRACT We study linear damped and viscoelastic wave equations evolving on a bounded domain. For both models, we assume that waves are subject to an inhomogeneous Neumann boundary condition on a portion of the domain's boundary. The analysis of these models presents additional interesting features and challenges compared to their homogeneous counterparts. In the present context, energy depends on the boundary trace of velocity. It is not clear in advance how this quantity should be controlled based on the given data, due to regularity issues. However, we establish global existence and also prove uniform stabilisation of solutions with decay rates characterised by the Neumann input. We supplement these results with numerical simulations in which the data do not necessarily satisfy the given assumptions for decay. These simulations provide, at a numerical level, insights into how energy could possibly change in the presence of, for example, improper data.

Neural network-augmented differentiable finite element method for boundary value problems

Classical numerical methods such as finite element method (FEM) face limitations due to their low efficiency when addressing large-scale problems. As a novel paradigm, the physics-informed neural network (PINN) has demonstrated significant potential to solve partial differential equations. However, conventional PINNs utilize meshless control at discrete sampling points, which limits their ability to effectively handle complex boundaries. Moreover, catastrophic failure may occur in the deep energy method (DEM, a specific type of PINN). To handle these challenges, this study proposes a Neural Network-augmented Differentiable Finite Element Method (NNDFEM) by combining PINN and finite element approximation. In NNDFEM, the neural network backend solely predicts nodal variables. Derivatives and complex boundary conditions can be well handled by the finite element frontend. The governing equation over the domain, Dirichlet, and Neumann boundary conditions are directly enforced on the finite element frontend. Thus, losses of boundary conditions in PINN are rendered unnecessary. The overfitting problem in DEM is also significantly mitigated. Fully connected neural network (FCNN), modified FCNN, and graph-convolutional network are tested as backends. NNDFEM circumvents nodal force calculation and matrix assembly in FEM. Functional losses of linear elasticity, finite strain nonlinear elasticity, heat conduction, and flow in porous media are validated. A systematic exploration unveils the role of 3D finite element mesh. For large-scale problems, a multi-fidelity learning strategy is employed. Thus, the three-dimensional case with over three million degrees of freedom trains well in two minutes. Benefiting from the fast inference of the neural network backend, the forward pass is 8,550 times faster than FEM.

Error estimates of effective boundary conditions for the heat equation with optimally aligned coatings

We are interested in the validity of effective boundary conditions for a heat equation on a coated body as the thickness of the coating shrinks to zero. The coating is optimally aligned in the sense that the normal vector in the coating is an eigenvector of the thermal tensor. If the heat equation satisfies Neumann boundary condition on the outer boundary of the coating, Chen et al. (Arch. Ration. Mech. Anal. 206 (2012) 911-951) derived the complete list of effective boundary conditions satisfied by the limiting model. In this paper we provide explicit error estimates between the full model and the effective model. Moreover, our error estimates are independent of time, which shows that the maximal time interval in which the effective boundary conditions remain valid are infinite. The proof is based on H2 estimates for solutions of the full model, characterization of large time behaviors for solutions of the effective model, and interaction estimates between the two models.

Dynamic behavior in a pursuit-evasion system with signaling mechanism

We mainly focus on the predator-prey problem associated with indirect predator taxis{ut=Δu+χ1∇⋅(u∇w)+u(1−μ1u−αv),vt=Δv−χ2∇⋅(ψ(v)∇u)+v(−1−μ2v+βu),0=Δw+v−w in a smooth bounded domain Ω⊂Rn(n≤3) with homogeneous Neumann boundary conditions, where the parameters χ1, χ2, μ1, μ2, α and β are positive. The chemotaxis sensitivity function with/without “volume-filling” effect ψ(v)=v1+σv, where σ≥0 is a constant. Compared with the well-known predator-prey system, the discussion of signaling mechanism w forms a novel feature of our system. For the case σ=0, we identify a positive constant χ0 such that the above system possesses global classical solutions with relative bound if χ2<χ0; For the case σ>0, we also show that the classical solutions of the system are globally bounded without any restrictions on system parameters. Furthermore, utilizing the Lyapunov functionals, we establish stability for the prey-only situation, the steady state of coexistence is also investigated.

Dynamics of a non-local intraspecific competition predator–prey model with memory effect

In the paper, we investigate a diffusive predator–prey model with nonlocal intraspecific prey competition and spatial memory under Neumann boundary conditions. Through stability and bifurcation analysis, we find that the memory-based diffusion coefficient and the spatiotemporal diffusive delay have important effects on the dynamics of the model. By using the spatiotemporal diffusive delay as a bifurcation parameter, the critical values are determined for the stability of the positive constant steady state and the associated Hopf bifurcation. We find that the system may admit no stability switch, one stability switch and multiple stability switches.

Mass ladder operators and quasinormal modes of the static BTZ-like black hole in the Einstein-bumblebee gravity

Abstract We investigate the mass ladder operators for the static BTZ-like black hole in the Einstein-bumblebee gravity, and probe the quasinormal frequencies of the mapped modes by the mass ladder operators for a scalar perturbation under the Dirichlet and Neumann boundary conditions. It is found that the mass ladder operators depend on the Lorentz symmetry breaking parameter, and the imaginary parts of the frequencies shifted by the mass ladder operators increase with the increase of the Lorentz symmetry breaking parameter under two boundary conditions. It should be noted that, under the Neumann boundary condition, the mapped modes caused by the mass ladder operator $D_{0,k_+}$ are unstable. Moreover, the mass ladder operators do not change the Breitenlohner-Freedman bound for the scalar modes, just as in the usual BTZ black hole. These results could help us to further understand the conformal symmetry and Lorentz symmetry breaking in the Einstein-bumblebee gravity.

Bayesian Inference for Estimating Heat Sources through Temperature Assimilation

Abstract This paper utilizes a Bayesian inference framework to address the two-dimensional steady-state heat conduction problem, focusing on the estimation of unknown distributed heat sources in a thermally-conducting medium with uniform conductivity. The goal is to infer the locations, strength, and shape of heaters by assimilating temperature data in Euclidean space, employing a Fourier series to represent each heater's shape. The Markov Chain Monte Carlo method, incorporating the random-walk Metropolis-Hasting algorithm and parallel tempering, is utilized for posterior distribution exploration in both unbounded and wall-bounded domains. It is found that multiple solutions arise in cases where the number of temperature sensors is less than the number of unknown states. Moreover, smaller heaters introduce greater uncertainty in estimated strength. To address the challenge of estimating the heater's strength and shape simultaneously due to their strong correlation, our method incorporates sharp priors on one to ensure accurate and feasible solutions of the other. The diffusive nature of heat conduction smooths out any deformations in the temperature contours, especially in the presence of multiple heaters positioned near each other, impacting convergence. In wall-bounded domains with Neumann boundary conditions, the inference of heater parameters tends to be more accurate than in unbounded domains.

Heat transfer performance of regenerative cooling in the presence of supersonic combustion based on two-way decoupling method

As a promising active cooling method for aero engines, regenerative cooling with endothermic hydrocarbon fuel plays an important role in thermal protection systems. However, simply assuming the thermal boundary conditions for the numerical simulation of regenerative cooling as either Dirichlet or Neumann boundary conditions is often unsuitable, as the combustion process exhibits complex three-dimensional (3D) characteristics. We propose a novel multi-step iterative calculation method in this work to simulate the coupled flow and heat transfer between supersonic combustion in the combustor and regenerative cooling outside. The method captures the main features of the supersonic combustion including different shockwave structures, and generates a realistic 3D distribution of heat flux as the boundary condition for the regenerative cooling study. With improved simulation, the result shows that regenerative cooling significantly reduces the peak temperature of the combustor by 58.5% with a mass flow rate of 1 g/s. A local high-temperature region is observed on the coupled wall inside the combustor near the injector, where the flame is primarily concentrated. Increasing the mass flow rate by 100%, from 1 g/s to 2 g/s, leads to a reduction in the maximum wall temperature by up to 15.7%.

Stochastic Control Problems with State Reflections Arising from Relaxed Benchmark Tracking

This paper studies stochastic control problems motivated by optimal consumption with wealth benchmark tracking. The benchmark process is modeled by a combination of a geometric Brownian motion and a running maximum process, indicating its increasing trend in the long run. We consider a relaxed tracking formulation such that the wealth compensated by the injected capital always dominates the benchmark process. The stochastic control problem is to maximize the expected utility of consumption deducted by the cost of the capital injection under the dynamic floor constraint. By introducing two auxiliary state processes with reflections, an equivalent auxiliary control problem is formulated and studied, which leads to the Hamilton-Jacobi-Bellman equation with two Neumann boundary conditions. We establish the existence of a unique classical solution to the dual partial differential equation using some novel probabilistic representations involving the local time of some dual processes together with a tailor-made decomposition-homogenization technique. The proof of the verification theorem on the optimal feedback control can be carried out by some stochastic flow analysis and technical estimations of the optimal control. Funding: L. Bo and Y. Huang are supported by the National Natural Science Foundation of China [Grant 12471451], the Natural Science Basic Research Program of Shaanxi [Grant 2023-JC-JQ-05], the Shaanxi Fundamental Science Research Project for Mathematics and Physics [Grant 23JSZ010], and Fundamental Research Funds for the Central Universities [Grant 20199235177]. X. Yu is supported by the Hong Kong RGC General Research Fund (GRF) [Grant 15304122] and the Research Centre for Quantitative Finance at the Hong Kong Polytechnic University [Grant P0042708].

A novel transformation free nonuniform higher order compact finite difference scheme for solving incompressible flows on circular geometries

This paper presents a newly developed higher order compact scheme that is designed on a polar grid by using an implicit form of first order derivatives on nonuniform grids. These derivatives are formulated compactly and implicitly with a relationship of the coefficients in terms of the unknown variables. The proposed scheme is free from transformation technique and third order accurate in space. The objective is to solve Stokes equations and Navier–Stokes equations on curvilinear grids using the polar nature of the coordinate system. Our newly developed scheme is used to solve several problems namely, a problem having an analytical solution, Stokes flow with different orientations of the lids for the half filled annular and wedge cavity, the lid driven polar cavity flow and flow past an impulsively started circular cylinder. Our newly developed scheme is used to analyze the flow structures for the flow governed by different physical control parameters: the cavity radius ratio, the cavity angle and the ratio of the upper and lower lid speeds with rotating coaxial cylinders and Reynolds number for the impulsively started circular cylinder. The Stokes equations and the Navier–Stokes equations are efficiently solved with Dirichlet as well as Neumann boundary conditions. The efficacy and robustness of our proposed scheme are shown through its applicability in all the complex fluid flow problems.

A diffusive predator-prey system with hunting cooperation in predators and prey-taxis: II stationary pattern formation

This paper is a continuation of the study conducted by Ko and Ryu (2024) [5], which introduces and analyzes a generalized predator-prey reaction-diffusion system incorporating (repulsive) prey-taxis and a hunting cooperation effect in predators, under homogeneous Neumann boundary conditions. In the study, the existence and uniqueness of global and classical solutions for the time- and space-dependent system are analytically examined. Furthermore, the study examines the local and global stability and convergence rate at the constant predator-extinction and coexistence states. In our paper, we analyze the stationary system corresponding to the system in [5], with a specific focus on examining the existence and nonexistence of positive and nonconstant solutions. The nonexistence occurs when the diffusion rate of prey is sufficiently high. On the other hand, the existence occurs when the prey-tactic rate is sufficiently high, indicating a strong repulsive prey-taxis, and the diffusion rate of prey is sufficiently low. For this investigation, we separately employ the energy method and the Leray-Schauder degree theory.

Strong Solutions of Brusselator System

The study involves a mathematical analysis of the Brusselator system on a convex bounded three-dimensional open domain, considering Neumann boundary conditions. We establish the global existence and uniqueness of the strong solution for this system. Achieving high regularity for the strong solution requires stringent conditions on the initial data. The study demonstrates the continuous dependence of the solution on the initial conditions.

Existence of global weak solution to tumor chemotaxis competition systems with loop and signal dependent sensitivity

This article examines the weak solution of a fully parabolic chemotaxis-competition system with loop and signal-dependent sensitivity. The system is subject to homogeneous Neumann boundary conditions within an open, bounded domain \(\Omega\subset\mathbb{R}^n\), where \(n\geq 1\) and \(\partial\Omega\) is smooth. We assume that the parameters in the system are positive constants. Additionally, the initial data \((u_{10}, u_{20}, v_{10}, v_{20})\in L^2(\Omega)\times L^2(\Omega) \times W^{1,2}(\Omega)\times W^{1,2}(\Omega)\) are non-negative. The existence of a weak solution to the problem is established using energy inequality method. For more information see https://ejde.math.txstate.edu/Volumes/2024/56/abstr.html

Null controllability for a strongly coupled stochastic degenerate reaction-diffusion system

This paper concerns the null controllability for a strongly coupled stochastic degenerate reaction-diffusion system in cardiac electrocardiology, which describes the electrical activity in the cardiac tissue with random effects. To deal with degeneracy of this system, we first consider an approximate problem of this coupled stochastic degenerate system. By a weighted identity method, we then prove a uniform Carleman estimate for the adjoint system of this approximate problem, which is a strongly coupled backward stochastic parabolic system with homogeneous Neumann boundary conditions. Based on this Carleman estimate and a limit process, we finally obtain the null controllability result.

Virtual body-fitted grid-based immersed boundary method for simulation of thermal flows with Dirichlet and Neumann boundary conditions

In this work, a virtual body-fitted grid-based immersed boundary method (IBM) combined with the thermal lattice Boltzmann flux solver (TLBFS) is proposed to efficiently simulate thermal flows with Dirichlet and Neumann boundary conditions. The method involves generating a virtual body-fitted grid along the immersed boundary and implementing the boundary conditions using a fractional step technique. In the prediction step, the finite-volume TLBFS is used to obtain the predicted flow field on the Eulerian mesh without considering the immersed boundary, while in the correction step, the correction process of the flow field is conducted on the virtual grid. Thus, the effect of the boundary is transferred to the fluid domain through the virtual grid indirectly. For Dirichlet boundary conditions, such as the no-slip condition and the specified temperature condition, the velocity and temperature corrections are considered unknowns and solved through a matrix system formed from enforcing the boundary conditions. As for the Neumann boundary condition, such as the specified heat flux condition, the temperatures at virtual layers inside the immersed body are regarded as unknowns and computed from those at virtual layers outside the wall according to the quadratic distribution. Several benchmark cases including natural, forced, and mixed convection problems are well simulated to validate the method. The numerical results are in good agreement with reference data, demonstrating the capacity of the present method to simulate thermal flows with Dirichlet and Neumann boundary conditions while requiring fewer Lagrangian points and achieving higher computational efficiency.