Neumann-type Boundary Conditions Research Articles

Beta critical of Schrödinger operator with delta potential in one and two dimension

We consider 1D and 2D Schrödinger equation with delta potential on the positive half-axis with Dirichlet, Neumann, and Robin type boundary conditions. We presented and estimated the exact values of the beta critical.

Half-space stationary Kardar–Parisi–Zhang equation beyond the Brownian case

We study the Kardar–Parisi–Zhang (KPZ) equation on the half-line x ⩾ 0 with Neumann type boundary condition. Stationary measures of the KPZ dynamics were characterized in recent work: they depend on two parameters, the boundary parameter u of the dynamics, and the drift −v of the initial condition at infinity. We consider the fluctuations of the height field when the initial condition is given by one of these stationary processes. At large time t, it is natural to rescale parameters as (u, v) = t −1/3(a, b) to study the critical region. In the special case a + b = 0, treated in previous works, the stationary process is simply Brownian. However, these Brownian stationary measures are particularly relevant in the bound phase (a < 0) but not in the unbound phase. For instance, starting from the flat or droplet initial condition, the height field near the boundary converges to the stationary process with a > 0 and b = 0, which is not Brownian. For a + b ⩾ 0, we determine exactly the large time distribution of the height function h(0, t). As an application, we obtain the exact covariance of the height field in a half-line at two times 1 ≪ t 1 ≪ t 2 starting from stationary initial condition, as well as estimates, when starting from droplet initial condition, in the limit t 1/t 2 → 1.

A double critical mass phenomenon in a no-flux-Dirichlet Keller-Segel system

Derived from a biophysical model for the motion of a crawling cell, the evolution system(⋆){ut=Δu−∇⋅(u∇v),0=Δv−kv+u, is investigated in a finite domain Ω⊂Rn, n≥2, with k≥0. Whereas a comprehensive literature is available for cases in which (⋆) describes chemotaxis-driven population dynamics and hence is accompanied by homogeneous Neumann-type boundary conditions for both components, the presently considered modeling context, besides yet requiring the flux ∂νu−u∂νv to vanish on ∂Ω, inherently involves homogeneous Dirichlet boundary conditions for the attractant v, which in the current setting corresponds to the cell's cytoskeleton being free of pressure at the boundary. This modification in the boundary setting is shown to go along with a substantial change with respect to the potential to support the emergence of singular structures: It is, inter alia, revealed that in contexts of radial solutions in balls there exist two critical mass levels, distinct from each other whenever k>0 or n≥3, that separate ranges within which (i) all solutions are global in time and remain bounded, (ii) both global bounded and exploding solutions exist, or (iii) all nontrivial solutions blow up. While critical mass phenomena distinguishing between regimes of type (i) and (ii) belong to the well-understood characteristics of (⋆) when posed under classical no-flux boundary conditions in planar domains, the discovery of a distinct secondary critical mass level related to the occurrence of (iii) seems to have no nearby precedent. In the planar case with the domain being a disk, the analytical results are supplemented with some numerical illustrations, and it is discussed how the findings can be interpreted biophysically for the situation of a cell on a flat substrate.

Steady state of the KPZ equation on an interval and Liouville quantum mechanics

We obtain a simple formula for the stationary measure of the height field evolving according to the Kardar-Parisi-Zhang equation on the interval with general Neumann-type boundary conditions and any interval size. This is achieved using the recent results of Corwin and Knizel (arXiv:2103.12253) together with Liouville quantum mechanics. Our formula allows to easily determine the stationary measure in various limits: KPZ fixed point on an interval, half-line KPZ equation, KPZ fixed point on a half-line, as well as the Edwards-Wilkinson equation on an interval.

Dynamics of a viscoelastic membrane with gradient constraint

Taking into account inertial and viscosity effects, we consider the dynamics of a two dimensional membrane subjected to an unilateral constraint on its deformation gradient. Specifically, due to the constitutive law, we assume that higher deformations lock the material, leading to the inequality |∇u|≤g, where u denotes the displacement of the membrane and g is a certain positive threshold. We then introduce the concepts of weak and generalised solutions to the associated wave equation, and prove the existence of them for rather general data and homogeneous Dirichlet boundary conditions. The presence of the gradient constraint provides the existence of a Lagrange multiplier λ related to the existence of a reaction term ϒ, which corresponds to a strongly nonlinear term in the wave equation. We then extend the existence result to a weak form of the Neumann type boundary condition αu+∂u∂ν+∂u˙∂ν+ϒ⋅ν=0, for any α≥0, and we show that these solutions tend, as α→∞, in a certain sense to a solution of the homogeneous Dirichlet constrained problem.

On coupled flows of micropolar heat conducting fluids with mixed boundary conditions

In this letter we are concerned with the initial boundary value problem for the micropolar heat conducting fluid in nonsmooth domains with mixed boundary conditions. The considered boundary conditions are of two types: Navier’s slip conditions on solid surfaces and Neumann-type boundary conditions on free surfaces. The present paper is devoted to the proof of the existence and regularity of the solution in distribution spaces under some restrictions on the data of the problem.

Boundary Conditions in a Lattice Boltzmann Method For Plane Strain Problems

AbstractThe Lattice Boltzmann Method (LBM), e.g. in [1] and [2], can be interpreted as an alternative method for the numerical solution of certain partial differential equations that is not restricted to its origin in computational fluid mechanics. The interpretation of the LBM as a general numerical tool allows to extend the LBM to solid mechanics as well, see e.g. [3], which is concerned with the simulation of elastic solids under simplified deformation assumptions, and [4] as well as [5] which propose LBMs for the general plane strain case. In previous works on a LBM for plain strain such as [5], the treatment of practically relevant boundary conditions like Neumann and Dirichlet type boundary conditions is not the main focus and thus periodic conditions or absorbing layers are specified to simulate numerical examples. In this work, we show how Neumann and Dirichlet type boundary conditions are implemented in our LBM for plane strain from [4].

Moving vesicles in elastic tissues: A model with existence and uniqueness of weak solutions

We describe the transport of vesicles in an elastic body in terms of a phase field. Besides standard forces implied by macroscopic strain, we account for a balance of microstructural actions induced by vesicle-to-vesicle interactions, vesicle self-actions, membrane bending and related nonlinear pressure-type confinement effects on vesicles. For the resulting balance equations, we prove existence and uniqueness of appropriate weak solutions (the so-called regular weak ones). Also, in the absence of bulk forces and under homogeneous Neumann-type boundary conditions, we prove a weak maximum principle for the vesicle phase-field.

A general blow-up result for a degenerate hyperbolic inequality in an exterior domain

In this paper, we consider a degenerate hyperbolic inequality in an exterior domain under three types of boundary conditions: Dirichlet-type, Neumann-type, and Robin-type boundary conditions. Using a unified approach, we show that all the considered problems have the same Fujita critical exponent. Moreover, we answer some open questions from the literature regarding the critical case.

On viscous incompressible flows of nonsymmetric fluids with mixed boundary conditions

In this paper we are concerned with the initial boundary value problem for the micropolar fluid system in nonsmooth domains with mixed boundary conditions. The considered boundary conditions are of two types: Navier’s slip conditions on solid surfaces and Neumann-type boundary conditions on free surfaces. The Dirichlet boundary condition for the microrotation of the fluid is commonly used in practice. However, the well-posedness of problems with different types of boundary conditions for microrotation are completely unexplored. The present paper is devoted to the proof of the existence, regularity and uniqueness of the solution in distribution spaces.

On a Boundary Value Problem for the Biharmonic Equation with Multiple Involutions

A nonlocal analogue of the biharmonic operator with involution-type transformations was considered. For the corresponding biharmonic equation with involution, we investigated the solvability of boundary value problems with a fractional-order boundary operator having a derivative of the Hadamard-type. First, transformations of the involution type were considered. The properties of the matrices of these transformations were investigated. As applications of the considered transformations, the questions about the solvability of a boundary value problem for a nonlocal biharmonic equation were studied. Modified Hadamard derivatives were considered as the boundary operator. The considered problems covered the Dirichlet and Neumann-type boundary conditions. Theorems on the existence and uniqueness of solutions to the studied problems were proven.

Galilean-Invariant Characteristic-Based Volume Penalization Method for Supersonic Flows with Moving Boundaries

This work extends the characteristic-based volume penalization method, originally developed and demonstrated for compressible subsonic viscous flows in (J. Comput. Phys. 262, 2014), to a hyperbolic system of partial differential equations involving complex domains with moving boundaries. The proposed methodology is shown to be Galilean-invariant and can be used to impose either homogeneous or inhomogeneous Dirichlet, Neumann, and Robin type boundary conditions on immersed boundaries. Both integrated and non-integrated variables can be treated in a systematic manner that parallels the prescription of exact boundary conditions with the approximation error rigorously controlled through an a priori penalization parameter. The proposed approach is well suited for use with adaptive mesh refinement, which allows adequate resolution of the geometry without over-resolving flow structures and minimizing the number of grid points inside the solid obstacle. The extended Galilean-invariant characteristic-based volume penalization method, while being generally applicable to both compressible Navier–Stokes and Euler equations across all speed regimes, is demonstrated for a number of supersonic benchmark flows around both stationary and moving obstacles of arbitrary shape.

A simple and efficient curved boundary scheme of the lattice Boltzmann method for Robin boundary conditions of convection–diffusion equations

This paper is concerned with boundary schemes of the lattice Boltzmann method for Robin boundary conditions of convection–diffusion equations on curved boundaries. For such boundary conditions, all the existing boundary schemes suffer from the possibility that the denominator in the scheme may become zero, which will lead to numerical instability. To avoid this possibility, we propose a boundary scheme by approximating the gradient along the outgoing discrete velocity at the boundary with the given Robin boundary condition and the gradient at the interior point next to the boundary. With this approximated gradient at the boundary, the classical bounce back scheme for Neumann-type boundary conditions is employed to obtain the unknown distribution function at the interior point. The scheme obtained has the first-order accuracy for curved boundaries and its advantages are: (1) the scheme is simple in form so that it can be easily implemented; (2) it avoids the denominator in the scheme to be zero, and (3) the scheme is single-node, i.e., it only involves the information at the present point. Numerical examples demonstrate the designed accuracy and good stability of our scheme for complex boundaries.

An Approximation of Solutions for the Problem with Quasistatic Contact in the Case of Dry Friction

In this paper, we discuss the question of finding an optimal control for the solutions of the problem with dry friction quasistatic contact, in the case that the friction law is modeled by a nonlocal version of Coulomb’s law. In order to get the necessary optimality conditions, we use some regularization techniques, and this leads us to a problem of control for an inequality of the variational type. The optimal control problem consists, in our case, of minimizing a sequence of optimal control problems, where the control variable is given by a Neumann-type boundary condition. The state system is represented by a limit of a sequence, whose terms are obtained from the discretization, in time with finite difference and space with the finite element method of a regularized quasistatic contact problem with Coulomb friction. The purpose of this optimal control problem is that the traction force (the control variable) acting on one side of the boundary (the Neumann boundary condition) of the elastic body produces a displacement field (the state system solution) close enough to the imposed displacement field, and the traction force from the boundary remains small enough.

Residual-type a posteriori error analysis of HDG methods for Neumann boundary control problems

We study a posteriori error analysis of linear-quadratic boundary control problems under bilateral box constraints on the control which acts through a Neumann-type boundary condition. We adopt the hybridizable discontinuous Galerkin method as the discretization technique, and the flux variables, the scalar variables, and the boundary trace variables are all approximated by polynomials of degree k. As for the control variable, it is discretized by the variational discretization concept. Then, an efficient and reliable a posteriori error estimator is introduced, and we prove that the error estimator provides an upper bound and a lower bound for the errors. Finally, numerical results are presented to illustrate the performance of the obtained a posteriori error estimator.

Schrödinger Equation in Moving Domains

We consider the Schr\''odinger equation \begin{equation}\label{eq_abstract} i\partial_t u(t)=-\Delta u(t)~~~~~\text{ on }\Omega(t) \tag{$\ast$} \end{equation}where $\Omega(t)\subset\mathbb{R}$ is a moving domain depending on the time $t\in [0,T]$. The aim of this work is to provide a meaning to the solutions of such an equation. We use the existence of a bounded reference domain $\Omega_0$ and a specific family of unitary maps $h^\sharp(t): L^2(\Omega(t),\mathbb{C})\longrightarrow L^2(\Omega_0,\mathbb{C})$. We show that the conjugation by $h^\sharp$ provides a newequation of the form \begin{equation}\label{eq_abstract2}i\partial_t v= h^\sharp(t)H(t)h_\sharp(t) v~~~~~\text{ on }\Omega_0\tag{$\ast\ast$} \end{equation} where $h_\sharp=(h^\sharp)^{-1}$. The Hamiltonian $H(t)$ is a magnetic Laplacian operator of the form$$H(t)=-(div+iA)\circ(grad+iA)-|A|^2$$where $A$ is an explicit magnetic potential depending on the deformation of the domain $\Omega(t)$. The formulation \eqref{eq_abstract2} enables to ensure the existence of weak and strong solutions of the initial problem \eqref{eq_abstract} on $\Omega(t)$ endowed with Dirichlet boundary conditions. In addition, it also indicates that the correct Neumann type boundary conditions for \eqref{eq_abstract} are not the homogeneous but the magnetic ones$$\partial_\nu u(t)+i\langle\nu| A\rangle u(t)=0,$$even though \eqref{eq_abstract} has no magnetic term. All the previous results are also studied in presence of diffusion coefficients as well as magnetic and electric potentials. Finally, we prove some associated byproducts as an adiabatic result for slow deformations of the domain and atime-dependent version of the so-called ``Moser's trick''. We use this outcome in order to simplify Equation \eqref{eq_abstract2} and to guarantee the well-posedness for slightly less regular deformations of $\Omega(t)$.

Pore-scale simulation of salt fingers in porous media using a coupled iterative source-correction immersed boundary-lattice Boltzmann solver

In the conventional representative elementary volume-scale models, details of fluid movement, thermal transfer and mass transport in the porous media are relatively hard to capture, while the understanding of them is crucial in many diverse engineering and environmental applications such as running nuclear power facilities or pollutant diffusion in soil environment. In this paper, a pore-scale lattice Boltzmann method (LBM) modeling of double diffusion systems in saturated porous media is developed to investigate the role played by pore structure in the behavior of salt finger phenomena together with their fluid flow mechanism. An iterative source-correction immersed boundary method is developed to describe the solid skeleton with different boundary conditions for velocity field, heat transfer, and mass transport, respectively. The natural convection in a vertical annulus and the conjugate heat transfer problem in a two-layer horizontal annulus are employed to verify the accuracy and the applicability of the immersed boundary method for Dirichlet-type, Neumann-type and conjugate heat transfer boundary conditions, respectively. Comparisons between the present results and existing investigation show a very good agreement and demonstrate that the double diffusive process in porous media can be predicted with a good accuracy by the proposed model. Subsequently, the instability evolution process of salt finger-type of double diffusive convection in porous media is investigated, with a focus on the structure and behavior of salt fingers and the kinetic energy evolution. The obtained simulation results show that in saturated porous media, relatively large vertical mass fluxes can be generated even if the density distribution is of dynamical stability.

Neumann and Robin type boundary conditions in Fractional Orlicz-Sobolev spaces

In the first part of this article we deal with the existence of at least three non-trivial weak solutions of a nonlocal problem with nonstandard growth involving a nonlocal Robin type boundary condition. The second part of the article is devoted to study eigenvalues and minimizers of several nonlocal problems for the fractionalg-Laplacian (-Δg)swith different boundary conditions, namely, Dirichlet, Neumann and Robin.

An asymptotically compatible approach for Neumann-type boundary condition on nonlocal problems

In this paper we consider 2D nonlocal diffusion models with a finite nonlocal horizon parameterδcharacterizing the range of nonlocal interactions, and consider the treatment of Neumann-like boundary conditions that have proven challenging for discretizations of nonlocal models. We propose a new generalization of classical local Neumann conditions by converting the local flux to a correction term in the nonlocal model, which provides an estimate for the nonlocal interactions of each point with points outside the domain. While existing 2D nonlocal flux boundary conditions have been shown to exhibit at most first order convergence to the local counter part asδ → 0, the proposed Neumann-type boundary formulation recovers the local case asO(δ2) in theL∞(Ω) norm, which is optimal considering theO(δ2) convergence of the nonlocal equation to its local limit away from the boundary. We analyze the application of this new boundary treatment to the nonlocal diffusion problem, and present conditions under which the solution of the nonlocal boundary value problem converges to the solution of the corresponding local Neumann problem as the horizon is reduced. To demonstrate the applicability of this nonlocal flux boundary condition to more complicated scenarios, we extend the approach to less regular domains, numerically verifying that we preserve second-order convergence for non-convex domains with corners. Based on the new formulation for nonlocal boundary condition, we develop an asymptotically compatible meshfree discretization, obtaining a solution to the nonlocal diffusion equation with mixed boundary conditions that converges withO(δ2) convergence.

On causality of wave motion in nonlocal theories of elasticity: a Kramers–Kronig relations study

In this paper, the subject of the principle of causality and the physical realizability in the wave motion characteristics within a linear nonlocal elastic medium is examined. The principle of primitive causality is examined via Kramers–Kronig (K–K) relations and the principle of relativistic causality or the Einstein causality is examined via wave motion responses. Gradient as well as integral type nonlocality has been considered. Methodology here involves a Fourier frequency domain based spectral analysis and the wave motion characteristics include: wave modes, group speeds and frequency response function. In general, due to existence of atleast one of the non-physical features in the characteristics, violation of the causality is observed. The non-physical features include: existence of infinitesimally small or zero speeds; existence of very large or infinite speeds; existence of negative speeds; and absence of attenuation of waves. Violation to the primitive causality takes place as a disagreement to the K–K relations, either due to existence of negative and/or zero group speeds or due to absence of wave attenuation in the possible wave modes. Violation to Einstein causality is observed due to existence of infinitely large group speeds. Agreement to the primitive causality is achieved due to the presence of both wave dispersion and wave attenuation in the wave modes. Although existence of infinitely large group speeds violates Einstein causality, however, violation of the primitive causality is not observed. Upon considering only the physically realizable wavemodes, it is observed that, a local Neumann type boundary condition may be sufficient to conduct a wave motion study in a class of nonlocal boundary value problems. As an application of the primitive causality to the Fourier domain analysis, the wavenumbers from the K–K relations are utilized to demonstrate a mitigation effect of certain non-physical features in the wave motion responses.