Dissipative Boundary Conditions Research Articles

On a class of third-order dissipative differential operator with distributional potentials

This paper aims to investigate a class of boundary value problems generated by a third-order differential equation with distributional potentials and some dissipative boundary conditions. We first construct an operator related to the problem, then prove that the operator is dissipative and show some eigenvalue properties of this operator. Further using Krein’s theorem we prove the completeness theorems of the operator.

Numerical analysis for nonlinear wave equations with boundary conditions: Dirichlet, Acoustics and Impenetrability

In this article, we present an error estimation in the L2 norm referring to three wave models with variable coefficients, supplemented with initial and boundary conditions. The first two models are nonlinear wave equations with Dirichlet, Acoustics, and nonlinear dissipative impenetrability boundary conditions, while the third model is a linear wave equation with Dirichlet, Acoustics, and linear dissipative impenetrability boundary conditions. In the field of numerical analysis, we establish two key theorems for estimating errors to the semi-discrete and totally discrete problems associated with each model. Such theorems provide theoretical results on the convergence rate in both space and time. For conducting numerical simulations, we employ linear, quadratic, and cubic polynomial basis functions for the finite element spaces in the Galerkin method, in conjunction with the Crank-Nicolson method for time discretization. For each time step, we apply Newton's method to the resulting nonlinear problem. The numerical results are presented for all three models in order to corroborate with the theoretical convergence order obtained.

Complete nonselfadjointness for Schrödinger operators on the semi-axis

This note is devoted to the study of complete nonselfadjointness for all maximally dissipative extensions of a Schrödinger operator on a half-line with dissipative bounded potential and dissipative boundary condition. It is shown that all maximally dissipative extensions that preserve the differential expression are completely nonselfadjoint. However, it is possible for maximally dissipative extensions to have a one-dimensional reducing subspace on which the operator is selfadjoint. A characterization of these extensions and the corresponding subspaces is given, accompanied by a specific example.

Laplace transform solution of the time-dependent annular Couette flow with dynamic wall slip

The annular Couette flow has several industrial applications, particularly for the characterization of the fluid flow and deformation behavior of fluids. The inclusion of the dynamic wall slip into the flow boundary conditions seems to be necessary for an efficient continuum description of motion of nanofluidics as it reflects the importance of fluid–structure interface related phenomena. Dynamic wall slip introduces a dissipative boundary condition and thus increases the difficulties of finding solutions to related problems. In the present work we investigate the behavior of fluid flow between two infinitely long coaxial circular cylinders, when the inner cylinder is axially moving due to sudden constant velocity, while the outer cylinder is held stationary. The boundary condition on the outer cylinder is that of dynamic wall slip, in addition to the usual Navier slip. The medium considered here is a Newtonian viscous fluid. The solution of the governing equations, initial and boundary conditions for this flow is obtained using the Laplace transform technique and inversion by Laguerre polynomials. This method may be useful, when applied in conjunction with perturbation methods, to solve nonlinear Couette flow problems involving temperature changes. Numerical results are presented and discussed.

An Explicit–Implicit Upwind Difference Splitting Scheme in Directions for a Mixed Boundary Control Problem for a Two-Dimensional Symmetric t-Hyperbolic System

In this paper, we introduce a numerical integration method for hyperbolic systems problems known as the splitting method, which serves as an effective tool for solving complex multidimensional problems in mathematical physics. The exponential stability of the upwind explicit–implicit difference scheme split into directions is established for the mixed problem of a linear two-dimensional symmetric t-hyperbolic system with variable coefficients and lower-order terms. It is noteworthy that there are control functions in the dissipative boundary conditions. A discrete quadratic Lyapunov function was devised to address this issue. A condition for the problem’s boundary data, ensuring the exponential stability of the difference scheme with directional splitting for the mixed problem in the l2 norm, has been identified.

Asymptotic of the dissipative eigenvalues of Maxwell’s equations

Let Ω = R 3 ∖ K ¯, where K is an open bounded domain with smooth boundary Γ. Let V ( t ) = e t G b , t ⩾ 0, be the semigroup related to Maxwell’s equations in Ω with dissipative boundary condition ν ∧ ( ν ∧ E ) + γ ( x ) ( ν ∧ H ) = 0, γ ( x ) > 0, ∀ x ∈ Γ. We study the case when γ ( x ) ≠ 1, ∀ x ∈ Γ, and we establish a Weyl formula for the counting function of the eigenvalues of G b in a polynomial neighbourhood of the negative real axis.

Evolution of scalar field resonances in a braneworld

In this work, we investigate the numerical evolution of massive Kaluza–Klein (KK) modes of a scalar field in a thick brane. We derive the Klein–Gordon equation in five-dimensional spacetime, and obtain the evolution equation and the Schrödinger-like equation. With the resonances of the scalar KK modes as the initial data, the scalar field is evolved with the maximally dissipative boundary condition. The results show that there are scalar KK resonant particles with long life on the brane, which indicates that these resonances might be regarded as a candidate for dark matter.

Subsonic time-periodic solution to compressible Euler equations with damping in a bounded domain

In this paper, we consider the one-dimensional isentropic compressible Euler equations with linear damping β(t,x)ρu in a bounded domain, which can be used to describe the process of compressible flows through a porous medium. And the model is imposed a dissipative subsonic time-periodic boundary condition. Our main results reveal that the time-periodic boundary can trigger a unique subsonic time-periodic smooth solution which is stable under small perturbations on initial data. Moreover, the time-periodic solution possesses higher regularity and stability provided a higher regular boundary condition.

FROM PROPAGATION SYSTEMS TO TIME DELAYS AND BACK. CRITICAL CASES

The paper originates from the early ideas of A. D. Myshkis and his co-workers and of K. L. Cooke and his co-worker. These ideas send to a one-to-one correspondence between lossless and/or distortionless propagation described by nonstandard boundary value problems and a system of coupled differential and difference equations with deviated argument. In this way any property obtained for one mathematical ob­ject is automatically projected back on the other one. This approach is considered here for certain engineering applications. The common feature of these applications is the critical stability of the difference operator associated with the system with deviated argument obtained for each of the aforementioned applications. In fact the associated sys­tems are of neutral type and, according to the assumption of Hale, only strong stability of the difference operator ensures robust asymp­totic stability with respect to the delays. If the difference operator is in the critical case, the stability becomes fragile with respect to the delays. Based on some old results in the field, a conjecture concerning the (quasi)-critical modes of the system is stated; also a connection with the so called dissipative boundary conditions is suggested.

A Discrete Analogue of the Lyapunov Function for Hyperbolic Systems

We study the difference splitting scheme for the numerical calculation of stable solutions of a two-dimensional linear hyperbolic system with dissipative boundary conditions in the case of constant coefficients with lower terms. A discrete analogue of the Lyapunov function is constructed and an a priori estimate is obtained for it. The obtained a priori estimate allows us to assert the exponential stability of the numerical solution.

An Algebraic Condition for the Exponential Stability of an Upwind Difference Scheme for Hyperbolic Systems

In the paper, we investigate the question of obtaining the algebraic condition for the exponential stability of the numerical solution of the upwind difference scheme for the mixed problem posed for onedimensional symmetric t-hyperbolic systems with constant coefficients and with dissipative boundary conditions. An a priori estimate for the numerical solution of the boundary-value difference problem is obtained. This estimate allows us to state the exponential stability of the numerical solution. A theorem on the exponential stability of the numerical solution of the boundary-value difference problem is proved. Easily verifiable algebraic conditions for the exponential stability of the numerical solution are given. The convergence of the numerical solution is proved.

Controlling the real-time dynamics of a spin coupled to the helical edge states of the Kane-Mele model

The time-dependent state of a classical spin locally exchange coupled to an edge site of a Kane-Mele model in the topologically non-trivial phase is studied numerically by solving the full set of coupled microscopic equations of motion for the spin and the electron system. Dynamics in the long-time limit is accessible thanks to dissipative boundary conditions, applied to all but the zigzag edge of interest. We study means to control the state of the spin via transport of a spin-polarization cloud through the helical edge states. The cloud is formed at a distant edge site using a local magnetic field to inject an electron spin density and released by suddenly switching off the injection field. This basic process, consisting of spin injection, propagation of the spin-polarization cloud, and scattering of the cloud from the classical spin, can be used to steer the spin state in a controlled way. We find that the effect of a single basic process can be reverted to a high degree with a subsequent process. Furthermore, we show that by concatenating several basic injection-propagation-scattering processes, the spin state can be switched completely and that a full reversal can be achieved.

The SMGT equation from the boundary: regularity and stabilization

We consider the third-order (in time) linear equation known as SMGT-equation, as defined on a multidimensional bounded domain. Part A gives optimal interior and boundary regularity results from – Dirichlet or Neumann boundary terms. Explicit representation formulas are given that can be taken to define the notion of solution in the canonical case , while the same regularity results hold for Part B considers the SMGT equation under Neumann dissipative boundary conditions and critical parameter and a.e. in Ω. We provide two results: (i) uniform stabilization under minimal checkable geometric conditions, and (ii) strong stabilization in the absence of geometrical conditions.

Exponential stability of a general slope limiter scheme for scalar conservation laws subject to a dissipative boundary condition

In this paper, we establish the exponential BV stability of general systems of discretized scalar conservation laws with positive speed. The focus is on numerical approximation of such systems using a wide class of slope limiter schemes built from the upwind monotone flux. The proof is based on a Lyapunov analysis taken from the continuous theory (Coron et al. in J Differ Equ 262(1):1–30, 2017) and a careful use of Harten formalism.

Exact boundary controllability of partial nodal profile for network of strings

Based on the theory of exact boundary controllability of nodal profile for hyperbolic systems, the authors propose the concept of exact boundary controllability of partial nodal profile to expand the scope of applications. With the new concept, we can shorten the controllability time, save the number of controls, and increase the number of charged nodes with given nodal profiles. Furthermore, we introduce the concept of in-situ controlled node to deal with a new situation that one node can be charged and controlled simultaneously. The minimum number of boundary controls on the entire tree-like network is determined by using the concept of ‘degree of freedom of charged nodes’ introduced. And the concept of ‘control path’ is introduced to appropriately divide the network, so that we can determine the infimum of controllability time. General frameworks of constructive proof are given on a single interval, a star-like network, a chain-like network and a planar tree-like network for linear wave equation(s) with Dirichlet, Neumann, Robin and dissipative boundary conditions to establish a complete theory on the exact boundary controllability of partial nodal profile.

Existence and regularity result for Stokes system with special inlet/outlet condition

Our aim is to analyze special type of boundary conditions, created to simulate flows like in cardiovascular and respiratory systems. First, we will describe model of viscous, incompressible fluid in a domain consisting many inlets and outlets with open dissipative boundary conditions. The conditions are augmented by the inertia terms. We are posing additional constrains on a fluid motion by a volumetric flow rates or inlet/outlet pressure. Afterwards, we will define weak formulation of the problem and its motivation. Then, we will prove mathematical correctness of proposed conditions by properly modified Galerkin method. Also, we will prove existence of a solution and its uniqueness.

Stability of Discontinuous Galerkin Spectral Element Schemes for Wave Propagation when the Coefficient Matrices have Jumps

We use the behavior of the L_{2} norm of the solutions of linear hyperbolic equations with discontinuous coefficient matrices as a surrogate to infer stability of discontinuous Galerkin spectral element methods (DGSEM). Although the L_{2} norm is not bounded in terms of the initial data for homogeneous and dissipative boundary conditions for such systems, the L_{2} norm is easier to work with than a norm that discounts growth due to the discontinuities. We show that the DGSEM with an upwind numerical flux that satisfies the Rankine–Hugoniot (or conservation) condition has the same energy bound as the partial differential equation does in the L_{2} norm, plus an added dissipation that depends on how much the approximate solution fails to satisfy the Rankine–Hugoniot jump.

The spectral analysis of a system of first‐order equations with dissipative boundary conditions

This paper aims to share some completeness theorems related with a boundary value problem generated by a system of equations and non‐self‐adjoint (dissipative) boundary conditions. Indeed, we consider a system of equations that contains a continuous analogous of the orthogonal polynomials on the unit circle. Constructing the characteristic function of the related dissipative operator, we share some completeness theorems. Moreover, we give an explicit form of the self‐adjoint dilation of the dissipative operator.

A new model for dissipative particle dynamics boundary condition of walls with different wettabilities

The implementation of solid-fluid boundary condition has been a major challenge for dissipative particle dynamics (DPD) method. Current implementations of boundary conditions usually try to approach a uniform density distribution and a velocity profile close to analytical solution. The density oscillations and slip velocity are intentionally eliminated, and different wall properties disappear in the same analytical solution. This paper develops a new wall model that combines image and frozen particles and a new strategy to emphasize different wall properties especially wettabilities. The strategy first studies the realistic wall-fluid system by molecular dynamics (MD) simulation depending on physical parameters. Then, a DPD simulation is used to match the density and velocity profiles with the new wall model. The obtained DPD parameters can simulate the systems with the same wall and fluid materials. With this method, a simulation of the Poiseuille flow of liquid argon with copper walls is presented. Other walls with super-hydrophilic, hydrophilic, and hydrophobic wettabilities are also simulated. The limitations of the analytical solution and the effect of the wall-fluid interaction are discussed. The results show that the method suggested in this paper can simulate the mesoscale behavior of the microchannel flow related to realistic systems.

Dissipative boundary conditions and entropic solutions in dynamical perfect plasticity

We prove the well–posedness of a dynamical perfect plasticity model under general assumptions on the stress constraint set and on the reference configuration. The problem is studied by combining both calculus of variations and hyperbolic methods. The hyperbolic point of view enables one to derive a class of dissipative boundary conditions, somehow intermediate between homogeneous Dirichlet and Neumann ones. By using variational methods, we show the existence and uniqueness of solutions. Then we establish the equivalence between the original variational solutions and generalized entropic–dissipative ones, derived from a weak hyperbolic formulation for initial–boundary value Friedrichs' systems with convex constraints.