Nonlinear Robin Boundary Conditions Research Articles

Second-order curved boundary treatments of the lattice Boltzmann method for convection–diffusion equations

In this paper, we present a kind of second-order curved boundary treatments for the lattice Boltzmann method solving two-dimensional convection-diffusion equations with general nonlinear Robin boundary conditions. The key idea is to derive approximate boundary values or normal derivatives on computational boundaries, with second-order accuracy, by using the prescribed boundary condition. Once the approximate information is known, the second-order bounce-back schemes can be perfectly adopted. Our boundary treatments are validated with a number of numerical examples. The results show the utility of our boundary treatments and very well support our theoretical predications on the second-order accuracy thereof. The idea is quite universal. It can be directly generalized to 3-dimensional problems, multiple-relaxation-time models, and the Navier-Stokes equations.

Linear and non-linear Robin boundary conditions for thermal lattice Boltzmann method: cases of convective and radiative heat transfer at interfaces

Despite the wide applications of the linear and non-linear Robin boundary constraints in thermal simulations, not much works are reported on their implementation in lattice Boltzmann framework. In present work, counter-slip energy approach is employed to derive kinetic level equations, representing two particular cases of Robin boundary conditions; convection and combined convection and surface radiation. Loss of generality is avoided in the study and the terms accounting for boundary movement or viscous dissipation effects are incorporated, too. Utilizing a D2Q9 lattice structure, the derived equations are validated with 1D and 2D analytical solutions for conduction heat transfer problems in a square slab. Results of analysis show a first order rate of convergence for the convective boundary condition, while second order rate is found for combined convection and surface radiation constraint.

Multiple positive solutions of a ( p 1 , p 2 ) $(p_{1},p_{2})$ -Laplacian system with nonlinear BCs

Using the theory of fixed point index, we discuss existence, non-existence, localization and multiplicity of positive solutions for a $(p_{1},p_{2})$ -Laplacian system with nonlinear Robin and/or Dirichlet type boundary conditions. We give an example to illustrate our theory.

$$L^2$$ L 2 -energy decay of convective nonlinear PDEs reaction–diffusion systems via auxiliary ODEs systems

The \(L^2\)-energy longtime behaviour of convective binary P.D.Es. reaction–diffusion systems under the action of: self and cross diffusion; nonlinear reaction terms and Robin boundary conditions, is investigated. An auxiliary ODEs system, depending on a positive parameter \(\mu \), is introduced. Via the energy decay of the auxiliary system, for general classes of nonlinear reaction terms, the absence of subcritical instabilities and the \(L^2\)-energy asymptotic decay, are obtained. Estimates of the attraction basin are furnished. Instead of Sobolev type inequalities, only algebraic inequalities are requested. Applications to a celebrated model are furnished.

Non-CMC solutions to the Einstein constraint equations on asymptotically Euclidean manifolds with apparent horizon boundaries

In this article we further develop the solution theory for the Einstein constraint equations on an n-dimensional, asymptotically Euclidean manifold with interior boundary Σ. Building on recent results for both the asymptotically Euclidean and compact with boundary settings, we show the existence of far-from-CMC and near-CMC solutions to the conformal formulation of the Einstein constraints when nonlinear Robin boundary conditions are imposed on Σ, similar to those analyzed previously by Dain (2004 Class. Quantum Grav. 21 555–73), by Maxwell (2004, 2005 Commun. Math. Phys. 253 561–83), and by Holst and Tsogtgerel (2013 Class. Quantum Grav. 30 205011) as a model of black holes in various CMC settings, and by Holst et al (2013 Non-CMC solutions to the einstein constraint equations with apparent horizon boundaries arXiv:1310.2302v1) in the setting of far-from-CMC solutions on compact manifolds with boundary. These ‘marginally trapped surface’ Robin conditions ensure that the expansion scalars along null geodesics perpendicular to the boundary region Σ are non-positive, which is considered the correct mathematical model for black holes in the context of the Einstein constraint equations. Assuming a suitable form of weak cosmic censorship, the results presented in this article guarantee the existence of initial data that will evolve into a space-time containing an arbitrary number of black holes. A particularly important feature of our results are the minimal restrictions we place on the mean curvature, giving both near- and far-from-CMC results that are new.

Asymptotic approximation for the solution to a semi-linear parabolic problem in a thick junction with the branched structure

We consider a semi-linear parabolic problem in a thick junction Ωε, which is the union of a domain Ω0 and a lot of joined thin trees situated ε-periodically along some manifold on the boundary of Ω0. The trees have finite number of branching levels. The following nonlinear Robin boundary condition ∂νvε+εαiμ(t,x2,vε)=εβgε is given on the boundaries of the branches from the i-th branching layer; {αi} and β are real parameters. The asymptotic analysis of this problem is made as ε→0, i.e., when the number of the thin trees infinitely increases and their thickness vanishes. In particular, the corresponding homogenized problem is found and the existence and uniqueness of its solution in an anisotropic Sobolev space of multi-sheeted functions is proved. We construct the asymptotic approximation for the solution vε and prove the corresponding asymptotic estimate in the space C([0,T];L2(Ωε))∩L2(0,T;H1(Ωε)), which shows the influence of the parameters {αi} and β on the asymptotic behavior of the solution.

Compensation of Wave Actuator Dynamics for Nonlinear Systems

The problem of stabilization of PDE-ODE cascades has been solved in the linear case for several PDE classes, whereas in the nonlinear case the problem has been solved only for the transport/delay PDE, namely for compensation of an arbitrary delay at the input of a nonlinear plant. Motivated by a specific engineering application in off-shore drilling, we solve the problem of stabilization of the cascade of a wave PDE with a general nonlinear ODE. Due to the presence of nonlinearities of arbitrary growth and the time-reversibility of the wave PDE, and due to the possibility of using arguments based on Lyapunov functionals or explicit solutions, several stability analysis approaches are possible. We present stability results in the H <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> × H <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> and C <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sup> × C <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</sup> norms for general nonlinear ODEs, as well as in the H <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> × L <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> norm for linear ODEs. We specialize our general design for wave PDE-ODE cascades to the case of a wave PDE whose uncontrolled end does not drive an ODE but is instead governed by a nonlinear Robin boundary condition (a “nonlinear spring,” as in the friction law in drilling). This is the first global stabilization result for wave equations that incorporate non-collocated destabilizing nonlinearities of superlinear growth. We present two numerical examples, one with a nonlinear ODE and one with a nonlinear spring at the uncontrolled boundary of the wave PDE.

Unsaturated subsurface flow with surface water and nonlinear in- and outflow conditions

We analytically and numerically analyze groundwater flow in a homogeneous soil described by the Richards equation, coupled to surface water represented by a set of ordinary differential equations (ODEs) on parts of the domain boundary, and with nonlinear outflow conditions of Signorini's type. The coupling of the partial differential equation (PDE) and the ODE's is given by nonlinear Robin boundary conditions. This paper provides two major new contributions regarding these infiltration conditions. First, an existence result for the continuous coupled problem is established with the help of a regularization technique. Second, we analyze and validate a solver-friendly discretization of the coupled problem based on an implicit–explicit time discretization and on finite elements in space. The discretized PDE leads to convex spatial minimization problems which can be solved efficiently by monotone multigrid. Numerical experiments are provided using the DUNE numerics framework.

Homogenization for the p-Laplace operator and nonlinear Robin boundary conditions in perforated media along (n − 1)-dimensional manifolds

Homogenization for the p-Laplace operator and nonlinear Robin boundary conditions in perforated media along (n − 1)-dimensional manifolds

Bounded solutions for a quasilinear singular problem with nonlinear Robin boundary conditions

The paper deals with the existence of bounded positive solutions for a quasilinear singular problem with nonlinear Robin conditions. The nonlinear term is singular with respect to the solution and has a quadratic growth with respect to its gradient. The result is obtained by approximation. We define an approximate problem which does not present the singularity and is bounded with respect to the gradient. In a first step we prove the existence of a bounded solution of this problem, using the Schauder fixed-point theorem. Then, we prove uniform a priori estimates for these solutions. Finally, by some equintegrability arguments, we pass to the limit in the approximate problem and show that the approximate solution converges to a solution of our problem. We also prove a strong-maximum property for the problem. In the last part of the paper, we consider the case of a nonsingular but more general quadratic growth with nonlinear Robin conditions.

A Spatially Extended Model of Kinase-Receptor Interaction

We perform a mathematical analysis of a spatially extended model describing mutual phosphorylation of cytosolic kinases and membrane receptors. The analyzed regulatory system is a part of signal transduction mechanisms, which enables communication of the cell with its extracellular environment or other cells. The mutual receptor-kinase interaction is characteristic for immune receptors and Src family kinases. From the mathematical viewpoint, the considered system is interesting because it couples differential equations defined in a domain $\Omega$ and on its boundary $\partial\Omega$ via nonlinear Robin boundary conditions. Assuming a spherically symmetric framework, our approach is to consider an auxiliary problem in which the Robin boundary condition on the external boundary of the spherical shell $\Omega$ is replaced by a uniform Dirichlet boundary condition. This method allows us to find the stationary spherically symmetric solutions, both stable and unstable. Interestingly, numerical computations sug...

Solutions of the Boussinesq equation subject to a nonlinear Robin boundary condition

Key Points The non‐linear Boussinesq equation is solved subject to a third kind BC An adequate non‐linear form of the Robin BC is used The accuracy of the semi‐analytical solution is checked using numerical results

Homogenization of a quasilinear elliptic problem with nonlinear Robin boundary conditions

This article is devoted to the homogenization of a quasilinear elliptic equation with oscillating coefficients in a periodically perforated domain. A nonlinear Robin condition is prescribed on the boundary of the holes, depending on a real parameter γ ≥ 1. We suppose that the data satisfy some suitable hypotheses which ensure, as proved by the authors in Cabarrubias and Donato [B. Cabarrubias and P. Donato, Existence and uniqueness for a quasilinear elliptic problem with nonlinear Robin conditions, Carpathian J. Math. (2) (2011) (to appear)], the existence and the uniqueness of a solution of the problem. In particular, suitable growth conditions are assumed on the nonlinear boundary term, as done in Cioranescu–Donato–Zaki [D. Cioranescu, P. Donato and R. Zaki, Asymptotic behavior of elliptic problems in perforated domains with nonlinear boundary conditions, Asymptot. Anal. 53 (2007), pp. 209–235]. On the quasilinear term, some assumptions on the modulus of continuity introduced in Chipot [M. Chipot, Elliptic Equations: An Introductory Course, Birkhauser Verlag AG, Germany, 2009] and weaker than a Lipschitz condition are prescribed. We study the convergence to a limit problem, which is identified by using the periodic unfolding method. We also prove the well-posedness of the limit system. To do that, we show that the homogenized operator inherits the modulus of continuity of the initial problem. As a consequence, the uniqueness of a solution of the homogenized quasilinear problem follows.

On the Laplace equation with a supercritical nonlinear Robin boundary condition in the half-space

We study the Laplace equation in the half-space \({\mathbb{R}_{+}^{n}}\) with a nonlinear supercritical Robin boundary condition \({\frac{\partial u}{\partial\eta }+\lambda u=u\left\vert u\right\vert^{\rho-1}+f(x)}\) on \({\partial \mathbb{R}_{+}^{n}=\mathbb{R}^{n-1}}\), where n ≥ 3 and λ ≥ 0. Existence of solutions \({u \in E_{pq}= \mathcal{D}^{1, p}(\mathbb{R}_{+}^{n}) \cap L^{q}(\mathbb{R}_{+}^{n})}\) is obtained by means of a fixed point argument for a small data \(f \in {L^{d}(\mathbb{R}^{n-1})}\). The indexes p, q are chosen for the norm \({\Vert\cdot\Vert_{E_{pq}}}\) to be invariant by scaling of the boundary problem. The solution u is positive whether f(x) > 0 a.e. \({x\in\mathbb{R}^{n-1}}\). When f is radially symmetric, u is invariant under rotations around the axis {xn = 0}. Moreover, in a certain Lq-norm, we show that solutions depend continuously on the parameter λ ≥ 0.

Global attractors in stronger norms for a class of parabolic systems with nonlinear boundary conditions

For a class of quasilinear parabolic systems with nonlinear Robin boundary conditions we construct a compact local solution semiflow in a nonlinear phase space of high regularity. We further show that a priori estimates in lower norms are sufficient for the existence of a global attractor in this phase space. The approach relies on maximal L p -regularity with temporal weights for the linearized problem. An inherent smoothing effect due to the weights is employed for obtaining gradient estimates. In several applications we can improve the convergence to an attractor by one regularity level.

Multiple solutions for an eigenvalue problem involving [formula omitted]-Laplacian type operators

In this paper we establish the existence of two nontrivial weak solutions of some eigenvalue problems, involving a general elliptic operator in divergence form. First we study a one parameter problem under homogeneous Dirichlet boundary conditions in bounded domains, and then a two parameters problem under inhomogeneous nonlinear Robin boundary conditions in regular bounded domains. These results complete in several directions some recent papers, where a Ricceri type critical point theorem was used. The main argument of this paper is based on two recent theorems due to Arcoya and Carmona (2007) in [2] which we use in a slightly modified version, see Section 2.

Asymptotic approximations for solutions to quasilinear and linear parabolic problems with different perturbed boundary conditions in perforated domains

We consider quasilinear and linear parabolic problems with rapidly oscillating coefficients in a domain Ω e that is e-periodically perforated by small holes of order . The holes are divided into three e-periodical sets depending on boundary conditions. The homogeneous Dirichlet boundary conditions are imposed for holes of one set, whereas, for holes in the remaining sets, different inhomogeneous Neumann and nonlinear Robin boundary conditions involving additional perturbation parameters are imposed. For a solution to the quasilinear problem we find the leading terms of the asymptotic expansion and prove asymptotic estimates that show the influence of perturbation parameters. In the linear case, we construct and justify a complete asymptotic expansion of the solution by using the two-scale asymptotic expansion method. Bibliography: 25 titles. Illustrations: 1 figure.

Existence results for quasilinear elliptic exterior problems involving convection term and nonlinear Robin boundary conditions

In this paper, the authors establish the existence of solutions for a class of elliptic exterior problems involving convection terms and nonlinear Robin boundary conditions. The proof of the result is made by combining Galerkin method with a priori estimates for this kind of problem.

Asymptotic analysis of a parabolic semilinear problem with nonlinear boundary multiphase interactions in a perforated domain

We consider a parabolic semilinear problem with rapidly oscillating coefficients in a domain Ωe that is e-periodically perforated by small holes of size $\mathcal {O}$ (e). The holes are divided into two e-periodical sets depending on the boundary interaction at their surfaces, and two different nonlinear Robin boundary conditions σe(u e) + eκ m (u e) = eg (m) e, m = 1, 2, are imposed on the boundaries of holes. We study the asymptotics as e → 0 and establish a convergence theorem without using extension operators. An asymptotic approximation of the solution and the corresponding error estimate are also obtained. Bibliography: 60 titles. Illustrations: 1 figure.

The heat equation with nonlinear generalized Robin boundary conditions

Let Ω ⊂ R N be a bounded domain and let μ be an admissible measure on ∂ Ω. We show in the first part that if Ω has the H 1 -extension property, then a realization of the Laplace operator with generalized nonlinear Robin boundary conditions, formally given by ∂ u ∂ ν d σ + β ( x , u ) d μ = 0 on ∂ Ω, generates a strongly continuous nonlinear submarkovian semigroup S B = ( S B ( t ) ) t ⩾ 0 on L 2 ( Ω ) . We also obtain that this semigroup is ultracontractive in the sense that for every u , v ∈ L p ( Ω ) , p ⩾ 2 and every t > 0 , one has ‖ S B ( t ) u − S B ( t ) v ‖ ∞ ⩽ C 1 e C 2 t t − N 2 p ‖ u − v ‖ p , for some constants C 1 , C 2 ⩾ 0 . In the second part, we prove that if Ω is a bounded Lipschitz domain, one can also define a realization of the Laplacian with nonlinear Robin boundary conditions on C ( Ω ¯ ) and this operator generates a strongly continuous and contractive nonlinear semigroup.