Nonlinear Robin Boundary Conditions Research Articles

Solvability analysis for the Boussinesq model of heat transfer under the nonlinear Robin boundary condition for the temperature.

We consider the new boundary value problem for the generalized Boussinesq model of heat transfer under the inhomogeneous Dirichlet boundary condition for the velocity and under mixed boundary conditions for the temperature. It is assumed that the viscosity, thermal conductivity and buoyancy force in the model equations, as well as the heat exchange boundary coefficient, depend on the temperature. The mathematical apparatus for studying the inhomogeneous boundary value problem under study based on the variational method is being developed. Using this apparatus, we prove the main theorem on the global existence of a weak solution of the mentioned boundary value problem and establish sufficient conditions for the problem data ensuring the local uniqueness of the weak solution that has the additional property of smoothness with respect to temperature. This article is part of the theme issue 'Non-smooth variational problems with applications in mechanics'.

Homogenization of a reaction-diffusion problem with large nonlinear drift and Robin boundary data

We study the periodic homogenization of a reaction-diffusion problem with large nonlinear drift and Robin boundary condition posed in an unbounded perforated domain. The nonlinear problem is associated with the hydrodynamic limit of a totally asymmetric simple exclusion process (TASEP) governing a population of interacting particles crossing a domain with obstacle. We are interested in deriving rigorously the upscaled model equations and the corresponding effective coefficients for the case when the microscopic dynamics are linked to a particular choice of characteristic length and time scales that lead to an exploding nonlinear drift. The main mathematical difficulty lies in proving the two-scale compactness and strong convergence results needed for the passage to the homogenization limit. To cope with the situation, we use the concept of two-scale compactness with drift, which is similar to the more classical two-scale compactness result but it is defined now in moving coordinates. We provide as well a strong convergence result for the corrector function, starting this way the search for the order of the convergence rate of the homogenization process for our target nonlinear drift problem.

Existence analysis of a cross-diffusion system with nonlinear Robin boundary conditions for vesicle transport in neurites

A one-dimensional cross-diffusion system modeling the transport of vesicles in neurites is analyzed. The equations are coupled via nonlinear Robin boundary conditions to ordinary differential equations for the number of vesicles in the reservoirs in the cell body and the growth cone at the end of the neurite. The existence of bounded weak solutions is proved by using the boundedness-by-entropy method. Numerical simulations show the dynamical behavior of the concentrations of anterograde and retrograde vesicles in the neurite.

Classical solutions to the one‐dimensional logarithmic diffusion equation with nonlinear Robin boundary conditions

AbstractIn this paper, we investigate the behavior of classical solutions to the one‐dimensional (1D) logarithmic diffusion equation with nonlinear Robin boundary conditions, namely, where γ is a constant. Let u0 > 0 be a smooth function defined on [ − l, l], and which satisfies the compatibility condition We show that for γ > 0, classical solutions to the logarithmic diffusion equation above with initial data u0 are global and blow‐up in infinite time, and that for p > 2 there is finite time blow‐up. Also, we show that in the case of γ < 0, , solutions to the logarithmic diffusion equation with initial data u0 are global and blow‐down in infinite time, but if p ⩽ 1 there is finite time blow‐down. For some of the cases mentioned above, and some particular families of examples, we provide blow‐up rates and blow‐down rates along sequences of times. Our approach is based on studying the Ricci flow on a cylinder endowed with a ‐symmetric metric, and some comparison arguments. Then, we bring our ideas full circle by proving a new long time existence result for the Ricci flow on a cylinder without any symmetry assumption. Finally, we show a blow‐down result for the logarithmic diffusion equation on a disc.

Analytical solution of Nye–Tinker–Barber model by Laplace transform

The Nye–Tinker–Barber model is a classical convection–diffusion model for nutrient uptake by plant roots in cylindrical coordinates and has one nonlinear left Robin boundary condition with Michaelis–Menten function of concentration. First the Michaelis–Menten function is fitted into a function of time by numerical concentration at root surface from difference scheme, and then the Laplace and numerical inverse Laplace transforms — Zakian inversion method are taken to obtain the approximate analytical solution. Compared with other solutions made by difference scheme, Stehfest inversion method and previous analytical methods, it is found that the analytical solution obtained by Laplace and Zakian inversion transforms has higher accuracy and computation efficiency. This analytical method can be extended to other nutrient uptake models with Michaelis–Menten function.

On the square-root approximation finite volume scheme for nonlinear drift-diffusion equations

We study a finite volume scheme for the approximation of the solution to convection diffusion equations with nonlinear convection and Robin boundary conditions. The scheme builds on the interpretation of such a continuous equation as the hydrodynamic limit of some simple exclusion jump process. We show that the scheme admits a unique discrete solution, that the natural bounds on the solution are preserved, and that it encodes the second principle of thermodynamics in the sense that some free energy is dissipated along time. The convergence of the scheme is then rigorously established thanks to compactness arguments. Numerical simulations are finally provided, highlighting the overall good behavior of the scheme.

Spatio-Temporal SIR Model with Robin Boundary Condition and Automatic Lockdown Policy.

This paper presents the SIR space-time model, which is a coupled reaction-diffusion system with nonlinear Robin boundary conditions. These boundary conditions are supposed to lock the border (no outflow neither immigration nor migration) when the number of infected individuals explode, and this may be considered as an automatic containment or lock-down. In practice, we can precise some threshold for the number of infected individuals and when it is reached the model locks the region automatically. This work provides a thorough study of the presented model, including the existence and uniqueness of the solution, its boundedness and its asymptotic behaviour. We end with some numerical experiments performed on the basis of the finite difference approach and Newton's method to highlight and validate the theoretical results.

Boundary homogenization with large reaction terms on a strainer-type wall

We consider a homogenization problem for the Laplace operator posed in a bounded domain of the upper half-space, a part of its boundary being in contact with the plane {x_3=0}. On this part, the boundary conditions alternate from Neumann to nonlinear-Robin, being of Dirichlet type outside. The nonlinear-Robin boundary conditions are imposed on small regions periodically placed along the plane and contain a Robin parameter that can be very large. We provide all the possible homogenized problems, depending on the relations between the three parameters: period varepsilon , size of the small regions r_varepsilon and Robin parameter beta (varepsilon ). In particular, we address the convergence, as varepsilon tends to zero, of the solutions for the critical size of the small regions r_varepsilon =O(varepsilon ^{ 2}). For certain beta (varepsilon ), a nonlinear capacity term arises in the strange term which depends on the macroscopic variable and allows us to extend the usual capacity definition to semilinear boundary conditions.

Interaction of scales for a singularly perturbed degenerating nonlinear Robinproblem.

We study the asymptotic behaviour of solutions of a boundary value problem for the Laplace equation in a perforated domain in , , with a (nonlinear) Robin boundary condition on the boundary of the small hole. The problem we wish to consider degenerates in three respects: in the limit case, the Robin boundary condition may degenerate into a Neumann boundary condition, the Robin datum may tend to infinity, and the size ϵ of the small hole where we consider the Robin condition collapses to 0. We study how these three singularities interact and affect the asymptotic behaviour as ϵ tends to 0, and we represent the solution and its energy integral in terms of real analytic maps and known functions of the singular perturbation parameters.This article is part of the theme issue ‘Non-smooth variational problems and applications’.

Convergence rates for the Allen–Cahn equation with boundary contact energy: the non-perturbative regime

We extend the recent rigorous convergence result of Abels and Moser (SIAM J Math Anal 54(1):114–172, 2022. https://doi.org/10.1137/21M1424925) concerning convergence rates for solutions of the Allen–Cahn equation with a nonlinear Robin boundary condition towards evolution by mean curvature flow with constant contact angle. More precisely, in the present work we manage to remove the perturbative assumption on the contact angle being close to 90^circ . We establish under usual double-well type assumptions on the potential and for a certain class of boundary energy densities the sub-optimal convergence rate of order {varepsilon ^{frac{1}{2}}} for general contact angles alpha in (0,pi ). For a very specific form of the boundary energy density, we even obtain from our methods a sharp convergence rate of order varepsilon ; again for general contact angles alpha in (0,pi ). Our proof deviates from the popular strategy based on rigorous asymptotic expansions and stability estimates for the linearized Allen–Cahn operator. Instead, we follow the recent approach by Fischer et al. (SIAM J Math Anal 52(6):6222–6233, 2020. https://doi.org/10.1137/20M1322182), thus relying on a relative entropy technique. We develop a careful adaptation of their approach in order to encode the constant contact angle condition. In fact, we perform this task at the level of the notion of gradient flow calibrations. This concept was recently introduced in the context of weak-strong uniqueness for multiphase mean curvature flow by Fischer et al. (arXiv:2003.05478v2).

Well-Posedness for Nonlinear Parabolic Stochastic Differential Equations with Nonlinear Robin Conditions

In this paper, we present the existence and uniqueness of strong probabilistic solutions for nonlinear parabolic Stochastic Partial Differential Equations (SPDEs) with nonlinear Robin boundary conditions in a domain with holes. On the boundary of the holes, a nonlinear Robin condition is imposed, while a homogeneous Dirichlet condition is prescribed on the exterior boundary. The coefficient matrix is assumed to be symmetric, while the nonlinear random forces are assumed to satisfy some types of regularities. We use Galerkin’s approximation method, probabilistic compactness results and some results from stochastic calculus.

Solutions for a quasilinear elliptic p⃗(x)${\vec{p}(x)}$‐Kirchhoff type problem with weight and nonlinear Robin boundary conditions

AbstractThis paper deals with the existence and multiplicity of weak solutions to a class of quasilinear elliptic ‐Kirchhoff type problems with weight and a nonlinear Robin boundary condition such as where Ω is a smooth bounded domain. Under suitable conditions on the data, we show the existence and multiplicity of weak solutions by means of a variational approach in the framework of anisotropic Sobolev spaces with variable exponents.

Convergence of the Allen--Cahn Equation with a Nonlinear Robin Boundary Condition to Mean Curvature Flow with Contact Angle Close to 90°

Convergence of the Allen--Cahn Equation with a Nonlinear Robin Boundary Condition to Mean Curvature Flow with Contact Angle Close to 90°

Sequential estimation of the time-dependent heat transfer coefficient using the method of fundamental solutions and particle filters

In many thermal engineering problems involving high temperatures/high pressures, the boundary conditions are not fully known since there are technical difficulties in obtaining such data in hostile conditions. To perform the task of estimating the desired parameters, inverse problem formulations are required, which entail to performing some extra measurements of certain accessible and relevant quantities. In this paper, justified also by uniqueness of solution conditions, this extra information is represented by either local or non-local boundary temperature measurements. Also, the development of numerical methods for the study of coefficient identification thermal problems is an important topic of research. In addition, in order to decrease the computational burden, meshless methods are becoming popular. In this article, we combine, for the first time, the method of fundamental solutions (MFS) with a particle filter sequential importance resampling (SIR) algorithm for estimating the time-dependent heat transfer coefficient in inverse heat conduction problems. Two different types of measurements are used. Numerical results indicate that the combination of MFS and SIR shows high performance on several test cases, which include both linear and nonlinear Robin boundary conditions, in comparison with other available methods.

Boundary optimal control of time-space SIR model with nonlinear Robin boundary condition.

A boundary optimal control problem arising in time-space SIR epidemic models is treated. In this work we aim with the control of the flux of infected individuals crossing part of boundary. On the other side of the domain, we suppose a nonlinear boundary condition of third kind: nonlinear Robin boundary condition, this condition models immersing individual crossing this part of the boundary of the domain of study. We give the existence and uniqueness of the solution of both state and optimal control problem ending some numerical tests throughout a simple example.

Existence and uniqueness of maximal strong solution of a 1D blood flow in a network of vessels

We study the well-posedness of a system of one-dimensional partial differential equations modeling blood flows in a network of vessels with viscoelastic walls. We prove the existence and uniqueness of maximal strong solution for this type of hyperbolic/parabolic model. We also prove a stability estimate under suitable nonlinear Robin boundary conditions.

Uniform convergence and asymptotics for problems in domains finely perforated along a prescribed manifold in the case of the homogenized Dirichlet condition

A boundary value problem for a second-order elliptic equation with variable coefficients is considered in a multidimensional domain which is perforated by small holes along a prescribed manifold. Minimal natural conditions are imposed on the holes. In particular, all of these are assumed to be of approximately the same size and have a prescribed minimal distance to neighbouring holes, which is also a small parameter. The shape of the holes and their distribution along the manifold are arbitrary. The holes are divided between two sets in an arbitrary way. The Dirichlet condition is imposed on the boundaries of holes in the first set and a nonlinear Robin boundary condition is imposed on the boundaries of holes in the second. The sizes and distribution of holes with the Dirichlet condition satisfy a simple and easily verifiable condition which ensures that these holes disappear after homogenization and a Dirichlet condition on the manifold in question arises instead. We prove that the solution of the perturbed problem converges to the solution of the homogenized one in the -norm uniformly with respect to the right-hand side of the equation, and an estimate for the rate of convergence that is sharp in order is deduced. The full asymptotic solution of the perturbed problem is also constructed in the case when the holes form a periodic set arranged along a prescribed hyperplane. Bibliography: 32 titles.

Homogenization of a quasilinear elliptic problem in domains with small holes

This work aims to provide the asymptotic behavior of some class of quasilinear problems posed in a domain with small holes in for N>2, as . A nonlinear Robin boundary condition is prescribed on some of the holes, while a Dirichlet boundary one is imposed on the other holes as well as on the outer boundary. We apply the Periodic Unfolding Method for the homogenization of this problem.

A Numerical Model for Steel Continuous Casting Problem in a Time-variable Domain

A mathematical model and numerical method for simulation of the continuous casting process in a variable in time domain are presented. The variable geometry of the slab is caused by the change in time of the width of the mould. The mathematical model of the process is a Stefan problem with prescribed convection and non-linear Robin boundary condition. Considered differential equation is approximated by a finite difference scheme, which is constructed in several steps. First, a semi-discrete problem is constructed using the method of characteristics with respect to the time variable. Then, at each time level, the current elliptic problem in a curvilinear domain is replaced by a problem in the parallelepiped domain using the fictitious domain method. Finally, the boundary-value problem in the parallelepiped domain is approximated by a finite-difference scheme. The constructed fully discrete problem in algebraic form is a system of nonlinear equations containing a diagonal monotone operator and a linear part with a symmetric and positive definite $$M$$ -matrix. To solve the resulting system of nonlinear algebraic equations, well-known iterative solution methods can be applied.

ISS-like estimates for nonlinear parabolic PDEs with variable coefficients on higher dimensional domains

This paper presents a maximum principle-based approach in the establishment of ISS-like estimates for a class of nonlinear parabolic partial differential equations (PDEs) with variable coefficients and different types of nonlinear boundary conditions on higher dimensional domains. Comparing with the existing literature, the ISS-like estimates established in this paper are independent of the nonlinear terms of the PDEs. Technical development on ISS analysis of the considered systems is detailed, and an example of establishing ISS-like estimates for a nonlinear parabolic equation with, respectively, a nonlinear Robin boundary condition and a nonlinear Dirichlet boundary condition is provided to illustrate the application of the developed method.