Inhomogeneous Boundary Conditions Research Articles

Well-posedness of a nonlinear second-order anisotropic reaction-diffusion problem with nonlinear and inhomogeneous dynamic boundary conditions

The paper is concerned with a qualitative analysis for a nonlinear second-order boundary value problem, endowed with nonlinear and inhomogeneous dynamic boundary conditions, extending the types of bounday conditions already studied. Under certain assumptions on the input data: $f_{_1}(t,x)$, $w(t,x)$ and $u_0(x)$, we prove the well-posedness (the existence, a priori estimates, regularity and uniqueness) of a classical solution in the Sobolev space $W^{1,2}_p(Q)$. This extends previous works concerned with nonlinear dynamic boundary conditions, allowing to the present mathematical model to better approximate the real physical phenomena (the anisotropy effects, phase change in $\Omega$ and at the boundary $\partial\Omega$, etc.).

Solvability Analysis of a Mixed Boundary Value Problem for Stationary Magnetohydrodynamic Equations of a Viscous Incompressible Fluid

We investigate the boundary value problem for steady-state magnetohydrodynamic (MHD) equations with inhomogeneous mixed boundary conditions for a velocity vector, given the tangential component of a magnetic field. The problem represents the flow of electrically conducting viscous fluid in a 3D-bounded domain, which has the boundary comprising several parts with different physical properties. The global solvability of the boundary value problem is proved, a priori estimates of the solutions are obtained, and the sufficient conditions on data, which guarantee a solution’s local uniqueness, are determined.

Boundary layer solutions in the Gierer–Meinhardt system with inhomogeneous boundary conditions

The use of inhomogeneous boundary conditions has become increasingly important in the study of pattern forming reaction diffusion systems. Such inhomogeneities can be used to describe, for example, the interaction between an individual cell with its environment. In this paper we consider arbitrary mixed inhomogeneous boundary conditions for the activator in the singularly perturbed Gierer-Meinhardt system. By using formal asymptotics we derive an algebraic system and nonlocal eigenvalue problem respectively describing the structure and linear stability of multi-spike solutions in the case of a one-dimensional domain, thereby extending previous results obtained in the case of inhomogeneous Neumann boundary conditions by Gomez and Wei (2021). We also rigorously prove partial stability results and provide detailed stability thresholds for two examples consisting of one- and two-spike solutions. In higher dimensions we restrict our attention to the shadow limit in which the inhibitor is well-mixed and for which we rigorously establish both the existence and stability of a boundary layer solution. In both the one- and higher-dimensional cases we find that when the boundary conditions are symmetric then a symmetric solution is stable only when the magnitude of the inhomogeneity exceeds some threshold. Below this threshold we demonstrate that a stable asymmetric two spike solution emerges in the case of a one-dimensional domain, while in the case of a two-dimensional domain we show numerical simulations illustrating the formation of a near-boundary spike.

New explicit exact traveling wave solutions of Camassa–Holm equation

In this paper, we construct two families of new explicit exact traveling wave solutions of the Camassa–Holm equation which solve an open problem in the previous paper [Zhang G, Qiao ZJ, Liu F. Cuspons and smooth solitons of the Camassa-Holm equation under an inhomogeneous boundary condition. Pac J Appl Math. 2007;1(1):113–130]. With the help of the numerical graphs and a profound mathematical analysis, we discovered the analytic formulas for those two explicit exact solutions.

Handling Neumann and Robin boundary conditions in a fictitious domain volume penalization framework

Sakurai et al. (2019) [14] presented a flux-based volume penalization (VP) approach for imposing inhomogeneous Neumann boundary conditions on embedded interfaces. The flux-based VP method modifies the diffusion coefficient of the original elliptic (Poisson) equation and uses a flux-forcing function as a source term in the equation to impose the Neumann boundary conditions. As such, the flux-based VP method can be easily incorporated into existing fictitious domain codes. Sakurai et al. relied on an analytical construction of flux-forcing functions, which limits the practicality of the approach. Because of the analytical approach taken in the prior work, only (spatially) constant flux values on simple interfaces were considered. In this paper, we present a numerical technique for constructing flux-forcing functions for arbitrarily complex boundaries. The imposed flux values are also allowed to vary spatially in our approach. Furthermore, the flux-based VP method is extended to include (spatially varying) Robin boundary conditions, which makes the flux-based VP method even more general. The numerical construction of the flux-forcing functions relies only on a signed distance function that describes the distance of a grid point from the interface and can be constructed for any irregular boundary. We consider several two- and three-dimensional test examples to access the spatial accuracy of the numerical solutions. The method is also used to simulate flux-driven thermal convection in a concentric annular domain. We formally derive the flux-based volume penalized Poisson equation satisfying Neumann/Robin boundary condition in strong form; such a derivation was not presented in Sakurai et al., where the equation first appeared for the Neumann problem. The derivation reveals that the flux-based VP approach relies on a surface delta function to impose inhomogeneous Neumann/Robin boundary conditions. However, explicit construction of the delta function is not necessary for the flux-based VP method, which makes it different from other diffuse domain equations presented in the literature.

Hydrodynamic dispersion in Hele-Shaw flows with inhomogeneous wall boundary conditions

Single-phase flow inside a Hele-Shaw cell can exhibit inhomogeneous flow fields, for example when actuated by electroosmosis with varying wall mobilities, leading to internal pressure gradients. We derive a two-dimensional dispersion model for a dissolved species in such a non-uniform flow field, utilizing a multiple-scale perturbation approach. The resulting two-dimensional transport equation is an advection–diffusion equation containing an effective dispersion tensor field and additional advection-correction terms. It can be viewed as a generalization of the well-known Taylor–Aris dispersion model. The dispersion model allows for flow fields with both stationary and oscillatory components. For the special case of non-uniform flow induced by both pressure gradients and electroosmosis, we derive expressions for the flow field in the long-wavelength limit. These include arbitrary, time-dependent functions for both the driving field as well as the wall mobilities. We discuss the general characteristics of the model using a sinusoidally varying wall mobility, and derive analytical expressions for the dispersion tensor. Then, in order to validate the model, we compare three-dimensional Lagrangian particle tracing simulations with the dispersion model for several test cases, including stationary and oscillatory shear flow as well as a recirculating flow field. For each test case, a good agreement between the full three-dimensional simulations and the results of the two-dimensional dispersion model is obtained. The presented model has the potential to significantly simplify computations of mass transport in Hele-Shaw flows.

Stationary Navier–Stokes equations under inhomogeneous boundary conditions in 3D exterior domains

Stationary Navier–Stokes equations under inhomogeneous boundary conditions in 3D exterior domains

Lp‐estimates of the Stokes resolvent with nonhomogeneous Dirichlet boundary conditions in 3D exterior domains

The article deals with the homogeneous Stokes resolvent system in a 3D exterior domain, under inhomogeneous Dirichet boundary conditions. Solutions to this boundary value problem are estimated in Lp‐norms, with the bounds in these estimates depending on the absolute value of the resolvent parameter λ in an explicit way. Two types of boundary data are considered, that is, Lp‐ and W2 − 1/p, p‐data. It is shown in particular that the Lp‐norm of the velocity outside a vicinity of the boundary, after subtraction of the gradient of a certain harmonic function, is bounded by a constant times |λ|−1 ‖b‖p. This estimate carries over to the Oseen resolvent system, leading to a result which has applications in the theory of spatial asymptotics of solutions to the 3D time‐dependent Navier‐Stokes system with Oseen term.

Numerical Solution of Fractional Elliptic Problems with Inhomogeneous Boundary Conditions

The numerical solution of fractional-order elliptic problems is investigated in bounded domains. According to real-life situations, we assumed inhomogeneous boundary terms, while the underlying equations contain the full-space fractional Laplacian operator. The basis of the convergence analysis for a lower-order boundary element approximation is the theory for the corresponding continuous problem. In particular, we need continuity results for Riesz potentials and the fractional-order extension of the theory for boundary integral equations with the Laplacian operator. Accordingly, the convergence is stated in fractional-order Sobolev norms. The results were confirmed in a numerical experiment.

On Hr(curl,Ω) estimates for a Maxwell type system in convex domains

In bounded convex domains, the regularity of a vector field u with its divu, curlu in Lr space and the tangential component or the normal component of u over the boundary in Lr space, is established for 1<r<∞. As an application, we derive an Hr(curl,Ω) estimate for solutions to a Maxwell type system with an inhomogeneous boundary condition in convex domains. In contrast to the well-posed region of r in the space Hr(curl,Ω) for the Maxwell type system in Lipschitz domains given by Kar and Sini (2016) [16], we extend the well-posed region to be optimal.

Ultraconvergence of finite element method by Richardson extrapolation for elliptic problems with inhomogeneous boundary conditions

AbstractIn this article, Richardson extrapolation technique is employed to investigate the local ultraconvergence properties of Lagrange finite element method using piecewise polynomials of degrees () for the second order elliptic problem with inhomogeneous boundary. A sequence of special graded partition are proposed and a new interpolation operator is introduced to achieve order local ultraconvergence for the displacement and derivative.

Elliptic and Parabolic Boundary Value Problems in Weighted Function Spaces

In this paper we study elliptic and parabolic boundary value problems with inhomogeneous boundary conditions in weighted function spaces of Sobolev, Bessel potential, Besov and Triebel-Lizorkin type. As one of the main results, we solve the problem of weighted Lq-maximal regularity in weighted Besov and Triebel-Lizorkin spaces for the parabolic case, where the spatial weight is a power weight in the Muckenhoupt A_{infty }-class. In the Besov space case we have the restriction that the microscopic parameter equals to q. Going beyond the Ap-range, where p is the integrability parameter of the Besov or Triebel-Lizorkin space under consideration, yields extra flexibility in the sharp regularity of the boundary inhomogeneities. This extra flexibility allows us to treat rougher boundary data and provides a quantitative smoothing effect on the interior of the domain. The main ingredient is an analysis of anisotropic Poisson operators.

Superconvergence of the Strang splitting when using the Crank-Nicolson scheme for parabolic PDEs with Dirichlet and oblique boundary conditions

We show that the Strang splitting method applied to a diffusion-reaction equation with inhomogeneous general oblique boundary conditions is of order two when the diffusion equation is solved with the Crank-Nicolson method, while order reduction occurs in general if using other Runge-Kutta schemes or even the exact flow itself for the diffusion part. We prove these results when the source term only depends on the space variable, an assumption which makes the splitting scheme equivalent to the Crank-Nicolson method itself applied to the whole problem. Numerical experiments suggest that the second order convergence persists with general nonlinearities.

The conormal and Robin boundary value problems in nonsmooth domains satisfying a measure condition

We consider elliptic equations and systems in divergence form with the conormal or the Robin boundary conditions, with small BMO (bounded mean oscillation) or variably partially small BMO coefficients. We propose a new class of domains which are locally close to a half space (or convex domains) with respect to the Lebesgue measure in the system (or scalar, respectively) case, and obtain the Wp1 estimate for the conormal problem with the homogeneous boundary condition. Such condition is weaker than the Reifenberg flatness condition, for which the closeness is measured in terms of the Hausdorff distance, and the semi-convexity condition. For the conormal problem with inhomogeneous boundary conditions, we also assume that the domain is Lipschitz. By using these results, we obtain the Wp1 and weighted Wp1 estimates for the Robin problem in these domains.

Well-posed variational formulations of Friedrichs-type systems

All finite element methods, as well as much of the Hilbert-space theory for partial differential equations, rely on variational formulations, that is, problems of the type: find $u\in V$ such that $a(v,u) = l(v)$ for each $v\in L$, where $V, L$ are Sobolev spaces. However, for systems of Friedrichs type, there is a sharp disparity between established well-posedness theories, which are not variational, and the very successful discontinuous Galerkin methods that have been developed for such systems, which are variational. In an attempt to override this dichotomy, we present, through three specific examples of increasing complexity, well-posed variational formulations of boundary and initial--boundary-value problems of Friedrichs type. The variational forms we introduce are generalizations of those used for discontinuous Galerkin methods, in the sense that inhomogeneous boundary and initial conditions are enforced weakly through integrals in the variational forms. In the variational forms we introduce, the solution space is defined as a subspace $V$ of the graph space associated with the differential operator in question, whereas the test function space $L$ is a tuple of $L^2$ spaces that separately enforce the equation, boundary conditions of characteristic type, and initial conditions.

Turbulent convection in a cube with mixed thermal boundary conditions: low Rayleigh number regime

Turbulent thermal convection was numerically investigated in a cubic cavity under homogeneous and inhomogeneous heating for Rayleigh number Ra=107 and Prandtl number Pr=6.46. Inhomogeneous heating was created only at the lower boundary using mixed boundary conditions. Three configurations of the heated regions distribution of the same heating area were considered. The total heat flux through the lower boundary substantially depends on the distribution of the heated regions and increases with decreasing of the heaters size, which is in a good agreement with Ripesi et al., 2014 and Bakhuis et al., 2018. It was found that in turbulent thermal convection under inhomogeneous boundary conditions large-scale circulation (LSC) is formed. Dynamics and structure of LSC strongly depends on the distribution of conducting regions. In case of Rayleigh-Bénard convection, the main flow is more complex with higher-order modes prevailing. Time-averaged velocity field revealed the existence of two vortex rings rotating in opposite directions and located near horizontal walls. Decomposing the velocity into free-slip modes allowed us to find that these vortex rings are stable and provide the most of system energy. The absence of mean LSC results in a specific convective heat transfer mechanism when mean and turbulent heat fluxes are spatially separated.

A fractional PDE model for turbulent velocity fields near solid walls

Abstract

Multi-spike Patterns in the Gierer–Meinhardt System with a Nonzero Activator Boundary Flux

The structure, linear stability, and dynamics of localized solutions to singularly perturbed reaction-diffusion equations has been the focus of numerous rigorous, asymptotic, and numerical studies in the last few decades. However, with a few exceptions, these studies have often assumed homogeneous boundary conditions. Motivated by the recent focus on the analysis of bulk-surface coupled problems we consider the effect of inhomogeneous Neumann boundary conditions for the activator in the singularly perturbed one-dimensional Gierer-Meinhardt reaction-diffusion system. We show that these boundary conditions necessitate the formation of spikes that concentrate in a boundary layer near the domain boundaries. Using the method of matched asymptotic expansions we construct boundary layer spikes and derive a new class of shifted Nonlocal Eigenvalue Problems for which we rigorously prove partial stability results. Moreover by using a combination of asymptotic, rigorous, and numerical methods we investigate the structure and linear stability of selected one- and two-spike patterns. In particular we find that inhomogeneous Neumann boundary conditions increase both the range of parameter values over which asymmetric two-spike patterns exist and are stable.

A Mixed Boundary Value Problem for the Laplace Equation in a semi-infinite Layer

The paper offers a solution of the mixed Dirichlet-Neumann and Dirichlet-Neumann-Robin boundary value problems for the Laplace equation in the semi-infinite layer, using the previously obtained solution of the mixed Dirichlet-Neumann boundary value problem for a layer.The functions on the right-hand sides of the boundary conditions are considered to be functions of slow growth, in particular, polynomials. The solution to boundary value problems is also sought in the class of functions of slow growth. Continuing the functions on the right-hand sides of the boundary conditions on the upper and lower sides of the semi-infinite layer from the semi-hyperplane to the entire hyperplane, we obtain the Dirichlet-Neumann problem for the layer, the solution of which is known and written in the form of a convolution. If the right-hand sides of the boundary conditions are polynomials, then the solution is also a polynomial. To the solution obtained it is necessary to add the solution of the problem for a semi-infinite layer with homogeneous boundary conditions on the upper and lower sides and with an inhomogeneous boundary condition of Dirichlet, Neumann or Robin on the lateral side. This solution is written as a series. If we take a finite segment of the series, then we obtain a solution that exactly satisfies the Laplace equation and the boundary conditions on the upper and lower sides of the semi-infinite layer and approximately satisfies the boundary condition on the lateral side.An example of solving the Dirichlet-Neumann and Dirichlet-Neumann-Robin problems is considered, describing the temperature field of a semi-infinite plate the upper side of which is heat-isolated, on the lower side the temperature is set in the form of a polynomial, and the lateral side is either heat-isolated, or holds a zero temperature, or has heat exchange with a zero-temperature environment. For the first two Dirichlet-Neumann problems, the solution is obtained in the form of polynomials. For the third Dirichlet-Neumann-Robin problem, the solution is obtained as a sum of a polynomial and a series. If in this solution the series is replaced by a finite segment, then an approximate solution of the problem will be obtained, which approximately satisfies the Robin condition on the lateral side of the semi-infinite layer.

On elliptic operators with Steklov condition perturbed by Dirichlet condition on a small part of boundary

We consider a boundary value problem for a homogeneous elliptic equation with an inhomogeneous Steklov boundary condition. The problem involves a singular perturbation, which is the Dirichlet condition imposed on a small piece of the boundary. We rewrite such problem to a resolvent equation for a self-adjoint operator in a fractional Sobolev space on the boundary of the domain. We prove the norm convergence of this operator to a limiting one associated with an unperturbed problem involving no Dirichlet condition. We also establish an order sharp estimate for the convergence rate. The established convergence implies the convergence of the spectra and spectral projectors. In the second part of the work we study perturbed eigenvalues converging to limiting simple discrete ones. We construct two-terms asymptotic expansions for such eigenvalues and for the associated eigenfunctions.