Nonlocal overdetermined boundary value problem for stationary Navier-Stokes equations

A stationary system of Stokes and Navier-Stokes equations describing the flow of a homogeneous incompressible fluid in a bounded domain is considered. The vector of the flow velocity and a finite number of nonlocal conditions are defined at a part of the domain boundary. It is proved that, in the linear case, the problem has at least one stable solution. In the nonlinear case, the local solvability of the problem is proved.     

On the Lp–Lq maximal regularity for Stokes equations with Robin boundary condition in a bounded domain

We obtain the L_p–L_q maximal regularity of the Stokes equations with Robin boundary condition in a bounded domain in ?ⁿ (n?2). The Robin condition consists of two conditions: v ? u=0 and αu+β(T(u, p)v – 〈T(u, p)v, v〉v)=h on the boundary of the domain with α, β?0 and α+β=1, where u and p denote a velocity vector and a pressure, T(u, p) the stress tensor for the Stokes flow and v the unit outer normal to the boundary of the domain. It presents the slip condition when β=1 and non‐slip one when α=1, respectively. The slip condition is appropriate for problems that involve free boundaries. Copyright © 2006 John Wiley & Sons, Ltd.     

On a problem of nonstationary two-dimensional motion of micropolar fluid when normal stress and tangential velocity are given on the boundary

The object of this paper is to investigate the solution of nonstationary motion of micropolar fluid in the half-plane when the normal stresses and tangential velocities are given on the boundary. The Laplace-Fourier transform technique is used to point out the solution by quadratures. Numerical results of the physical quantities such as tangential and normal velocities, pressure, microrotation, stresses and momentums are obtained and displayed graphically. The problem could be met in the study of the vibrations of a memberance or a plate contacting with the fluid.     

Non-homogeneous boundary value problem for one-dimensional compressible viscous micropolar fluid model: a local existence theorem

An initial-boundary value problem for 1-D flow of a compressible viscous heat-conducting micropolar fluid is considered; the fluid is assumed thermodynamically perfect and polytropic. The original problem is transformed into homogeneous one and studied the Faedo-Galerkin method. A local-in-time existence of generalized solution is proved.      

Determination of the source parameter in a heat equation with a non-local boundary condition

In this article we consider the inverse problem of identifying a time dependent unknown coefficient in a parabolic problem subject to initial and non-local boundary conditions along with an overspecified condition defined at a specific point in the spatial domain. Due to the non-local boundary condition, the system of linear equations resulting from the backward Euler approximation have a coefficient matrix that is a quasi-tridiagonal matrix. We consider an efficient method for solving the linear system and the predictor–corrector method for calculating the solution and updating the estimate of the unknown coefficient. Two model problems are solved to demonstrate the performance of the methods.     

1-D compressible viscous micropolar fluid model with non-homogeneous boundary conditions for temperature: A local existence theorem

We consider non-stationary 1-D flow of a compressible viscous and heat-conducting micropolar fluid, assuming that it is in thermodynamical sense perfect and polytropic. The homogeneous boundary conditions for velocity and microrotation, as well as non-homogeneous boundary conditions for temperature are assumed. Using the Faedo-Galerkin method we prove a local-in-time existence of a generalized solution.     

Riemann-Hilbert approach to TD equation with nonzero boundary condition

We extend the Riemann-Hilbert approach to the TD equation, which is a highly nonlinear differential integrable equation. Zero boundary condition at infinity for the TD equation is not suitable. Inverse scattering transform for this equation involves the singular Riemann-Hilbert problem, which means that the sectionally analytic functions have singularities on the boundary curve. Regularization procedures of the singular Riemann-Hilbert problem for two cases, the general case and the case for reflectionless potentials, are considered. Solitonic solutions to the TD equation are given.     

A free boundary problem of diffusion equation with integral condition

We consider a free boundary problem of heat equation with integral condition on the unknown free boundary. Results of solution regularity and problem well-posedness are presented.     

Two-dimensional inverse quasilinear parabolic problem with periodic boundary condition

In this study, we consider a coefficient problem of a quasi-linear two-dimensional parabolic inverse problem with periodic boundary and integral over determination conditions. We prove the existence, uniqueness and continuously dependence upon the data of the solution by iteration method. Also, we consider numerical solution for this inverse problem by using linearization and the implicit finite-difference scheme.     

On a resolvent estimate of the Stokes equation with Neumann–Dirichlet‐type boundary condition on an infinite layer

This paper is concerned with the standard Lp estimate of solutions to the resolvent problem for the Stokes operator on an infinite layer with ‘Neumann–Dirichlet‐type’ boundary condition. Copyright © 2004 John Wiley & Sons, Ltd.     

Shape-dependent natural boundary condition of Lagrangian field

In the framework of the Lagrangian field theory, we derive the equations characterizing shape-dependent natural boundary conditions from the Hamilton’s principle. Of these equations, one exhibits mathematical pattern similar to general relativity. In this equation, one side of the sign of equality is the energy–momentum tensor of field and another side is the combination of mean curvature and Gaussian curvature of boundary surface. Meanwhile, we verify that the shape-dependent natural boundary condition can be simplified into the shape equation of lipid vesicle or the generalized Young–Laplace’s equation under different condition.     

Analysis of the immersed boundary method for a finite element Stokes problem

Convergence results are presented for the immersed boundary (IB) method applied to a model Stokes problem. As a discretization method, we use the finite element method. First, the immersed force field is approximated using a regularized delta function. Its error in the W^{?1, p} norm is examined for 1 ≤ p < n/(n ? 1), with n representing the space dimension. Subsequently, we consider IB discretization of the Stokes problem and examine the regularization and discretization errors separately. Consequently, error estimate of order h^{1 ? α} in the W^{1, 1} × L¹ norm for the velocity and pressure is derived, where α is an arbitrary small positive number. The validity of those theoretical results is confirmed from numerical examples.     

The thermostat problem with a nonlocal nonlinear boundary condition

The thermostat controller for an air-conditioning system isusually placed in a position at some distance from the unitand this can lead to large swings in temperature. This paperaddresses this question by studying a paradigm—a one-dimensionalheat conduction equation with and without heat loss, and wherethe flux of heat extracted or input by the unit is consideredto be a function of the temperature at the other end. The essential results are that the system can be unstable andthat this is exacerbated both by a more powerful air-conditioningunit and by more efficient insulation.     

The Sturm-Liouville problem with a nonlocal boundary condition

In this paper, we consider the Sturm-Liouville problem with one classical and another nonlocal boundary condition. We investigate general properties of the characteristic function and spectrum for such a problem in the complex case. In the second part, we investigate the case of real eigenvalues, analyze the dependence of the spectrum on parameters of the boundary condition, and describe the qualitative behavior of all eigenvalues subject to of the nonlocal boundary condition. Dedicated to N. S. Bakhvalov (1934–2005) Published in Lietuvos Matematikos Rinkinys, Vol. 47, No. 3, pp. 410–428, July–September, 2007.     

Generalized impedance boundary condition at high frequency for a domain with thin layer: the circular case

Consider a conducting disk surrounded by a thin dielectric layer submitted to an electric field at the pulsation ω. The conductivity of the layer grows like ω^1?γ, γ∈[0,1], when the pulsation ω?tends to infinity. Using a pseudodifferential approach on the torus, we build an equivalent boundary condition with the help of an appropriate factorization of Helmholtz operator in the layer. This generalized impedance condition approximates the thin membrane in the high frequency limit for small thickness of the layer. L ²-error estimates are given and we illustrate our results with numerical simulations. This work extends, in the circular geometry, previous works of Lafitte and Lebeau (Lafitte O. Lebeau G. 1993, Équations de Maxwell et opérateur d’impédance sur le bord d’un obstacle convexe absorbant. Comptes Rendus de l ' Académic dis Science, Paris, Série I, Mathématiques, 316(11), 1177–1182); (Lafitte O.D., 1999, Diffraction in the high frequency regime by a thin layer of dielectric material. I. The equivalent impedance boundary condition. SIAM Journal on Applied Mathematics, 59(3), 1028–1052 (electronic)) in which γ?identically equals zero.