Radiation boundary conditions for wave-like equations

In the numerical computation of hyperbolic equations it is not practical to use infinite domains. Instead, one truncates the domain with an artificial boundary. In this study we construct a sequence of radiating boundary conditions for wave-like equations. We prove that as the artificial boundary is moved to infinity the solution approaches the solution of the infinite domain as O(r^?m?1/2) for the m-th boundary condition. Numerical experiments with problems in jet acoustics verify the practical nature and utility of the boundary conditions.     

The boundary elements method for magneto-hydrodynamic (MHD) channel flows at high Hartmann numbers

In this article the constant and the continuous linear boundary elements methods (BEMs) are given to obtain the numerical solution of the coupled equations in velocity and induced magnetic field for the steady magneto-hydrodynamic (MHD) flow through a pipe of rectangular and circular sections having arbitrary conducting walls. In recent decades, the MHD problem has been solved using some variations of BEM for some special boundary conditions at moderate Hartmann numbers up to 300. In this paper we develop this technique for a general boundary condition (arbitrary wall conductivity) at Hartmann numbers up to 10⁵${10}^5$ by applying some new ideas. Numerical examples show the behavior of velocity and induced magnetic field across the sections. Results are also compared with the exact values and the results of some other numerical methods.     

A three field stabilized finite element method for the Stokes equationsUne méthode d'éléments finis stabilisée à trois champs pour les équations de Stokes

We consider in this work the boundary value problem for Stokes equations on a two dimensional domain in cases where non-standard boundary conditions are given. We study the cases where pressure and normal or tangential components of the velocity are given in different parts of the boundary and solve the problem with a minimal regularity. We introduce the problem and its variational formulation which is a mixed one. The principal unknowns are the pressure and the vorticity, the multiplier is the velocity. We present the numerical discretization which needs some stabilization. We prove the convergence and the behavior of the a priori error estimates. Some numerical tests are also presented. To cite this article: M. Amara et al., C. R. Acad. Sci. Paris, Ser. I 334 (2002) 603–608.     

A stable boundary elements method for magnetohydrodynamic channel flows at high Hartmann numbers

The article is devoted to extension of boundary element method (BEM) for solving coupled equations in velocity and induced magnetic field for time dependent magnetohydrodynamic (MHD) flows through a rectangular pipe. The BEM is equipped with finite difference approach to solve MHD problem at high Hartmann numbers up to 10⁶. In fact, the finite difference approach is used to approximate partial derivatives of unknown functions at boundary points respect to outward normal vector. It yields a numerical method with no singular boundary integrals. Besides, a new approach is suggested in this article where transforms 2D singular BEM's integrals to 1D nonsingular ones. The new approach reduces computational cost, significantly. Note that the stability of the numerical scheme is proved mathematically when computational domain is discretized uniformly and Hartmann number is 40 times bigger than length of boundary elements. Numerical examples show behavior of velocity and induced magnetic field across the sections.     

A domain embedding method for mixed boundary value problems

We propose a domain embedding (fictitious domain) method for elliptic equations subject to mixed boundary conditions, and prove the sharp convergence rate. The theory provides a unified treatment for Dirichlet, Neumann, and Robin boundary conditions. To cite this article: S. Zhang, C. R. Acad. Sci. Paris, Ser. I 343 (2006).     

Classical solutions to Navier–Stokes equations for nonhomogeneous incompressible fluids with non-negative densities

We study the Navier–Stokes equations for nonhomogeneous incompressible fluids in a bounded domain Ω of R³. We first prove the existence and uniqueness of local classical solutions to the initial boundary value problem of linear Stokes equations and then we obtain the existence and uniqueness of local classical solutions to the Navier–Stokes equations with vacuum under the assumption that the data satisfies a natural compatibility condition.     

Riemann-Hilbert and Poincaré problems with discontinuous boundary conditions for some model systems of partial differential equations

We study the solvability of the Riemann-Hilbert and Poincaré problems for systems of Cauchy-Riemann and Bitsadze equations in Sobolev spaces. For a generalized system of Cauchy-Riemann equations, we pose a boundary value problem and prove its unique solvability in the Sobolev space W ₂¹ (D). By supplementing the Riemann-Hilbert boundary conditions with some new conditions, we obtain a statement of the Poincaré problem with discontinuous boundary conditions for a system of second-order Bitsadze equations; we also prove the unique solvability of this problem in Sobolev spaces.     

Numerical methods for the solution of a system of Eikonal equations with Dirichlet boundary conditions

In this Note, we discuss the numerical solution of a system of Eikonal equations with Dirichlet boundary conditions. Since the problem under consideration has infinitely many solutions, we look for those solutions which are nonnegative and of maximal (or nearly maximal) L¹-norm. The computational methodology combines penalty, biharmonic regularization, operator splitting, and finite element approximations. Its practical implementation requires essentially the solution of cubic equations in one variable and of discrete linear elliptic problems of the Poisson and Helmholtz type. As expected, when the spatial domain is a square whose sides are parallel to the coordinate axes, and when the Dirichlet data vanishes at the boundary, the computed solutions show a fractal behavior near the boundary, and particularly, close to the corners. To cite this article: B. Dacorogna et al., C. R. Acad. Sci. Paris, Ser. I 336 (2003).     

Navier–Stokes equations with nonhomogeneous boundary conditions in a convex bi-dimensional domain

We prove the existence of a weak solution to Navier–Stokes equations describing the isentropic flow of a gas in a convex and bounded region, ΩR², with nonhomogeneous Dirichlet boundary conditions on ∂Ω. These results are also extended to flow domain surrounding an obstacle.     

Contrôlabilité exacte frontière pour les équations des ondes quasi linéaires unidimensionnelles

By means of the existence and uniqueness of semi-global C¹ solution to the mixed initial-boundary value problem with general nonlinear boundary conditions for first order quasilinear hyperbolic systems with zero eigenvalues, we present a unified method to establish the exact boundary controllability for 1-D quasilinear wave equations with boundary conditions of different types. To cite this article: T.T. Li, L.X. Yu, C. R. Acad. Sci. Paris, Ser. I 337 (2003).     

Eigenvalue bifurcation in multiparameter families of non-self-adjoint operator matrices

We consider two-point non-self-adjoint boundary eigenvalue problems for linear matrix differential operators. The coefficient matrices in the differential expressions and the matrix boundary conditions are assumed to depend analytically on the complex spectral parameter λ and on the vector of real physical parameters p. We study perturbations of semi-simple multiple eigenvalues as well as perturbations of non-derogatory eigenvalues under small variations of p. Explicit formulae describing the bifurcation of the eigenvalues are derived. Application to the problem of excitation of unstable modes in rotating continua such as spherically symmetric MHD α ²-dynamo and circular string demonstrates the efficiency and applicability of the approach.     

Maximum Principles for Boundary-Degenerate Second Order Linear Elliptic Differential Operators

We prove weak and strong maximum principles, including a Hopf lemma, for C ² subsolutions to equations defined by linear, second-order, linear, elliptic partial differential operators whose principal symbols vanish along a portion of the domain boundary. The boundary regularity property of the C ² subsolutions along this boundary vanishing locus ensures that these maximum principles hold irrespective of the sign of the Fichera function. Boundary conditions need only be prescribed on the complement in the domain boundary of the principal symbol's vanishing locus. We obtain uniqueness and a priori maximum principle estimates for C ² solutions to boundary value and obstacle problems defined by these boundary-degenerate elliptic operators with partial Dirichlet or Neumann boundary conditions. We also prove weak maximum principles and uniqueness for W ^{1, 2} solutions to the corresponding variational equations and inequalities defined with the aide of weighted Sobolev spaces. The domain is allowed to be unbounded when the operator coefficients and solutions obey certain growth conditions.     

Energy Equality and Uniqueness of Weak Solutions to MHD Equations in L∞（0, T; L^n（Ω））

In this paper, we study the energy equality and the uniqueness of weak solutions to the MHD equations in the critical space L∞（0, T; L^n（Ω）. We prove that if the velocity u belongs to the critical space L∞（0, T; L^n（Ω）, the energy equality holds. On the basis of the energy equality, we further prove that the weak solution to the MHD equations is unique.     

非特征边界的MHD方程的边界层

考查了小粘性时非特征边界情况下MHD方程在边界附近的性质,说明速度在边界上不为零.源于之前非特征边界条件下不可压缩Navier-Stokes方程边界层的工作,证明了边界层的存在性,并得到了当粘性收敛于零时,MHD方程的解收敛于理想MHD方程的解.     

Existence of weak solutions to the three-dimensional steady compressible magnetohydrodynamic equations

The purpose of this paper is to prove the existence of a spatially periodic weak solution to the steady compressible isentropic MHD equations in R3 for any specific heat ratio γ 1.The proof is based on the weighted estimates of both pressure and kinetic energy for the approximate system which result in some higher integrability of the density,and the method of weak convergence.According to the author's knowledge,it is the first result that treats in three dimensions the existence of weak solutions to the steady compressible MHD equations with γ 1.     

Stokes and Navier–Stokes equations with nonhomogeneous boundary conditions

In this paper, we study the existence and regularity of solutions to the Stokes and Oseen equations with nonhomogeneous Dirichlet boundary conditions with low regularity. We consider boundary conditions for which the normal component is not equal to zero. We rewrite the Stokes and the Oseen equations in the form of a system of two equations. The first one is an evolution equation satisfied by Pu, the projection of the solution on the Stokes space – the space of divergence free vector fields with a normal trace equal to zero – and the second one is a quasi-stationary elliptic equation satisfied by (I−P)u, the projection of the solution on the orthogonal complement of the Stokes space. We establish optimal regularity results for Pu and (I−P)u. We also study the existence of weak solutions to the three-dimensional instationary Navier–Stokes equations for more regular data, but without any smallness assumption on the initial and boundary conditions.     

Martin Kernels for Markov Processes with Jumps

We prove the existence of boundary limits of ratios of positive harmonic functions for a wide class of Markov processes with jumps and irregular (possibly disconnected) domains of harmonicity, in the context of general metric measure spaces. As a corollary, we prove the uniqueness of the Martin kernel at each boundary point, that is, we identify the Martin boundary with the topological boundary. We also prove a Martin representation theorem for harmonic functions. Examples covered by our results include: strictly stable Lévy processes in R ^d with positive continuous density of the Lévy measure; stable-like processes in R ^d and in domains; and stable-like subordinate diffusions in metric measure spaces.     

Vanishing viscosity limit for the 3D magnetohydrodynamic system with generalized Navier slip boundary conditions

In this paper, we investigate the vanishing viscosity limit problem for the 3D incompressible magnetohydrodynamic (MHD) system in a general bounded smooth domain of R ³ with the generalized Navier slip boundary conditions. We also obtain rates of convergence of the solution of viscous MHD to the corresponding ideal MHD. Copyright © 2016 John Wiley & Sons, Ltd.     

Following homology in singularly perturbed systems

The fast systems associated to a system of first-order equations where a small parameter multiplies the derivatives (hereinafter, an SPMD system) are obtained by stretching time by a factor of one over the small parameter and then reducing the small parameter to zero. Curves of isolated invariant sets associated to the fast system serve as approximate solutions to the SPMD system over finite intervals in the following sense: A manifold of initial points representing non-zero homology α in the Morse-Conley index of the invariant set at the left endpoint has a submanifold which is carried by the flow of the SPMD system to a manifold representing that non-zero homology class in the index of the invariant set at the right endpint which is the continuation of α along the curve of invariants sets. This is a type of linearization of the approximation problem since the continuation of homology classes along the curve of invariant sets is given by a linear transformation which varies functorially with the curve of invariant sets. Examples are given for systems in R² and R³. The results are used elsewhere to prove existence theorems for two point boundary value problems to obtain solutions with boundary and interior transition layers.     

Boundary Layers Associated with Incompressible Navier-Stokes Equations: The Noncharacteristic Boundary Case

The goal of this article is to study the boundary layer of wall bounded flows in a channel at small viscosity when the boundaries are uniformly noncharacteristic, i.e., there is injection and/or suction everywhere at the boundary. Following earlier work on the boundary layer for linearized Navier-Stokes equations in the case where the boundaries are characteristic (no-slip at the boundary and non-permeable), we consider here the case where the boundary is permeable and thus noncharacteristic. The form of the boundary layer and convergence results are derived in two cases: linearized equation and full nonlinear equations. We prove that there exists a boundary layer at the outlet (downwind) of the form e^−Uz/ε where U is the speed of injection/suction at the boundary, z is the distance to the outlet of the channel, and ε is the kinematic viscosity. We improve an earlier result of S. N. Alekseenko (1994, Siberian Math. J.35, No. 2, 209-230) where the convergence in L² of the solutions of the Navier-Stokes equations to that of the Euler equations at vanishing viscosity was established. In the two dimensional case we are able to derive the physically relevant uniform in space (L^∞ norm) estimates of the boundary layer. The uniform in space estimate is derived by properly developing our previous idea of better control on the tangential derivative and the use of an anisotropic Sobolev imbedding. To the best of our knowledge this is the first rigorously proved result concerning boundary layers for the full (nonlinear) Navier-Stokes equations for incompressible fluids.