Exterior Domain Research Articles

The scalar Oseen problem in two-dimensional exterior domains

In the present paper, we investigate the scalar Oseen equation in a two-dimensional exterior domain Ω, with Dirichlet boundary condition. This problem represents a linearized form of the steady Navier-Stokes equations. In order to better control the behavior at infinity of functions, the use of weighted Sobolev spaces as a functional framework turns out to be adequate. The established results are related to the existence and the uniqueness of weak solutions on Lp-theory for any real 1<p<∞. Our analysis is mainly based on the principle that linear exterior problems can be solved by combining their properties in bounded domains which are also established in this paper and their properties well known in the whole space R2.

Solvability of Hessian quotient equations in exterior domains

Abstract In this paper, we study the Dirichlet problem of Hessian quotient equations of the form $S_k(D^2u)/S_l(D^2u)=g(x)$ in exterior domains. For $g\equiv \mbox {const.}$ , we obtain the necessary and sufficient conditions on the existence of radially symmetric solutions. For g being a perturbation of a generalized symmetric function at infinity, we obtain the existence of viscosity solutions by Perron’s method. The key technique we develop is the construction of sub- and supersolutions to deal with the non-constant right-hand side g.

Radial solutions of p-Laplace equations with nonlinear gradient terms on exterior domains

This paper studies the existence of radial solutions of the boundary value problem of p-Laplace equation with gradient term {−Δpu=K(|x|)f(|x|,u,|∇u|),x∈Ω,∂u∂n=0,x∈∂Ω,lim|x|→∞u(x)=0,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \ extstyle\\begin{cases} -\\Delta_{p} u= K( \\vert x \\vert ) f( \\vert x \\vert , u, \\vert \ abla u \\vert ) ,\\quad x\\in\\Omega , \\\\ \\frac{\\partial u}{\\partial n}=0 ,\\quad x\\in\\partial\\Omega, \\\\ \\lim_{ \\vert x \\vert \ o\\infty}u(x)=0 , \\end{cases} $$\\end{document} where Ω={x∈RN:|x|>r0}\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\Omega=\\{x\\in\\mathbb{R}^{N}: |x|>r_{0}\\}$\\end{document}, N≥3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$N\\ge3$\\end{document}, 1<p≤2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$1< p\\le2$\\end{document}, K:[r0,∞)→R+\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$K: [r_{0}, \\infty)\ o\\mathbb{R}^{+}$\\end{document}, and f:[r0,∞)×R×R+→R\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$f: [r_{0}, \\infty)\ imes\\mathbb{R}\ imes\\mathbb{R}^{+}\ o \\mathbb{R}$\\end{document} are continuous. Under certain inequality conditions of f, the existence results of radial solutions are obtained.

Q-Laplace equation involving the gradient on general bounded and exterior domains

q-Laplace equation involving the gradient on general bounded and exterior domains

Scattering of water waves by multiple rows of vertical thin barriers

The reflection and transmission of waves by a periodic array of thin fixed identical vertical barriers extending uniformly through the depth of the fluid is considered. The water wave problem, which also has analogues in acoustics and electromagnetics, involves scattering by a large finite number of equally spaced rows, each consisting of an array of barriers with a linear periodic arrangement extending to infinity. The main purpose of the work is to compare the results of two approximate methods of solution based on homogenisation techniques with exact results allowing us to determine when the results can be expected to be in good agreement. When no approximation is made, a method of solution (Discrete Model) describing the propagation of incident waves through the exact barrier arrangement is derived using Floquet–Bloch eigensolutions for the corresponding infinite periodic array. It is shown how the solution can be reduced to a simple pair of integral equations which results only from matching the region containing barrier arrays to the two exterior domains. Numerical solutions are, nevertheless, relatively complicated and two approximate models are developed which considerably simplify the numerical effort required. In one approximation (Continuum Model I) it is assumed only that the spacing between two adjacent rows of barriers is small and is complementary to the commonly-used wide-spacing approximation. The complication of the geometry is removed by replacing the governing equations and boundary conditions with an effective continuous medium and a simplified pair of integral equations now govern the solutions. In a second approximation (Continuum Model II), a low-frequency homogenisation approach based on a long-wave assumption is invoked and matching now leads to closed-form expressions for the reflection and transmission coefficients. Results concentrate on the comparison between different methods which allows us to assess the validity of each continuum model. This understanding will be important in developing approximate models that describe the operation of wave energy farms consisting of large arrays of oscillating flaps designed to absorb energy.

Finite-time blow-up for the compressible Euler system in the exterior domain

Finite-time blow-up for the compressible Euler system in the exterior domain

Global strong solutions to full compressible magnetohydrodynamic system with vacuum in 3D exterior domains

This paper investigates the full compressible magnetohydrodynamic system in three-dimensional exterior domains. For the initial-boundary-value problem of this system with slip boundary condition for the velocity, adiabatic one for the temperature, and perfect one for the magnetic field, we establish the global existence and uniqueness of strong solutions, under the condition that the initial data are of small energy but possibly large oscillations, where the initial density and temperature are both allowed to vanish. Moreover, the large-time behavior of the strong solutions is also shown.

Low-frequency radiation from a vibrating cap on a rigid spherical shell with a circular aperture.

Theoretical models based on spherical geometries have long provided essential insights into the directional behavior of sound sources such as loudspeakers and human speech. Because commonly applied models predict omnidirectional radiation at low frequencies and increasing directionality at higher frequencies, they fail to predict the directional characteristics of certain sources with different source geometries. These sources include violins and open-back guitar amplifiers that have openings or ports connecting a cavity or enclosure to the exterior domain. This work presents the low-frequency radiation from a vibrating cap on a rigid spherical shell with a circular aperture to study the directional characteristics of such sources. The proposed model predicts dipolar radiation at very low frequencies, monopolar radiation near the Helmholtz resonance, and increasing directionality at higher frequencies. Experimental results based on measuring the sound field of an open-back spherical loudspeaker validate the theoretical model and highlight its utility in predicting directional behavior.

Detection of scatterers using an XFEM-BEM level set solver based on the topological derivative

A numerical method is proposed for the solution of the inverse scattering problem. This problem consists of determining the location and shape of an unknown number of inclusions composed by a homogeneous material with known mechanical properties different that those of the surrounding medium. The information available to solve the inverse problem are measurements of the fundamental mechanical magnitude of the wave propagation problem. At the boundary of the scatterers, transmission conditions depending on the material properties are considered. For the solution of the forward problem, a coupled extended finite element method (XFEM)-boundary element method (BEM) is proposed, where the XFEM is used for the bounded region where the scatterers are supposed to be located, and the BEM is used for the exterior domain. The inverse problem is formulated as a topology optimization problem, and solved by means of a heuristic algorithm based on the topological derivative and a level set representation of the scatterers.

Stability of planar exterior stationary flows with suction

We consider the two-dimensional Navier-Stokes system in a domain exterior to a disk. The system admits a stationary solution with critical decay O(|x|−1) written as a linear combination of the pure rotating flow and the flux carrier. We prove its nonlinear stability in large time for initial disturbances in L2 under smallness conditions, assuming that there is suction across the boundary, namely that the sign of coefficients of the flux carrier is negative. This result partially solves an open problem in the literature.

Numerical solution of the cavity scattering problem for flexural waves on thin plates: Linear finite element methods

Flexural wave scattering plays a crucial role in optimizing and designing structures for various engineering applications. Mathematically, the flexural wave scattering problem on an infinite thin plate is described by a fourth-order plate wave equation on an unbounded domain, making it challenging to solve directly using the regular linear finite element method (FEM). In this paper, we propose two numerical methods, the interior penalty FEM (IP-FEM) and the boundary penalty FEM (BP-FEM) with a transparent boundary condition (TBC), to study flexural wave scattering by an arbitrary-shaped cavity on an infinite thin plate. Both methods decompose the fourth-order plate wave equation into the Helmholtz and modified Helmholtz equations with coupled conditions on the cavity boundary. A TBC is then constructed based on the analytical solutions of the Helmholtz and modified Helmholtz equations in the exterior domain, effectively truncating the unbounded domain into a bounded one. Using linear triangular elements, the IP-FEM and BP-FEM successfully suppress the oscillation of the bending moment of the solution on the cavity boundary, demonstrating superior stability and accuracy compared to the regular linear FEM when applied to this problem.

Numerical analysis for electromagnetic scattering with nonlinear boundary conditions

This work studies time-dependent electromagnetic scattering from obstacles whose interaction with the wave is fully determined by a nonlinear boundary condition. In particular, the boundary condition studied in this work enforces a power law type relation between the electric and magnetic fields along the boundary. Based on time-dependent jump relations of classical boundary operators, we derive a nonlinear system of time-dependent boundary integral equations that determines the tangential traces of the scattered electric and magnetic fields. These fields can subsequently be computed at arbitrary points in the exterior domain by evaluating a time-dependent representation formula. Fully discrete schemes are obtained by discretizing the nonlinear system of boundary integral equations with Runge–Kutta based convolution quadrature in time and Raviart–Thomas boundary elements in space. Error bounds with explicitly stated convergence rates are proven, under the assumption of sufficient regularity of the exact solution. The error analysis is conducted through novel techniques based on time-discrete transmission problems and the use of a new discrete partial integration inequality. Numerical experiments illustrate the use of the proposed method and provide empirical convergence rates.

Periodic Lp\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\mathrm{L}_{p}$\\end{document} Estimates by ℛ-Boundedness: Applications to the Navier-Stokes Equations

General evolution equations in Banach spaces are investigated. Based on an operator-valued version of de Leeuw’s transference principle, time-periodic Lp\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\mathrm {L}_{p}$\\end{document} estimates of maximal regularity type are carried over from ℛ-bounds of the family of solution operators (ℛ-solvers) to the corresponding resolvent problems. With this method, existence of time-periodic solutions to the Navier-Stokes equations is shown for two configurations: in a periodically moving bounded domain and in an exterior domain, subject to prescribed time-periodic forcing and boundary data.

Soft Riemann‐Hilbert problems and planar orthogonal polynomials

AbstractRiemann‐Hilbert problems are jump problems for holomorphic functions along given interfaces. They arise in various contexts, for example, in the asymptotic study of certain nonlinear partial differential equations and in the asymptotic analysis of orthogonal polynomials. Matrix‐valued Riemann‐Hilbert problems were considered by Deift et al. in the 1990s with a noncommutative adaptation of the steepest descent method. For orthogonal polynomials on the line or on the circle with respect to exponentially varying weights, this led to a strong asymptotic expansion in the given parameters. For orthogonal polynomials with respect to exponentially varying weights in the plane, the corresponding asymptotics was obtained by Hedenmalm and Wennman (2017), based on the technically involved construction of an invariant foliation for the orthogonality. Planar orthogonal polynomials are characterized in terms of a certain matrix ‐problem (Its, Takhtajan), which we refer to as a soft Riemann‐Hilbert problem. Here, we use this perspective to offer a simplified approach based not on foliations but instead on the ad hoc insertion of an algebraic ansatz for the Cauchy potential in the soft Riemann‐Hilbert problem. This allows the problem to decompose into a hierarchy of scalar Riemann‐Hilbert problems along the interface (the free boundary for a related obstacle problem). Inspired by microlocal analysis, the method allows for control of the solution in such a way that for real‐analytic weights, the asymptotics holds in the L2 sense with error in a fixed neighborhood of the closed exterior of the interface, for some constant , where . Here, m is the degree of the polynomial, and in terms of pointwise asymptotics, the expansion dominates the error term in the exterior domain and across the interface (by a distance proportional to ). In particular, the zeros of the orthogonal polynomial are located in the interior of the spectral droplet, away from the droplet boundary by a distance at least proportional to .

Exponential decay estimates for the damped defocusing Schrödinger equation in exterior domains

Exponential decay estimates for the damped defocusing Schrödinger equation in exterior domains

Fourier method for the Neumann problem on a torus

The Fourier method approach to the Neumann problem for the Laplacian operator in the case of a solid torus contrasts in many respects with the much more straight forward situation of a ball in 3-space. Although the Dirichlet-to-Neumann map can be readily expressed in terms of series expansions with toroidal harmonics, we show that the resulting equations contain undetermined parameters which cannot be calculated algebraically. A method for rapidly computing numerical solutions of the Neumann problem is presented with numerical illustrations. The results for interior and exterior domains combine to provide a solution for the Neumann problem for the case of a shell between two tori.

Local absorbing boundary conditions on fixed domains give order-one errors for high-frequency waves

Abstract We consider approximating the solution of the Helmholtz exterior Dirichlet problem for a nontrapping obstacle, with boundary data coming from plane-wave incidence, by the solution of the corresponding boundary value problem where the exterior domain is truncated and a local absorbing boundary condition coming from a Padé approximation (of arbitrary order) of the Dirichlet-to-Neumann map is imposed on the artificial boundary (recall that the simplest such boundary condition is the impedance boundary condition). We prove upper- and lower-bounds on the relative error incurred by this approximation, both in the whole domain and in a fixed neighbourhood of the obstacle (i.e., away from the artificial boundary). Our bounds are valid for arbitrarily-high frequency, with the artificial boundary fixed, and show that the relative error is bounded away from zero, independent of the frequency, and regardless of the geometry of the artificial boundary.

Bifurcation and stability for charged drops

In this paper, we investigate the Laplace’s equation for the electrical potential of charge drops on exterior domain, and overdetermined boundary conditions are prescribed. We determine the local bifurcation structure with respect to the surface tension coefficient as bifurcation parameter. Furthermore, we establish the stability and the instability near the bifurcation point.

Integral equation methods for the Morse-Ingard equations

We present two (a decoupled and a coupled) integral-equation-based methods for the Morse-Ingard equations subject to Neumann boundary conditions on the exterior domain. Both methods are based on second-kind integral equation (SKIE) formulations. The coupled method is well-conditioned and can achieve high accuracy. The decoupled method has lower computational cost and more flexibility in dealing with the boundary layer; however, it is prone to the ill-conditioning of the decoupling transform and cannot achieve as high accuracy as the coupled method. We show numerical examples using a Nyström method based on quadrature-by-expansion (QBX) with fast-multipole acceleration. We demonstrate the accuracy and efficiency of the solvers in both two and three dimensions with complex geometry.

Stabilization of the critical semilinear wave equation on non-compact Riemannian manifold

We study the stability of the defocusing critical semilinear wave equation with locally distributed damping in an exterior domain. Under some assumptions on the Riemannian metric, the geometric multiplier technique and the compactness-uniqueness arguments were used to prove the unique continuation, the observability inequality respectively. Then the energy decay of the nonlinear system was obtained.