Interaction of acoustic waves and piezoelectric structures

In the paper, we investigate the basic transmission problems arising in the model of fluid‐solid acoustic interaction when a piezo‐ceramic elastic body ( Ω ⁺ ) is embedded in an unbounded fluid domain ( Ω ^? ). The corresponding physical process is described by boundary‐transmission problems for second order partial differential equations. In particular, in the bounded domain Ω ⁺ , we have 4 × 4 dimensional matrix strongly elliptic second order partial differential equation, while in the unbounded complement domain Ω ^? , we have a scalar Helmholtz equation describing acoustic wave propagation. The physical kinematic and dynamic relations mathematically are described by appropriate boundary and transmission conditions. With the help of the potential method and theory of pseudodifferential equations, the uniqueness and existence theorems are proved in Sobolev–Slobodetskii spaces. Copyright © 2014 John Wiley & Sons, Ltd.     

Non‐local approach in mathematical problems of fluid–structure interaction

Three‐dimensional mathematical problems of interaction between elastic and scalar oscillation fields are investigated. An elastic field is to be defined in a bounded inhomogeneous anisotropic body occupying the domain Ω¯₁⊂ℝ³ while a physical (acoustic) scalar field is to be defined in the exterior domain Ω¯₂=ℝ³\Ω₁ which is filled up also by an anisotropic (fluid) medium. These two fields satisfy the governing equations of steady‐state oscillations in the corresponding domains together with special kinematic and dynamic transmission conditions on the interface ∂Ω₁. The problems are studied by the so‐called non‐local approach, which is the coupling of the boundary integral equation method (in the unbounded domain) and the functional‐variational method (in the bounded domain). The uniqueness and existence theorems are proved and the regularity of solutions are established with the help of the corresponding Steklov–Poincaré type operators and on the basis of the Gårding inequality and the Lax–Milgram theorem. In particular, it is shown that the physical fluid–solid acoustic interaction problem is solvable for arbitrary values of the frequency parameter. Copyright © 1999 John Wiley & Sons, Ltd.     

A singular Monge-Ampère equation on unbounded domains

In this paper, we study the Dirichlet problem for a singular Monge-Amp`ere type equation on unbounded domains. For a few special kinds of unbounded convex domains, we find the explicit formulas of the solutions to the problem. For general unbounded convex domain ?, we prove the existence for solutions to the problem in the space C∞(?) ∩ C(?). We also obtain the local C1/2-estimate up to the ?? and the estimate for the lower bound of the solutions.     

Approximation by Maxwell–Herglotz‐fields

By a general argument, it is shown that Maxwell–Herglotz‐fields are dense (with respect to the C^∞(Ω)‐topology) in the space of all solutions to Maxwell's equations in Ω. This is used to provide corresponding approximation results in global spaces (e.g. in L²‐Sobolev‐spaces H^m(Ω)) and for boundary data. Proofs are given within the framework of generalized Maxwell's equations using differential forms. Copyright © 2004 John Wiley & Sons, Ltd.     

On the nonexistence of a global nontrivial subsonic solution in a 3D unbounded angular domain

In this paper, under some assumptions on the flow with a low Mach number, we study the nonexistence of a global nontrivial subsonic solution in an unbounded domain Ω which is one part of a 3D ramp. The flow is assumed to be steady, isentropic and irrotational, namely, the movement of the flow is described by the potential equation. By establishing a fundamental a priori estimate on the solution of a second order linear elliptic equation in Ω with Neumann boundary conditions on Ω and Dirichlet boundary value at some point of Ω, we show that there is no global nontrivial subsonic flow with a low Mach number in such a domain Ω.     

Interaction of Elastic and Scalar Fields

The three-dimensional mathematical problems of the interaction of an elastic and some scalar fields are investigated. It is assumed that the elastic structure under consideration is a bounded homogeneous anisotropic body occupying domain Ω¯⁺⊂ℝ³ and the physical scalar field is defined in the exterior domain Ω⁻ = ℝ³\Ω⁺. These two fields satisfy the governing equations in the corresponding domains together with the transmission conditions on the interface ∂Ω⁺. The problems are studied by the potential method and the existence and uniqueness theorems are proved.     

An exterior boundary value problem in Minkowski space

In three‐dimensional Lorentz–Minkowski space ??³, we consider a spacelike plane Π and a round disc Ω over Π. In this article we seek the shapes of unbounded surfaces whose boundary is ? Ω and its mean curvature is a linear function of the distance to Π. These surfaces, called stationary surfaces, are solutions of a variational problem and governed by the Young–Laplace equation. In this sense, they generalize the surfaces with constant mean curvature in ??³. We shall describe all axially symmetric unbounded stationary surfaces with special attention in the case that the surface is asymptotic to Π at the infinity. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Asymptotics for models of non‐stationary diffusion in domains with a surface distribution of obstacles

We consider a time‐dependent model for the diffusion of a substance through an incompressible fluid in a perforated domain Ω_?, with n = 3,4. The fluid flows in a domain containing a periodical set of “obstacles” (Ω\Ω_?) placed along an inner (n ? 1)‐dimensional manifold . The size of the obstacles is much smaller than the size of the characteristic period ?. An advection term appears in the partial differential equation linking the fluid velocity with the concentration, while we assume a nonlinear adsorption law on the boundary of the obstacles. This law involves a monotone nonlinear function σ of the concentration and a large adsorption parameter. The “critical adsorption parameter” depends on the size of the obstacles , and, for different sizes, we derive the time‐dependent homogenized models. These models contain a “strange term” in the transmission conditions on Σ, which is a nonlinear function and inherits the properties of σ. The case in which the fluid velocity and the concentration do not interact is also considered for n ≥ 3.     

Spectral theory and iterative methods for the Maxwell system in nonsmooth domains

We study spectral properties of boundary integral operators which naturally arise in the study of the Maxwell system of equations in a Lipschitz domain Ω ? ?³. By employing Rellich‐type identities we show that the spectrum of the magnetic dipole boundary integral operator (composed with an appropriate projection) acting on L²(?Ω) lies in the exterior of a hyperbola whose shape depends only on the Lipschitz constant of Ω. These spectral theory results are then used to construct generalized Neumann series solutions for boundary value problems associated with the Maxwell system and to study their rates of convergence (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Analytic semigroups generated in L 1(Ω) by second order elliptic operators via duality methods

Given an open domain (possibly unbounded) Ω?R ⁿ , we prove that uniformly elliptic second order differential operators, under nontangential boundary conditions, generate analytic semigroups in L ¹(Ω). We use a duality method, and, further, give estimates of first order derivatives for the resolvent and the semigroup, through properties of the generator in Sobolev spaces of negative order.     

On the asymptotic stability of steady solutions of the Navier–Stokes equations in unbounded domains

We consider the problem of the asymptotic behaviour in the L²‐norm of solutions of the Navier–Stokes equations. We consider perturbations to the rest state and to stationary motions. In both cases we study the initial‐boundary value problem in unbounded domains with non‐compact boundary. In particular, we deal with domains with varying and possibly divergent exits to infinity and aperture domains. Copyright © 2007 John Wiley & Sons, Ltd.     

Boundary difference-integral equation method and its error estimates for second order hyperbolic partial differential equation

Combining difference method and boundary integral equation method, we propose a new numerical method for solving initial-boundary value problem of second order hyperbolic partial differential equations defined on a bounded or unbounded domain inR ³ and obtain the error estimates of the approximate solution in energy norm and local maximum norm.China State Major Key Project for Basic Researches.     

Generalized Stokes resolvent equations in an infinite layer with mixed boundary conditions

In this paper we prove unique solvability of the generalized Stokes resolvent equations in an infinite layer Ω₀ = ℝ^{n –1} × (–1, 1), n ≥ 2, in L^q ‐Sobolev spaces, 1 < q < ∞, with slip boundary condition of on the “upper boundary” ∂Ω⁺₀ = ℝ^{n –1} × {1} and non‐slip boundary condition on the “lower boundary” ∂Ω^–₀ = ℝ^{n –1} × {–1}. The solution operator to the Stokes system will be expressed with the aid of the solution operators of the Laplace resolvent equation and a Mikhlin multiplier operator acting on the boundary. The present result is the first step to establish an L^q ‐theory for the free boundary value problem studied by Beale [9] and Sylvester [22] in L ²‐spaces. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Local Change of Coordinates for Grid-Functions on Regions with Curved Boundary

Local change of coordinates, providing an appropriate transformation of functions on a domain Ω ∩ R ⁿ into functions on R ⁿ and R ⁺_n, is a well-known and frequently used technical tool in the theory of Sobolev-spaces W^m,p (Ω) and partial differential equations. In this paper we propose a corresponding transformation mapping grid-functions on regular grids Ω_h into functions on R ⁿ_h and R ⁿ_h,+ which as in the continuous case can be used to remedy various difficulties arising the curved boundary of Ω.     

Existence of a weak solution to the Navier–Stokes equation in a general time‐varying domain by the Rothe method

We assume that Ω^t is a domain in ?³, arbitrarily (but continuously) varying for 0?t?T. We impose no conditions on smoothness or shape of Ω^t. We prove the global in time existence of a weak solution of the Navier–Stokes equation with Dirichlet's homogeneous or inhomogeneous boundary condition in Q_{[0, T)} := {( x , t);0?t?T, x ∈Ω^t}. The solution satisfies the energy‐type inequality and is weakly continuous in dependence of time in a certain sense. As particular examples, we consider flows around rotating bodies and around a body striking a rigid wall. Copyright © 2008 John Wiley & Sons, Ltd.     

Finite element solutions for three‐dimensional elliptic boundary value problems on unbounded domains

The finite element (FE) solutions of a general elliptic equation ?div([a_ij] ??u) + u = f in an exterior domain Ω, which is the complement of a bounded subset of R ³, is considered. The most common approach to deal with exterior domain problems is truncating an unbounded subdomain Ω_∞, so that the remaining part Ω_B = Ω\Ω _∞ is bounded, and imposing an artificial boundary condition on the resulted artificial boundary Γ_a = Ω _∞ ∩ Ω _B. In this article, instead of discarding an unbounded subdomain Ω_∞ and introducing an artificial boundary condition, the unbounded domain is mapped to a unit ball by an auxiliary mapping. Then, a similar technique to the method of auxiliary mapping, introduced by Babu?ka and Oh for handling the domain singularities, is applied to obtain an accurate FE solution of this problem at low cost. This method thus does have neither artificial boundary nor any restrictions on f. © 2006 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2006     

Approximation by herglotz wave functions

By a general argument, it is shown that Herglotz wave functions are dense (with respect to the C^∞(Ω)‐topology) in the space of all solutions to the reduced wave equation in Ω. This is used to provide corresponding approximation results in global spaces (eg. in L2‐Sobolev‐spaces H^m(Ω)) and for boundary data. Copyright © 2004 John Wiley & Sons, Ltd.     

Measure-valued Solutions of the Euler Equations for Ideal Compressible Polytropic Fluids

The existence of global measure-valued solutions to the Euler equations describing the motion of an ideal compressible and heat conducting fluid is proved. The motion is considered in a bounded domain Ω⊂ℝ³ with impermeable boundary. The solution is a limit of an approximate solution obtained by adding the sixth-order elliptic operator in the equation of momentum.     

On the homogenization of some linear problems in domains weakly connected by a system of traps

The aim of the paper is to study the asymptotic behaviour of solutions of second‐order elliptic and parabolic equations, arising in modelling of flow in cavernous porous media, in a domain Ω^ε weakly connected by a system of traps ??^ε, where ε is the parameter that characterizes the scale of the microstructure. Namely, we consider a strongly perforated domain Ω^ε ?Ω a bounded open set of ?³ such that Ω^ε =Ω₁^ε ∪Ω₂^ε ∪??^ε ∪ W^ε, where Ω₁^ε, Ω₂^ε are non‐intersecting subdomains strongly connected with respect to Ω, ??^ε is a system of traps and meas W^ε → 0 as ε → 0. Without any periodicity assumption, for a large range of perforated media and by means of variational homogenization, we find the homogenized models. The effective coefficients are described in terms of local energy characteristics of the domain Ω^ε associated with the problem under consideration. The resulting homogenized problem in the parabolic case is a vector model with memory terms. An example is presented to illustrate the methodology. Copyright © 2007 John Wiley & Sons, Ltd.     

Moving mesh method for problems with blow-up on unbounded domains

This paper studies the numerical solutions of semilinear parabolic partial differential equations (PDEs) on unbounded spatial domains whose solutions blow up in finite time. There are two major difficulties usually in numerical solutions: the singularity of blow-up and the unboundedness. We propose local absorbing boundary conditions (LABCs) on the selected artificial boundaries by using the idea of unified approach (Brunner et al., SIAM J Sci Comput 31:4478–4496, (2010). Since the uniform fixed spatial meshes may be inefficient, we adopt moving mesh partial differential equation (MMPDE) method to adapt the spatial mesh as the singularity develops. Combining LABCs and MMPDE, we can effectively capture the qualitative behavior of the blow-up singularities in the unbounded domain. Moreover, the implementation of the combination consists of two independent parts. Numerical examples also illustrate the efficiency and the accuracy of the new method.