Showing papers in "Mathematical Methods in The Applied Sciences in 1989"

A global existence theorem for the general coagulation-fragmentation equation with unbounded kernels

Abstract: In this article an existence theorem is proved for the coagulation–fragmentation equation with unbounded kernel rates. Solutions are shown to be in the space X+ = {c∈L1: ∫ (1 + x)∣c(x)∣dx < ∞} whenever the kernels satisfy certain growth properties and the non-negative initial data belong to X+. The proof is based on weak L1 compactness methods applied to suitably chosen approximating equations.

Boundary integral equations for mixed Dirichlet, Neumann and transmission problems

Abstract: We consider a Helmholtz equation in a number of Lipschitz domains in n ≥ 2 dimensions, on the boundaries of which Dirichlet, Neumann and transmission conditions are imposed. For this problem an equivalent system of boundary integral equations is derived which directly yields the Cauchy data of the solutions. The operator of this system is proved to be injective and strongly elliptic, hence it is also bijective and the original problem has a unique solution. For two examples (a mixed Dirichlet and transmission problem and the transmission problem for four quadrants in the plane) the boundary integral operators and the treatment of the compatibility conditions are described.

Stationary solutions of the linearized Boltzmann equation in a half-space

Abstract: Stationary half-space solutions of the linearized Boltzmann equation are studied by energy estimates methods. We extend the results of Bardos, Caflisch and Nicolaenko for a gas of hard spheres to a general potential. Asymptotic behaviour is obtained for hard as well as soft potentials and compared to the case of hard spheres.

On the equivalence of the Schrödinger and the quantum Liouville equations

Abstract: We present a semigroup analysis of the quantum Liouville equation, which models the temporal evolution of the (quasi) distribution of an electron ensemble under the action of a scalar potential. By employing the density matrix formulation of quantum physics we prove that the quantum Liouville operator generates a unitary group on L2 if the corresponding Hamiltonian is essentially self-adjoint. Also, we analyse the existence and non-negativity of the particale density and prove that the solutions of the quantum Liouville equation converge to weak solutions of the classical Liouville equation as the Planck constant tends to zero (assuming that the potential is sufficiently smooth).

Surface gradients and continuity properties for some integral operators in classical scattering theory

Abstract: The surface gradients of some of the most important boundary integral operators for the time-harmonic Helmholtz and Maxwell equations are computed. These results are used to give new and elementary proofs of the continuity properties of these boundary operators in Sobolev and Holder spaces of arbitrary order.

Duality theorems in some overdetermined boundary value problems

Abstract: We consider several elliptic boundary value problems for which there is an overspecification of data on the boundary of the domain. After reformulating the problems in an equivalent integral form, we use the alternate integral formulation to deduce that if a solution exists, then the domain must be an N-ball. Various Green's functions and classical boundary value problems of second, fourth and higher order are included among the problems considered here.

Regular solutions of transmission and interaction problems for wave equations

Abstract: Consider n bounded domains Ω ⊆ ℝ and elliptic formally symmetric differential operators A1 of second order on Ωi Choose any closed subspace V in , and extend (Ai)i=1,…,n by Friedrich's theorem to a self-adjoint operator A with D(A1/2) = V (interaction operator). We give asymptotic estimates for the eigenvalues of A and consider wave equations with interaction. With this concept, we solve a large class of problems including interface problems and transmission problems on ramified spaces.25,32 We also treat non-linear interaction, using a theorem of Minty29.

On the exponential convergence of the h-p version for boundary element Galerkin methods on polygons

Abstract: : This paper applies the technique of the h-p version to the boundary element method for boundary value problems on non-smooth, plane domains with piecewise analytic boundary and data. The exponential rate of convergence of the boundary element Galerkin solution is proven when a geometric mesh refinement towards the vertices is used. (KR)

On properties of the iterative maximum likelihood reconstruction method

Abstract: In this paper, we continue our investigations6 on the iterative maximum likelihood reconstruction method applied to a special class of integral equations of the first kind, where one of the essential assumptions is the positivity of the kernel and the given right-hand side. Equations of this type often occur in connection with the determination of density functions from measured data. There are certain relations between the directed Kullback–Leibler divergence and the iterative maximum likelihood reconstruction method some of which were already observed by other authors. Using these relations, further properties of the iterative scheme are shown and, in particular, a new short and elementary proof of convergence of the iterative method is given for the discrete case. Numerical examples have already been given in References 6. Here, an example is considered which can be worked out analytically and which demonstrates fundamental properties of the algorithm.

On a Galerkin-averaging method for weakly non-linear wave equations

Abstract: Solutions of weakly non-linear wave equations can be approximated using Galerkin's procedure combined with the averaging method In this paper existence and uniqueness of solutions are proved in suitably chosen function spaces Error-estimates lead us to results on asymptotic validity of the approximations Some applications are indicated

Bifurcation from the essential spectrum for some non-compact non-linearities

Abstract: Keywords: non-compact nonlinearities ; essential spectrum ; bifurcation ; semilinear elliptic equation Reference ANA-ARTICLE-1989-001doi:10.1002/mma.1670110408View record in Web of Science Record created on 2008-12-10, modified on 2016-08-08

Interpolation of Coefficients and Transformation of the Dependent Variable in Finite Element Methods for the Non-linear Heat Equation

Abstract: Error estimates are shown for some spatially discrete Galerkin finite element methods for a non-linear heat equation. The approximation schemes studied are based on the introduction of the enthalpy as a new dependent variable, and also on the application of the Kirchhoff transformation and on interpolation of the non-linear coefficients into standard Lagrangian finite element spaces.

General conical singularities in three-dimensional Poisson problems

Abstract: On considere des problemes de Poisson a 3 dimensions tels que −Δ[u(x)]=f(x), x∈Ω, u(x)=0, x∈δΩ D , δu(x)/δγ=0, x∈δΩ N ou f est sur Ω lisse, ΩCR 3 est un domaine simplement connexe avec δΩ=δΩ D UδΩ N , δΩ D ∩δΩ N =Φ, et δ/δν est la derivee dans la direction de la normale sortante a δΩ N

Properties of far‐field operators in acoustic scattering

Abstract: The far-field operator for an exterior boundary value problem for the Helmholtz equation maps the boundary data onto the far-field pattern of the solution. This paper computes the L2-adjoint of this operator for various choices of boundary conditions. In scattering theory the boundary data are given by the traces of plane waves. We characterize the closure of the span of the images of these plane waves under the far field operator. Finally, the results are extended to more general topologies including Sobolev and Holder norms.

On the instability of resonances in parallel‐plane waveguides

Abstract: It has been observed13 that the propagation of acoustic waves in the region Ω0= ℝ2 × (0, 1), which are generated by a time-harmonic force density with compact support, leads to logarithmic resonances at the frequencies ω = 1, 2,… As we have shown9 in the case of Dirichlet's boundary condition U = 0 on ∂Ω, the resonance at the smallest frequency ω = 1 is unstable and can be removed by a suitable small perturbation of the region. This paper contains similar instability results for all resonance frequencies ω = 1, 2,… under more restrictive assumptions on the perturbations Ω of Ω0. By using integral equation methods, we prove that absence of admissible standing waves in the sense of Reference 7 implies the validity of the principle of limit amplitude for every frequency ω ≥ 0 in the region Ω =Ω0 −B, where B is a smooth bounded domain with B⊂Ω0. In particular, it follows from Reference 7 in the case of Dirichlet's boundary condition that the principle of limit amplitude holds for every frequency ω ≥ 0 if n·x′ ⩽ 0 on ∂ B, where x′ = (x1, x2, 0) and n is the normal unit vector pointing into the interior B of ∂ B. In the case of Neumann's boundary condition, the logarithmic resonance at ω = 0 is stable under the perturbations considered in this paper. The asymptotic behaviour of the solution for arbitary local perturbations of Ω0 will be discussed in a subsequent paper.

Boundary control of a vibrating plate with internal damping

Abstract: It is shown that finite energy states of a vibrating viscoelastic plate of the Kelvin–Voigt type are, in general, not exactly controllable by L2-boundary controls. Accordingly, we present a result on approximative controllability. The method is general.

Some remarks on Saint‐Venant's principle

Abstract: We consider a three-dimensional hyperelastic cylinder in Ω = D × [0, ∞]. We study the asymptotic behaviour of the deformations of the cross-sections in an equilibrium state. In this case we show that the solutions either have exponential decay or exponential growth. We give some initial conditions such that the latter case occurs.

A trace theorem for solutions of linear partial differential equations

Abstract: : This paper proves a trace regularity theorem for the solutions of general linear partial differential equations with smooth coefficients. Our result shows that by imposing additional microlocal smoothness along certain directions. the trace of the solution on a codimension one hypersurface will be just as regular as the solution itself. The proof is based on the Hoermander-Nirenberg pseudo-differential cut-off technique and a "fattening" lemma, together with standard energy estimates.

A penalty approximation for a unilateral contact problem in non-linear elasticity

Abstract: Using Ball's approach to non-linear elasticity, and in particular his concept of polyconvexity, we treat a unilateral three-dimensional contact problem for a hyperelastic body under volume and surface forces. Here the unilateral constraint is described by a sublinear function which can model the contact with a rigid convex cone. We obtain a solution to this generally non-convex, semicoercive Signorinin problem as a limit of solutions of related energy minimization problems involving friction normal to the contact surface where the friction coefficient goes to infinity. Thus we extend an approximation result of Duvaut and Lions for linear-elastic unilateral contact problems to finite deformations and to a class of non-linear elastic materials including the material models of Ogden and of Mooney-Rivlin for rubberlike materials. Moreover, the underlying penalty method is shown to be exact, that is a sufficiently large friction coefficient in the auxiliary energy minimization problems suffices to produce a solution of the original unilateral problem, provided a Lagrange multiplier to the unilateral constraint exists.

On the continuation for variational inequalities depending on an eigenvalue parameter

Abstract: In this paper we generalize recent theoretical results on the local continuation of parameter-dependent non-linear variational inequalities. The variational inequalities are rather general and describe, for example, the buckling of beams, plates or shells subject to obstacles. Under a technical hypothesis that is satisfied by the simply supported beam, we obtain the existence of a continuation of both the solution and the eigenvalue with respect to a local parameter. A numerical continuation method is presented that easily overcomes turning points. Numerical results are presented for the non-linear beam.

Three-dimensional stochastic vortex flows

Abstract: We discuss the stochastic structure of the Navier–Stokes flow in ℝ3 and prove that it can be approximated by means of a finite-dimensional stochastic process. Such a process reduces to an algorithm already discussed in Reference 4 for the Euler case, when the viscosity coefficient vanishes.

The electric field in the conductive half space as a model in mining and petroleum prospecting

Abstract: We present a formulation of Maxwell's equations in a conductive medium, in the time domain. In order to restrict the equations to the conductive half space, the solution in the air is represented by an integral boundary operator on the interface. The problem admits a variational formulation, allowing a finite element solution. A mathematical analysis is described for 2D and 3D models, and numerical results are presented.

Pseudo-transparent boundary conditions for the diffusion equation. Part I

Abstract: In this article, we introduce and study a family of artificial boundary conditions for the diffusion equation. These conditions are local in time and non-local in space. We describe the principle of the approximation, show that the corresponding approximate problems are mathematically well-posed and give convergence results, as well as error estimates, of the approximate solution to the exact one.

Resonance phenomena for a class of partial differential equations of higher order in cylindrical waveguides

Abstract: We consider a domain Ω in ℝn of the form Ω = ℝl × Ω′ with bounded Ω′ ⊂ ℝn−l. In Ω we study the Dirichlet initial and boundary value problem for the equation ∂u + [(− ∂ −… − ∂)m + (− ∂ −… − ∂)m]u = fe−iωt. We show that resonances can occur if 2m ≥ l. In particular, the amplitude of u may increase like tα (α rational, 0<α<1) or like in t as t∞∞. Furthermore, we prove that the limiting amplitude principle holds in the remaining cases.

Stability and optimal control of thermomechanical processes with non-convex free energies of Ginzburg-Landau type

Abstract: In this paper we study a coupled non-linear system of partial differential equations that models the dynamics of structural phase transitions in a one-dimensional non-viscous and heat-conducting solid. The corresponding Helmholtz free energy density is assumed in Ginzburg–Landau form; to allow for phase transitions and hysteresis phenomena, it is not assumed convex in the order parameter. It is shown that the solution of the system depends continuously upon the data, and we prove an existence result for an associated optimal control problem.

On stationary flows of asymmetric fluids with heat convection

Abstract: Existence, uniqueness and regularity of solutions of equations describing stationary flows of viscous incompressible isotropic fluids with an asymmetric stress tensor have been considered recently.5 In this paper we extend the results of Reference 5 to include heat convection in the hydrodynamic model. We show that the boundary value problem (1.1)–(1.6) below has solutions in appropriate Sobolev spaces, provided the viscosities v and ca + cd are sufficiently large. The proof is based on a fixed point argument. Moreover, we show that the solutions are unique if the heat conductivity κ is large enough.

Canonical difference scheme for the Hamiltonian equation

Abstract: On construit des schemas canoniques aux differences bases sur une approximation de Pade pour des systemes canoniques lineaires a coefficients constants. Pour des equations hamiltoniennes non lineaires on utilise une transformation infinitesimalement canonique pour construire des schemas canoniques de precision arbitraire

A free boundary problem of biological interest

Abstract: During fertilization of certain echinoderms, a long actin-filled tube is extended by the sperm towards the interior of the egg. This yields a parabolic free boundary problem, which differs from the classical one-phase Stefan problem by the presence of a convective term in the partial differential equation, and because the equilibrium interface condition θ(s(t),t) = 0 is here replaced by a kinetic law s′(t) = vθ(s(t),t). This problem is set in variational form and the existence of a solution is proved by means of a Faedo–Galerkin approximation procedure.

Finite element solution for the transcritical shallow-water equations

Abstract: A new solution of the two-dimensional shallow-water equations, using a finite element method is described. The formulation is based on the velocity and height variables and follows two steps. In the first step, the convective terms are solved by a characteristic method and in the second step the propagative and diffusion terms are taken into account. The close relation between the latter operator and a Stokes penalized problem is shown; an algorithm of Uzawa type is then used for its solution. Acceleration convergence is obtained by preconditioning, and new methods are presented, the quality of the convergence being shown on application to some sample tests. The overall method has revealed quite efficient and application to industrial cases is already planned.

Quasilinear hyperbolic-hyperbolic singular perturbation problem: study of shock layer

Abstract: We present the asymptotic analysis of a quasilinear hyperbolic–hyperbolic singular perturbation problem in one dimension. The leading part of the analysis concerns the construction of some shock layers associated with discontinuities of a hyperbolic problem. This study is a generalization of the case of viscous perturbation for a hyperbolic problem.