IMA Journal of Applied Mathematics - current issue

A Papkovich-Neuber-based numerical approach to cracks with contact in 3D

Hintermuller, M., Kovtunenko, V. A., Kunisch, K. — 2009-05-26

The mathematical model of a crack with non-penetration conditions is considered in the framework of 3D elasticity. The spatial crack problem is investigated with respect to its numerical realization in the context of constrained optimization. Specifically, for homogeneous isotropic solids with planar cracks, a Papkovich–Neuber-based representation is adopted. It allows to employ a primal–dual active set strategy with an unconditional global and monotone convergence property. The iterates turn out to be primally feasible. Illustrative numerical examples are presented.

Energy decay in a transmission problem in thermoelasticity of type III

Messaoudi, S. A., Said-Houari, B. — 2009-05-26

In this paper, we consider a 1D linear thermoelastic transmission problem, where the heat conduction is described by the theories of Green and Naghdi. By using the energy method, we prove that the thermal effect is strong enough to produce an exponential stability of the solution, no matter how small the action domain is.

Stability of equilibria of some switched non-linear systems with applications to control synthesis for electrohydraulic servomechanisms

Halanay, A., Ursu, I. — 2009-05-26

Stability of the zero solution is analyzed for a family of switched systems indexed by a parameter, each system having = 0 in the spectrum of the Jacobian matrix calculated in zero. It is proved that existence of a common quadratic Lyapunov function for some lower dimensional linear systems is sufficient to ensure local uniform stability of the zero solution of the switched non-linear system and a regular asymptotic behaviour. An application to control synthesis for stabilizing equilibria in a switched non-linear control system modelling an electrohydraulic servomechanism is given.

Global solution for a quasi-linear plate system with boundary memory damping

Zhang, Q. — 2009-05-26

In this work, we consider a quasi-linear plate model with boundary memory damping. We prove that this system has a unique global solution when the initial data are small enough and the non-linear coefficient function, the memory damping as well as the geometry of the domain satisfy suitable assumptions. We also prove the exponential decay of the energy of the system.

Non-existence of global solutions of a class of coupled non-linear Klein-Gordon equations with non-negative potentials and arbitrary initial energy

Wang, Y. — 2009-05-26

In the paper, we consider the non-existence of global solutions of Cauchy problem for coupled Klein–Gordon equations of the form on R x Rⁿ. First, for the case n = 2, 3, we prove the existence of ground state of the corresponding Lagrange–Euler equations of the above equations. Then, we establish a blow-up result with low initial energy, which leads to instability of standing waves of the system above. Moreover, as a byproduct we also discuss the global existence. Next, based on concavity method, we prove the blow-up result for the system with non-positive initial energy in the general case: 1 <= n < 6. Finally, when the initial energy is given arbitrarily positive, we show that if the initial datum satisfies some conditions, the corresponding solution blows up in a finite time. In other words, in this paper we establish the complete blow-up result for the Klein–Gordon equation above in the sense of the initial energy, – < E(0) < + .

An extended notion of enthalpy. Electromagnetic solids

Trimarco, C. — 2009-05-26

Some procedures are here expounded in order to introduce the physical and the material or configurational stress in deformable and moving solids. In electromagnetic materials, the electromagnetic fields are preliminarily introduced in the Lagrangian form. Then a variational approach is proposed through two different procedures. Two energy–momentum tensors are excerpted from the suggested procedures. One of them corresponds to a Cauchy-like stress and the other one to the material stress. The latter rules the balance law for the material momentum. This balance law across a first-order discontinuity surface addresses an extended notion of enthalpy and also an extension of the Maxwell condition in thermodynamical phase transitions of electromagnetic solids.

The Kelvin transformation in potential theory and Stokes flow

Dassios, G. — 2009-05-26

Kelvin's transformation is a non-linear map that, in some sense, preserves harmonicity. This property, which was the content of a letter sent by Kelvin to Liouville in 1845, provides a powerful machinery for solving particular potential problems in a very effective way. In the present work, we show that the basic theory can be extended to the biharmonic equation as well to the equations for irrotational and rotational Stokes flow. Hence, biharmonicity, stream functions and bistream functions are also preserved, in some sense, under the Kelvin transformation. We also demonstrate how the Kelvin-type theorems are interconnected with the relative Almansi-type decompositions. These results provide a way to solve analytically many problems in potential theory and Stokes flow which it is impossible to solve by the classical spectral method.

Micro/nanoparticle melting with spherical symmetry and surface tension

McCue, S. W., Wu, B., Hill, J. M. — 2009-05-26

The process of melting a small spherical particle is treated by setting up a two-phase Stefan problem. Surface tension is included through the Gibbs–Thomson condition, the effect of which is to decrease the melting temperature as the particle radius decreases. Analytical results are derived via a small-time expansion and also through large Stefan number asymptotics. Numerical solutions are computed with a front-fixing scheme, and these results suggest that the model exhibits finite-time blow-up, in the sense that both the interface speed and the temperature gradient in the solid phase (at the interface) will become unbounded at some time before complete melting. The near-blow-up behaviour appears to be similar to that encountered in the ill-posed problem of melting a superheated solid (without surface tension), and may help explain the onset of abrupt melting observed in some experiments with nanoscaled particles.

Stability of sticky particle dynamics and related scalar conservation laws

Moutsinga, O. — 2009-05-26

We show the stability of the sticky particle forward flow (x, s, t) ↦ (x, s, P_t, u_t) w.r.t. perturbations of the initial mass distribution P₀ and velocity function u₀. Then, we deduce the stability of related scalar conservation laws and pressureless gas system.

Asymptotic behaviour of ground state solutions for the Henon equation

Cao, D., Peng, S., Yan, S. — 2009-05-26

Let B₁(0) R^N be the unit ball centred at the origin, N ≥ 3. In this paper, we analyse the profile of the ground state solution of the Hénon equation – u = |x|u^p^–1 in B₁(0), u = 0 onB₁(0). We prove that for fixed p (2, 2^*), (2^* = 2N/(N – 2)), the maximum point x of the ground state solution u satisfies (1 – |x|) -> l (0, +) as -> . We also obtain the asymptotic behaviour of u, which shows that the ground state solution is non-radial. Moreover, we prove the existence of multi-peaked solutions and give their asymptotic behaviour.