A. van Harten

On the validity of the Ginzburg-Landau equation

SummaryThe famous Ginzburg-Landau equation describes nonlinear amplitude modulations of a wave perturbation of a basic pattern when a control parameterR lies in the unstable regionO(ε2) away from the critical valueRc for which the system loses stability. Hereε>0 is a small parameter. G-Ls equation is found for a general class of nonlinear evolution problems including several classical problems from hydrodynamics and other fields of physics and chemistry. Up to now, the rigorous derivation of G-Ls equation for general situations is not yet completed. This was only demonstrated for special types of solutions (steady, time periodic) or for special problems (the Swift-Hohenberg equation). Here a mathematically rigorous proof of the validity of G-Ls equation is given for a general situation of one space variable and a quadratic nonlinearity. Validity is meant in the following sense. For each given initial condition in a suitable Banach space there exists a unique bounded solution of the initial value problem for G-Ls equation on a finite interval of theO(1/ε2)-long time scale intrinsic to the modulation. For such a finite time interval of the intrinsic modulation time scale on which the initial value problem for G-Ls equation has a bounded solution, the initial value problem for the original evolution equation with corresponding initial conditions, has a unique solutionO(ε2) — close to the approximation induced by the solution of G-Ls equation. This property guarantees that, for rather general initial conditions on the intrinsic modulation time scale, the behavior of solutions of G-Ls equation is really inherited from solutions of the original problem, and the other way around: to a solution of G-Ls equation corresponds a nearby exact solution with a relatively small error.

Acoustic Coupling of Flames

In the limits of large activation energy and small Mach number, the full equations of reactive gas dynamics are reduced to a simpler set which is appropriate for studying acoustic interaction with slender flames. The model is used to study the interaction of a plane, steady flame with a normally incident acoustic wave. Explicit analytical expressions are obtained for the reflection and transmission coefficients, and, in two limiting cases, for the acoustically induced disturbance in the flame speed.

A New Model in Flame Theory

In a recent paper [SIAM J. Appl. Math., 37 (1979), pp. 686–699] Matkowsky and Sivashinsky employed asymptotic methods to derive a simplified model in flame theory from the general equations of combustion. The model was derived under the assumptions of (i) large activation energy, (ii) closeness to similarity and (iii) weak thermal expansion, and the resulting model was associated with the constant density approximation. In this paper, assumption (iii) is relaxed somewhat and in addition a more general temperature dependence of the thermal conductivity; heat capacity and diffusivity is assumed. The new model now exhibits nonconstant density effects. A number of solutions of the model, representing various types of flames are presented, and their stability is analyzed.

An Asymptotic Theory of Deflagrations and Detonations I. The Steady Solutions

Combustion waves propagating through a reactive gas are studied in the plane, one-dimensional geometry. On a length scale large compared to the diffusion length, the waves are treated as exothermic discontinuities in an ideal, nonreactive gas. An asymptotic theory is developed which yields the steady structures of the waves in a simple, analytical form. The theory, based on limits of large activation energy and small heat release, treats all possible deflagrations and detonations.

A Quasi-linear, Singular Perturbation Problem of Hyperbolic Type

Using matched asymptotic expansions, a formal approximation can be constructed for an initial value problem of singularly perturbed, hyperbolic type in two independent variables. Under a time-like condition for the subcharacteristics of the unperturbed operator the correctness of the formal approximation is shown. Because of the nonlinearity of the perturbing hyperbolic operator, this work generalizes Geel [4]. The correctness proof is based on Schauder’s fixed point theorem; it uses existence, uniqueness and regularity theory for hyperbolic systems and a priori estimates for a solution analogous to Geel [4] as ingredients.

Paper: Singularly perturbed systems of diffusion type and feedback control

Asymptotic approximations describing the behaviour of linear systems of diffusion type (convective or non-convective) with a small diffusivity, to which a feedback control of distributed or boundary type based on point sensors is applied, are constructed and proven to be correct. As a consequence one can find a near-optimal feedback control for a cost minimization problem with a quadratic performance index measuring the deviation of the stationary state from an ideal state, under the restriction of a prescribed exponential degree of stability of the stationary state.

Plasma produced by a laser in a medium with convection and free surface satisfying a Hamiltonian-Jacobi equation

A nonlinear diffusion equation is considered which models the temperature distribution in a laser-sustained plasma subject to wind. As certain parameters are small a singular perturbation problem arises and the method of matched asymptotic expansions is applied to approximate the solution. An essential role in this problem is played by the plasma front. This is a free surface separating the plasma and non-plasma phases. One of the main results is that we derive a 1st order nonlinear P.D.E. of Hamilton-Jacobi type, which describes certain parts of the free surface in the stationary case. This equation is analysed for various wind directions. It appears that for the initial conditions for these parts of this free surface there are different possibilities depending on the wind direction. We show further that the solution of the Hamilton-Jacobi equation can contain singularities of corner type. Furthermore, the effect of wind on the stability region of the stationary full plasma solution is analysed. The method of analysis presented in this paper is not restricted to the cone-like geometry or the specific form of the non-linearity of the problem considered here, but has potentially a much wider scope. However, the case under study in this paper is certainly representative for the effects that have to be taken into account.

On the spectral properties of a class of elliptic functional differential operators arising in feedback control theory for diffusion processes

In this paper Dirichlet boundary value problems are considered for certain operators of the form

Asymptotics of a rather unsual type in a free boundary problem

L + \Pi

Singular Perturbation Problems for Non-Linear Elliptic Second Order Equations.

, where L is a 2nd order, elliptic, formally self-adjoint