Stability of travelling waves with interface conditions

Brauner, Claude‐Michel; Lunardi, Alessandra; Schmidt-Lainé, Claudine

doi:10.1016/0362-546x(92)90085-s

Cited by 21 publications

(13 citation statements)

References 16 publications

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“…Then P(X) and (I − P)(D) are the linear unstable manifold and the linear stable manifold of w = 0. In view of F being smooth on a neighbourhood of w = 0 in D(L), we may apply Theorem 9.1.4 in [13] to conclude that for every k ≥ 1 there exist R > 0 and C k -maps 5) with ϕ, ϕ , ψ and ψ vanishing in the origin, such that the local unstable and stable manifolds of w = 0 are given by the graphs of ϕ and ψ. The unstable manifold contains solutions which, in terms of the original variables, do not focus at t = T but at some other time value.…”

Section: The Saddle Point Propertymentioning

confidence: 99%

“…The stability properties of such travelling waves have been studied from a global point of view in the one-dimensional case in [10]. In [5] two of the authors have developed with A. Lunardi a new method based on the elimination of the front and the reformulation of the problem as a fully nonlinear equation in a suitable Banach space. The method turns out to be quite efficient for multidimensional stability problems [6,7,8].…”

Section: 3)mentioning

confidence: 99%

“…In this note we adapt the methods in [5,6] to linearize the free boundary problem in its selfsimilar variables around the selfsimilar profile f and establish the existence of a stable and unstable manifold. The unstable manifold is not empty, because the free parameter T in (1.4) always leads to λ = 1 being an eigenvalue.…”

Section: 3)mentioning

confidence: 99%

See 2 more Smart Citations

The saddle point property for focusing selfsimilar solutions in a free boundary problem

Brauner¹,

Hulshof

Schmidt-Lainé

1999

Proc. Amer. Math. Soc.

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Section: The Saddle Point Propertymentioning

confidence: 99%

Section: 3)mentioning

confidence: 99%

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The saddle point property for focusing selfsimilar solutions in a free boundary problem

Brauner¹,

Hulshof

Schmidt-Lainé

1999

Proc. Amer. Math. Soc.

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“…We proceed below with equations for the curves y 1 (x, t), dropping the tilde's on the x variables. Following [6,7], we write the perturbation of the wave as…”

Section: Simple Equal-order Modelmentioning

confidence: 99%

“…As discussed in §2 below, this leads to the presence of a free boundary in the linearized problem, which can potentially create additional difficulties and new phenomenon in the analysis. Stability theory in the presence of free boundaries has been studied rigorously in [6,7], and below we will discuss what these results imply about the current models. Our theoretical discussion is less rigorous for the PDAE formulation of the mixed-order problem, although we observe good agreement between theoretical predictions of the absolute spectrum and numerical results.…”

Section: Introductionmentioning

confidence: 99%

Stability of travelling wave solutions for coupled surface and grain boundary motion

Beck

Pan

Wetton

2010

Physica D: Nonlinear Phenomena

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We investigate the spectral stability of the travelling wave solution for the coupled motion of a free surface and grain boundary that arises in materials science. In this problem a grain boundary, which separates two materials that are identical except for their crystalline orientation, evolves according to mean curvature. At a triple junction, this boundary meets the free surfaces of the two crystals, which move according to surface diffusion. The model is known to possess a unique travelling wave solution. We study the linearization about the wave, which necessarily includes a free boundary at the location of the triple junction. This makes the analysis more complex than that of standard travelling waves, and we discuss how existing theory applies in this context. Furthermore, we compute numerically the associated point spectrum by restricting the problem to a finite computational domain with appropriate physical boundary conditions. Numerical results strongly suggest that the two-dimensional wave is stable with respect to both two-and three-dimensional perturbations.

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A General Approach to Stability in Free Boundary Problems

Brauner

Hulshof²,

Lunardi

2000

Journal of Differential Equations

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Every solution of a linear elliptic equation on a bounded domain may be considered as an equilibrium of a free boundary problem. The free boundary problem consists of the corresponding parabolic equation on a variable unknown domain with free boundary conditions prescribing both Dirichlet and Neumann data. We establish a rigorous stability analysis of such equilibria, including the construction of stable and unstable manifolds. For this purpose we transform the free boundary problem to a fully nonlinear and nonlocal parabolic problem on a fixed domain with fully nonlinear lateral boundary conditions and we develop the general theory for such problems. As an illustration we give two examples, the second being the focussing flame problem in combustion theory. Academic Press

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Stability of travelling waves with interface conditions

Cited by 21 publications

References 16 publications

The saddle point property for focusing selfsimilar solutions in a free boundary problem

The saddle point property for focusing selfsimilar solutions in a free boundary problem

Stability of travelling wave solutions for coupled surface and grain boundary motion

A General Approach to Stability in Free Boundary Problems

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