On a Hele--Shaw-Type Domain Evolution with Convected Surface Energy Density

Günther, Matthias; Prokert, G Georg

doi:10.1137/s0036141004444846

Cited by 8 publications

(15 citation statements)

References 19 publications

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“…In this way, we prove our main result (Theorems 2.1 and 2.2) on short-time wellposedness of the moving boundary problem (1.1), (1.4), (1.5). We will omit certain details as they are parallel to the discussion in [10]. However, the right-hand side of the evolution problem obtained there is of order two.…”

Section: δϕ(· T) = 0 In ω(T) δψ(· T) = 0 In ω(T) ∂ N ψ(· T) = δ mentioning

confidence: 98%

“…For more details and references, see [1,7,10]. Kinetic undercooling regularization corresponds to adding in (1.3) a boundary integral term β Γ v 1 v 2 dΓ with β > 0, this case is discussed in [10].…”

Section: δϕ(· T) = 0 In ω(T) δψ(· T) = 0 In ω(T) ∂ N ψ(· T) = δ mentioning

confidence: 99%

“…However, the right-hand side of the evolution problem obtained there is of order two. As we are concerned here with an evolution equation whose right-hand side is of order three, we have to refine the construction of our variable inner product from [10] by including certain lower-order terms. Differing from the situation there, here we have to demand strict positivity of γ because its inverse γ −1 enters one of these terms.…”

Section: δϕ(· T) = 0 In ω(T) δψ(· T) = 0 In ω(T) ∂ N ψ(· T) = δ mentioning

confidence: 99%

“…The statements and their proofs are essentially parallel to Corollary 4.4 and Lemma 4.5 in [10], therefore proofs will be omitted. Note, however, that f → F (u)f is an operator of order one here as (2.6) is a Dirichlet problem.…”

Section: Smooth Domain Dependence Of the Nonlocal Operatorsmentioning

confidence: 99%

“…While experiments on such situations have been reported in the literature (e.g., [3,12]), theoretical investigations of this seem to be lacking. A first step in this direction has been made in [10] where shorttime solvability was proved for a Hele-Shaw problem with nonconstant surface tension coefficient and so-called kinetic undercooling. Here we discuss the problem without this regularization, using again the simple assumption that the surface energy density is convectively transported along the moving boundary.…”

Section: Introductionmentioning

confidence: 99%

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On a Hele–Shaw Type Domain Evolution with Convected Surface Energy Density: The Third‐Order Problem

Günther¹,

Prokert²

2006

SIAM J. Math. Anal.

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Abstract. We investigate a moving boundary problem with a gradient flow structure which generalizes Hele-Shaw flow driven solely by surface tension to the case of nonconstant surface tension coefficient taken along with the liquid particles at boundary. The resulting evolution problem is first order in time, contains a third-order nonlinear pseudodifferential operator and is degenerate parabolic. Well-posedness of this problem in Sobolev scales is proved. The main tool is the construction of a variable symmetric bilinear form so that the third-order operator is semibounded with respect to it. Moreover, we show global existence and convergence to an equilibrium for solutions near trivial equilibria (balls with constant surface tension coefficient). Finally, numerical examples in 2D and 3D are given.Key words. free boundary motion, degenerate nonlocal parabolic evolution AMS subject classifications. 35R35, 76B07DOI. 10.1137/0506269951. Introduction. It is the aim of the present paper to consider the generalization of the well-investigated Hele-Shaw flow problem to the case of nonconstant surface tension coefficient (or surface energy density). While experiments on such situations have been reported in the literature (e.g., [3,12]), theoretical investigations of this seem to be lacking. A first step in this direction has been made in [10] where shorttime solvability was proved for a Hele-Shaw problem with nonconstant surface tension coefficient and so-called kinetic undercooling. Here we discuss the problem without this regularization, using again the simple assumption that the surface energy density is convectively transported along the moving boundary.This leads to the following moving boundary problem: For a given bounded domain Ω(0) ⊂ R m and a given nonnegative function γ 0 defined on ∂Ω(0) one looks for a family of C 2 -domains Ω(t) ⊆ R m , t > 0 and functions ϕ(·, t) ∈ C 2 Ω(t) ,

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Section: δϕ(· T) = 0 In ω(T) δψ(· T) = 0 In ω(T) ∂ N ψ(· T) = δ mentioning

confidence: 98%

Section: δϕ(· T) = 0 In ω(T) δψ(· T) = 0 In ω(T) ∂ N ψ(· T) = δ mentioning

confidence: 99%

Section: δϕ(· T) = 0 In ω(T) δψ(· T) = 0 In ω(T) ∂ N ψ(· T) = δ mentioning

confidence: 99%

Section: Smooth Domain Dependence Of the Nonlocal Operatorsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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On a Hele–Shaw Type Domain Evolution with Convected Surface Energy Density: The Third‐Order Problem

Günther¹,

Prokert²

2006

SIAM J. Math. Anal.

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Micro Structures in Thin Coating Layers: Micro Structure Evolution and Macroscopic Contact Angle

Dohmen

Grunewald

Otto

et al.

Mathematics – Key Technology for the Future

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Paralinearization of the Dirichlet-Neumann operator and applications to progressive gravity waves

Alazard¹

2012

Sci. China Math.

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This note presents a paradifferential approach to the analysis of the water waves equations. Keywordsincompressible Euler equation, free surface, paradifferential calculus MSC(2010) 35B65, 35H20Citation: Alazard T. Paralinearization of the Dirichlet-Neumann operator and applications to progressive gravity waves.This note presents a paradifferential approach to the analysis of water wave equations. The main issue is that, thanks to this approach, one can conjugate nonlinear equations to linear equations, to the price of error terms which are smoother than the main terms (and thus presumed to be harmless in the derivation of estimates). In the first section I will recall the water wave equations for gravity waves and introduce the Dirichlet-Neumann operator. Then I will recall basic results about paradifferential operators. The main goal of this note is to state a result proved with Guy Métivier about the paralinearization of this operator, together with some extensions obtained in collaboration with Nicolas Burq and Claude Zuily. This strategy has a number of consequences since it reduces the proof of some difficult nonlinear estimates to easy symbolic calculus questions for symbols. In collaboration with Burq and Zuily, we have obtained sharp results about the well-posedness of the water wave system. These results include a local in time Cauchy theory for rough data corresponding to a natural threshold (cf. [1,3]), and the proofs of several dispersive estimates (cf. [1,2]).As was shown by Zakharov, the water wave system is a Hamiltonian one of the formwhere H is the total energy of the system. Denoting by H 0 the Hamiltonian associated with the linearized system at the origin, we have (for gravity waves)

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On a Hele--Shaw-Type Domain Evolution with Convected Surface Energy Density

Cited by 8 publications

References 19 publications

On a Hele–Shaw Type Domain Evolution with Convected Surface Energy Density: The Third‐Order Problem

On a Hele–Shaw Type Domain Evolution with Convected Surface Energy Density: The Third‐Order Problem

Micro Structures in Thin Coating Layers: Micro Structure Evolution and Macroscopic Contact Angle

Paralinearization of the Dirichlet-Neumann operator and applications to progressive gravity waves

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