On periodic Stokesian Hele-Shaw flows with surface tension

Escher, Joachim; Matioc, Bogdan–Vasile

doi:10.1017/s0956792508007699

Cited by 10 publications

(13 citation statements)

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“…These have been generalized to non-Newtonian one phase fluids [18]. There are similar results available for the two phase Stefan problem [19,20] which is mathematically related to, yet distinct from, the two-phase Hele-Shaw problem (also called the Muskat problem) being studied here.…”

Section: Introductionmentioning

confidence: 67%

Global solutions for a two-phase Hele-Shaw bubble for a near-circular initial shape

Tanveer

2012

Complex Variables and Elliptic Equations

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Section: Introductionmentioning

confidence: 67%

Global solutions for a two-phase Hele-Shaw bubble for a near-circular initial shape

Tanveer

2012

Complex Variables and Elliptic Equations

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“…To the best of the author's awareness, the work contained herein constitutes the first systematic treatment of semigroup generation in the case of variable coefficients for elliptic operators with periodic boundary conditions. A related result for constant coefficients in the periodic setting was proved by Escher and Matioc [14], see also [19], where they considered a specific abstract operator of third order, in the periodic little-Hölder setting.…”

Section: Introductionmentioning

confidence: 85%

Elliptic operators and maximal regularity on periodic little-Hölder spaces

LeCrone

2011

J. Evol. Equ.

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We consider one-dimensional inhomogeneous parabolic equations with higher-order elliptic differential operators subject to periodic boundary conditions. In our main result we show that the property of continuous maximal regularity is satisfied in the setting of periodic little-Hölder spaces, provided the coefficients of the differential operator satisfy minimal regularity assumptions. We address parameter-dependent elliptic equations, deriving invertibility and resolvent bounds which lead to results on generation of analytic semigroups. We also demonstrate that the techniques and results of the paper hold for elliptic differential operators with operator-valued coefficients, in the setting of vector-valued functions. IntroductionIn this paper we consider the following abstract periodic inhomogeneous equationwhere A(·, D) := 2m k=0 b k (·)D k is a differential operator of order 2m, with variable coefficients b k : R → C. Further, we enforce periodic boundary conditions on the problem by assuming that the given functions u 0 , b k , f (t, ·), for t ≥ 0, k = 0, . . . , 2m, are all 2π -periodic in x ∈ R. Hence, we will be looking for solutions u(t, ·) which also exhibit 2π -periodicity on R, for t > 0. We will also consider the more general setting of vector-valued functions u 0 , f (t, ·), u(t, ·) : R → E, and operator-valued coefficients b k : R → L(E), for an arbitrary Banach space E over C. This more general setting is discussed in Sect. 6.Understanding the nature of solutions (i.e., existence, uniqueness and regularity) to inhomogeneous equations of this form is integral to the study of abstract quasilinear equations. In the quasilinear setting, we see that (0.1) takes the form ∂ t u + A(u, D)u = F(u), (2000): Primary 35G16, 35J30; Secondary 35K59, 42A45. Mathematics Subject Classification

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“…Well-posedness results for (1.1) (in slightly different geometric settings and various classes of functions) have been proved in e.g. [1,8,13] and in [5,6] for non-Newtonian fluids.…”

Section: H(x T)mentioning

confidence: 99%