Relaxation to a planar interface in the Mullins–Sekerka problem

Chugreeva, Olga; Otto, Félix; Westdickenberg, María G.

doi:10.4171/ifb/415

Cited by 5 publications

(6 citation statements)

References 23 publications

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“…This is becasue it is harder to differentiate these equations. For the Mullins-Sekerka problem, one can quote two recent papers by Chugreeva-Otto-Westdickenberg [29] and Acerbi-Fusco-Julin-Morini [1]. In both papers, the authors compute the second derivative in time of some coercive quantities to study the long time behavior of the solutions, in perturbative regimes.…”

Section: Statements Of the Main Resultsmentioning

confidence: 99%

Functional inequalities and strong Lyapunov functionals for free surface flows in fluid dynamics

Alazard¹,

Bresch²

2020

Preprint

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This paper is motivated by the study of Lyapunov functionals for four equations describing free surface flows in fluid dynamics: the Hele-Shaw and Mullins-Sekerka equations together with their lubrication approximations, the Boussinesq and thin-film equations. We identify new Lyapunov functionals, including some which decay in a convex manner (these are called strong Lyapunov functionals). For the Hele-Shaw equation and the Mullins-Sekerka equation, we prove that the L 2 -norm of the free surface elevation and the area of the free surface are Lyapunov functionals, together with parallel results for the thinfilm and Boussinesq equations. The proofs combine exact identities for the dissipation rates with functional inequalities. For the thin-film and Boussinesq equations, we introduce a Sobolev inequality of independent interest which revisits some known results and exhibits strong Lyapunov functionals. For the Hele-Shaw and Mullins-Sekerka equations, we introduce a functional which controls the L 2 -norm of three-half spatial derivative. Under a mild smallness assumption on the initial data, we show that the latter quantity is also a Lyapunov functional for the Hele-Shaw equation, implying that the area functional is a strong Lyapunov functional. Precise lower bounds for the dissipation rates are established, showing that these Lyapunov functionals are in fact entropies. Other quantities are also studied such as Lebesgue norms or the Boltzmann's entropy.

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Section: Statements Of the Main Resultsmentioning

confidence: 99%

Functional inequalities and strong Lyapunov functionals for free surface flows in fluid dynamics

Alazard¹,

Bresch²

2020

Preprint

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“…Recall that the L 1 -norm of jrhj 2 = h, called the Fisher information, plays a key role in entropy methods and information theory (see Villani's lecture notes [40] and his book [41,Chapters 20,21,and 22]).…”

Section: Examplesmentioning

confidence: 99%

Functional inequalities and strong Lyapunov functionals for free surface flows in fluid dynamics

Alazard,

Bresch

2023

Interfaces Free Bound.

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This paper is motivated by the study of Lyapunov functionals for the Hele-Shaw and Mullins-Sekerka equations describing free surface flows in fluid dynamics. We prove that the L 2norm of the free surface elevation and the area of the free surface are Lyapunov functionals. The proofs combine exact identities for the dissipation rates with functional inequalities. We introduce a functional which controls the L 2 -norm of three-half spatial derivative. Under a mild smallness assumption on the initial data, we show that the latter quantity is also a Lyapunov functional for the Hele-Shaw equation, implying that the area functional is a strong Lyapunov functional. Precise lower bounds for the dissipation rates are established, showing that these Lyapunov functionals are in fact entropies.

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“…where 𝑓(𝑡) ∶ ℝ → ℝ, 𝑡 ≥ 0, is an unknown function. The same setting has been also considered in [13], where the authors establish convergence rates to a planar interface for global solutions (assuming they exist). Our goal is to establish the well-posedness of the Mullins-Sekerka problem in this unbounded regime for initial data whose regularity is close of being optimal.…”

Section: Introductionmentioning

confidence: 99%

“…To be more precise, we assume that at each time instant

t\ge 0

we have

\begin{equation*} \Omega ^\pm (t)={\left\lbrace (x,y)\in {\mathbb {R}}^2\,:\, y\gtrless f(t,x)\right\rbrace} \qquad \text{and}\qquad \Gamma (t):=\lbrace (x,f(t,x)):\, x\in {\mathbb {R}}\rbrace, \end{equation*}

where

f(t):{\mathbb {R}}\rightarrow {\mathbb {R}}

t\ge 0

, is an unknown function. The same setting has been also considered in [13], where the authors establish convergence rates to a planar interface for global solutions (assuming they exist). Our goal is to establish the well‐posedness of the Mullins–Sekerka problem in this unbounded regime for initial data whose regularity is close of being optimal.…”

Section: Introductionmentioning

confidence: 99%