Lyapunov functions, Identities and the Cauchy problem for the Hele-Shaw equation

Alazard, Thomas; Meunier, Nicolas; Smets, Didier

doi:10.48550/arxiv.1907.03691

Cited by 7 publications

(29 citation statements)

References 33 publications

(53 reference statements)

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“…It is known that Cauchy problem for the latter equation is well-posed on the Sobolev spaces H s (T d ) provided that s > 1 + d/2, and moreover the critical Sobolev exponent is 1 + d/2 (see [27,54,5,55]). On the other hand, the natural energy estimate only controls the L 2 -norm.…”

Section: Statements Of the Main Resultsmentioning

confidence: 99%

“…It is thus natural to seek higher order energies, which are bounded in time and which control Sobolev norms H µ (T d ) of order µ > 0. It was proved in [5,2] that one can control one-half derivative of h by exploiting some convexity argument. More precisely, it is proved in the previous references that…”

Section: Statements Of the Main Resultsmentioning

confidence: 99%

“…Many maximum principles exist also for the Hele-Shaw equations (see [51,24]). In particular, we will use the maximum principle for spacetime derivatives proved in [5]. Sea also [33] for related models.…”

Section: Equationsmentioning

confidence: 99%

“…(i) The Cauchy problems for the previous free boundary equations have been studied by different techniques, for weak solutions, viscosity solutions or also classical solutions. We refer the reader to [4,5,24,25,26,27,28,34,39,42,43,46,47,51,52,55,56]. Thanks to the parabolic smoothing effect, classical solutions are smooth for positive times (the elevation h belongs to C ∞ ((0, T ]×T d )).…”

Section: Introductionmentioning

confidence: 99%

“…This shows that T d h 2 dx is a Lyapunov functional. In [5], it is proved that in fact T d h 2 dx is a strong Lyapunov functional and also an entropy. This result is generalized in [2] to functionals of the form T d Φ(h) dx where Φ is a convex function whose derivative is also convex.…”

Section: Introductionmentioning

confidence: 99%

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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%

Section: Statements Of the Main Resultsmentioning

confidence: 99%

Section: Equationsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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

Alazard¹,

Bresch²

2020

Preprint

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Convexity and the Hele–Shaw Equation

Alazard

2020

Water Waves

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Walter Craig's seminal works on the water-waves problem established the importance of several exact identities: Zakharov's hamiltonian formulation, shape derivative formula for the Dirichlet to Neumann operator, normal forms transformations. In this paper, we introduce several identities for the Hele-Shaw equation which are inspired by his nonlinear approach. Firstly, we study convex changes of unknowns and obtain a large class of strong Lyapunov functions; in addition to be non-increasing, these Lyapunov functions are convex functions of time. The analysis relies on a compact elliptic formulation of the Hele-Shaw equation, which is of independent interest. Then we study the role of convexity to control the spatial derivatives of the solutions. We consider the evolution equation for the Rayleigh-Taylor coefficient a (this is a positive function proportional to the opposite of the normal derivative of the pressure at the free surface). Inspired by the study of entropies for elliptic or parabolic equations, we consider the special function ϕ(x) = x log x and find that ϕ(1/ √ a) is a sub-solution of a well-posed equation.

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Surface tension stabilization of the Rayleigh-Taylor instability for a fluid layer in a porous medium

Gancedo

Granero-Belinchón

Scrobogna

2020

Ann. Inst. H. Poincaré Anal. Non Linéaire

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This paper studies the dynamics of an incompressible fluid driven by gravity and capillarity forces in a porous medium. The main interest is the stabilization of the fluid in Rayleigh-Taylor unstable situations where the fluid lays on top of a dry region. An important feature considered here is that the layer of fluid is under an impervious wall. This physical situation has been widely study by mean of thin film approximations in the case of small characteristic high of the fluid considering its strong interaction with the fixed boundary. Here, instead of considering any simplification leading to asymptotic models, we deal with the complete free boundary problem. We prove that, if the fluid interface is smaller than an explicit constant, the solution is global in time and it becomes instantly analytic. In particular, the fluid does not form drops in finite time. Our results are stated in terms of Wiener spaces for the interface together with some non-standard Wiener-Sobolev anisotropic spaces required to describe the regularity of the fluid pressure and velocity. These Wiener-Sobolev spaces are of independent interest as they can be useful in other problems. Finally, let us remark that our techniques do not rely on the irrotational character of the fluid in the bulk and they can be applied to other free boundary problems.

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Lyapunov functions, Identities and the Cauchy problem for the Hele-Shaw equation

Cited by 7 publications

References 33 publications

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

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

Convexity and the Hele–Shaw Equation

Surface tension stabilization of the Rayleigh-Taylor instability for a fluid layer in a porous medium

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