Self-similar solutions for the Muskat equation

Garcia-Juarez, Eduardo; Gómez-Serrano, Javier; Nguyen, Huy Quang; Pausader, Benoît

doi:10.48550/arxiv.2109.02565

Cited by 2 publications

(2 citation statements)

References 39 publications

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“…These obtuse angle corners are shown to smooth out instantly. A more recent self-similar argument [30] in the two-phase regime shows this smoothing explicitly for very small slopes, or in other words, corners of the interface of large angle very close to π.…”

Section: Introductionmentioning

confidence: 84%

Rigidity of Acute Angled Corners for One Phase Muskat Interfaces

Agrawal¹,

Patel²,

Wu³

2021

Preprint

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We consider the one-phase Muskat problem modeling the dynamics of the free boundary of a single fluid in porous media. We prove local well-posedness for fluid interfaces that are general curves and can have singularities. In particular, the free boundary can have acute angle corners or cusps. Moreover, we show that isolated corners/cusps on the interface must be rigid, meaning the angle of the corner is preserved for a finite time, there is no rotation at the tip, the particle at the tip remains at the tip and the velocity of that particle at the tip points vertically downward.

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Section: Introductionmentioning

confidence: 84%

Rigidity of Acute Angled Corners for One Phase Muskat Interfaces

Agrawal¹,

Patel²,

Wu³

2021

Preprint

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“…Initial graphs can turn to non-graph in finite time [13]. See [28] for a review and [32,4,34] for recent developments of the two-phase case.…”

Section: Introductionmentioning

confidence: 99%

Rigorous thin film approximations of the one-phase unstable Muskat problem

Bocchi¹,

Gancedo²

2022

Preprint

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This paper studies the one-phase Muskat problem driven by gravity and surface tension. The regime considered here is unstable with the fluid on top of a dry region. By a novel approach using a depth-averaged formulation, we derive two asymptotic approximations for this scenario. The lower order approximation is the classical thin film equation, while the higher order approximation provides a new refined thin film equation. We prove optimal order of convergence in the shallowness parameter to the original Muskat solutions for both models with lowregular initial data.

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Self-similar solutions for the Muskat equation

Abstract: We show the existence of self-similar solutions for the Muskat equation. These solutions are parameterized by 0 ă s ! 1; they are exact corners of slope s at t " 0 and become smooth in x for t ą 0.:,;

Cited by 2 publications

References 39 publications

Rigidity of Acute Angled Corners for One Phase Muskat Interfaces

Rigidity of Acute Angled Corners for One Phase Muskat Interfaces

Rigorous thin film approximations of the one-phase unstable Muskat problem

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