The free boundary Euler equations with large surface tension

Disconzi, Marcelo M.; Ebin, David G.

doi:10.1016/j.jde.2016.03.029

Cited by 19 publications

(22 citation statements)

References 57 publications

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“…The assumption ǫ 0 > 0 in Theorem 2.2 (which implies a uniform bound from below away from zero by the compactness of Σ), however, is crucial. This is apparent from expression (1.1), but it is worth mentioning that allowing ǫ 0 to vanish leads to severe technical difficulties even in the better studied case of the Einstein-Euler system (see [18,24,36] for the known results and [13] for a discussion; in fact, the difficulties with vanishing density are present already in the non-relativistic case, see the discussion in [14,31]). In particular, if we were dealing with a non-compact Σ and had chosen an asymptotic condition where ǫ 0 approaches zero, the techniques here employed would not directly apply.…”

Section: Statement Of the Resultsmentioning

confidence: 99%

On the existence of solutions and causality for relativistic viscous conformal fluids

Disconzi¹

2019

Communications on Pure &Amp; Applied Analysis

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Section: Statement Of the Resultsmentioning

confidence: 99%

On the existence of solutions and causality for relativistic viscous conformal fluids

Disconzi¹

2019

Communications on Pure &Amp; Applied Analysis

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“…This is all sensible, but it is ultimately, at best, a general philosophy that can provide hints on how to deal with each particular set of equations. The point is that one cannot know how much of the linear behavior will be suppressed by the non-linearities (a case in point is studied, for example, in [26,27], where the particular form of the non-linearity plays a crucial role in guaranteeing that certain smallness conditions are propagated, which would not be the case for the linearized system).…”

Section: The Naive Linear Analysis and Its Limitationsmentioning

confidence: 99%

On a viable first-order formulation of relativistic viscous fluids and its applications to cosmology

Disconzi

Kephart

Scherrer

2017

Int. J. Mod. Phys. D

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We consider a first order formulation of relativistic fluids with bulk viscosity based on a stress-energy tensor introduced by Lichnerowicz. Choosing a barotropic equation of state, we show that this theory satisfies basic physical requirements and, under the further assumption of vanishing vorticity, that the equations of motion are causal, both in the case of a fixed background and when the equations are coupled to Einstein's equations. Furthermore, Lichnerowicz's proposal does not fit into the general framework of first order theories studied by Hiscock and Lindblom, and hence their instability results do not apply. These conclusions apply to the full-fledged non-linear theory, without any equilibrium or near equilibrium assumptions. Similarities and differences between the approach explored here and other theories of relativistic viscosity, including the Mueller-Israel-Stewart formulation, are addressed. Cosmological models based on the Lichnerowicz stress-energy tensor are studied. As the topic of (relativistic) viscous fluids is also of interest outside the general relativity and cosmology communities, such as, for instance, in applications involving heavy-ion collisions, we make our presentation is largely self-contained.

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“…Later Ambrose & Masmoudi [6], proved local well-posedness of the 2-D water waves problem as the limit of zero surface tension. Disconzi & Ebin [22,23] have considered the limit of surface tension tending to infinity. For 3-D fluids (and 2-D interfaces), Wu [44] used Clifford analysis to prove local existence of the 3-D water waves problem with infinite depth, again showing that the Rayleigh-Taylor sign condition is always satisfied in the irrotational case by virtue of the maximum principle holding for the potential flow.…”

Section: A)mentioning

confidence: 99%

On the Impossibility of Finite-Time Splash Singularities for Vortex Sheets

Coutand

Shkoller

2016

Arch Rational Mech Anal

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In fluid dynamics, an interface splash singularity occurs when a locally smooth interface self-intersects in finite time. By means of elementary arguments, we prove that such a singularity cannot occur in finite time for vortex sheet evolution, i.e. for the two-phase incompressible Euler equations. We prove this by contradiction; we assume that a splash singularity does indeed occur in finite time. Based on this assumption, we find precise blow-up rates for the components of the velocity gradient which, in turn, allow us to characterize the geometry of the evolving interface just prior to self-intersection. The constraints on the geometry then lead to an impossible outcome, showing that our assumption of a finite-time splash singularity was false.Date: June 30, 2015. Key words and phrases. vortex sheets, Euler equations, water waves, blow-up, interface singularity, splash singularity, splat singularity. 1 4 D. COUTAND AND S. SHKOLLERThe first local well-posedness result for the 3-D incompressible Euler equations without the irrotationality assumption was obtained by Lindblad [34] in the case that the domain is diffeomorphic to the unit ball using a Nash-Moser iteration. Coutand & Shkoller [18] proved local well-posedness for arbitrary initial geometries that have at least H 3 -class boundaries without derivative loss. Shatah & Zeng [40] established a priori estimates for this problem using an infinite-dimensional geometric formulation, and Zhang & Zhang proved well-poseness by extending the complex-analytic method of Wu [44] to allow for vorticity. Again, in the latter case the domain was with infinite depth.1.4.2. Long-time existence. It is of great interest to understand if solutions to the Euler equations can be extended for all time when the data is sufficiently smooth and small, or if a finite-time singularity can be predicted for other types of initial conditions.Because of irrotationality, the water waves problem does not suffer from vorticity concentration; therefore, singularity formation involves only the loss of regularity of the interface or interface collision. In the case that the irrotational fluid is infinite in the horizontal directions, certain dispersivetype properties can be made use of. For sufficiently smooth and small data, Alvarez-Samaniego & Lannes [5] proved existence of solutions to the water waves problem on large time-intervals (larger than predicted by energy estimates), and provided a rigorous justification for a variety of asymptotic regimes. By constructing a transformation to remove the quadratic nonlinearity, combined with decay estimates for the linearized problem (on the infinite half-space domain), Wu [45] established an almost global existence result (existence on time intervals which are exponential in the size of the data) for the 2-D water waves problem with sufficiently small data. In a different framework, Alazard, Burq & Zuily [2] have also proven this result. Using position-velocity potential holomorphic coordinates, Hunter, Ifrim, & Tataru [28] have also proved almo...

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The free boundary Euler equations with large surface tension

Cited by 19 publications

References 57 publications

On the existence of solutions and causality for relativistic viscous conformal fluids

On the existence of solutions and causality for relativistic viscous conformal fluids

On a viable first-order formulation of relativistic viscous fluids and its applications to cosmology

On the Impossibility of Finite-Time Splash Singularities for Vortex Sheets

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