Well Posedness for the Motion of a Compressible Liquid with Free Surface Boundary

Lindblad, Hans

doi:10.1007/s00220-005-1406-6

Cited by 104 publications

(157 citation statements)

References 21 publications

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“…The one-phase problem (where one of the two phases is replaced by vacuum) has been solved for the case of an incompressible liquid by Coutand and Shkoller [12] and [11] with and without surface tension and previously by Lindblad [18] without surface tension, and for the case of a compressible liquid by Coutand et al [10] with and without surface tension and previously by Lindblad [17] without surface tension. Also, the case of a gas-vacuum boundary has been solved under certain boundary conditions by Coutand and Shkoller [13] and Jang and Masmoudi [16].…”

Section: Existing Results and Methodology For Similar Problemsmentioning

confidence: 99%

“…Rather than follow the Nash-Moser scheme of [4,9,17,20], which can often be very technical, we introduce a degenerate artificial viscosity type regularisation inspired by [10,13,16], and prove a priori estimates on the nonlinear equations in a somewhat similar manner to [6], but using a change of coordinates which is a simple lift of the graph of the free-surface, similar to that of [7], rather than Lagrangian coordinates.…”

Section: Existing Results and Methodology For Similar Problemsmentioning

confidence: 99%

“…The one-phase problem with and without surface tension has been solved by Coutand et al [10], and earlier by Lindblad without surface tension in both the incompressible [18] and compressible [17] cases, so we do not devote a large section to this problem. However, we sketch the details of the results we would obtain for the one-phase problem using the above methods since the exact statements are slightly different to those in these references and the methods used are somewhat different.…”

Section: The One-phase Problem With and Without Surface Tensionmentioning

confidence: 99%

“…then for j 0 we inductively define (17) where the right hand side means we think of u ± (t) etc. as functions of t and formally differentiate with respect to t j times according to the chain and Leibniz rules, then replace ∂ l t u ± (t) for 0 l j by ∂ l 0 u 0 ± etc.…”

Section: Definitionmentioning

confidence: 99%

“…We remind the reader that this problem was originally solved by Lindblad [17]. estimate for the corrected unknowns is simple given that we have 4 derivatives in our energy, and is obtained as follows using the Sobolev embedding theorem:…”

Section: The One-phase Problem Without Surface Tensionmentioning

confidence: 99%

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Short-Time Structural Stability of Compressible Vortex Sheets with Surface Tension

Stevens

2016

Arch Rational Mech Anal

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Assume we start with an initial vortex-sheet configuration which consists of two inviscid fluids with density bounded below flowing smoothly past each other, where a strictly positive fixed coefficient of surface tension produces a surface tension force across the common interface, balanced by the pressure jump. We model the fluids by the compressible Euler equations in three space dimensions with a very general equation of state relating the pressure, entropy and density such that the sound speed is positive. We prove that, for a short time, there exists a unique solution of the equations with the same structure.The mathematical approach consists of introducing a carefully chosen artificial viscosity-type regularisation which allows one to linearise the system so as to obtain a collection of transport equations for the entropy, pressure and curl together with a parabolic-type equation for the velocity which becomes fairly standard after rotating the velocity according to the interface normal. We prove a high order energy estimate for the non-linear equations that is independent of the artificial viscosity parameter which allows us to send it to zero. This approach loosely follows that introduced by Shkoller et al. in the setting of a compressible liquid-vacuum interface.Although already considered by Coutand et al. [10] and Lindblad [17], we also make some brief comments on the case of a compressible liquid-vacuum interface, which is obtained from the vortex sheets problem by replacing one of the fluids by vacuum, where it is possible to obtain a structural stability result even without surface tension.

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Section: Existing Results and Methodology For Similar Problemsmentioning

confidence: 99%

Section: Existing Results and Methodology For Similar Problemsmentioning

confidence: 99%

Section: The One-phase Problem With and Without Surface Tensionmentioning

confidence: 99%

Section: Definitionmentioning

confidence: 99%

Section: The One-phase Problem Without Surface Tensionmentioning

confidence: 99%

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Short-Time Structural Stability of Compressible Vortex Sheets with Surface Tension

Stevens

2016

Arch Rational Mech Anal

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Static self‐gravitating elastic bodies in Einstein gravity

Andersson

Beig

Schmidt

2007

Comm Pure Appl Math

112

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ABSTRACT. We prove that given a stress-free elastic body there exists, for sufficiently small values of the gravitational constant, a unique static solution of the Einstein equations coupled to the equations of relativistic elasticity. The solution constructed is a small deformation of the relaxed configuration. This result yields the first proof of existence of static solutions of the Einstein equations without symmetries.

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Local existence for the free boundary problem for nonrelativistic and Relativistic compressible Euler equations with a vacuum boundary condition

Trakhinin

2009

Comm Pure Appl Math

238

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We study the free boundary problem for the equations of compressible Euler equations with a vacuum boundary condition. Our main goal is to recover in Eulerian coordinates the earlier well-posedness result obtained by Lindblad [11] for the isentropic Euler equations and extend it to the case of full gas dynamics.For technical simplicity we consider the case of an unbounded domain whose boundary has the form of a graph and make short comments about the case of a bounded domain. We prove the local-in-time existence in Sobolev spaces by the technique applied earlier to weakly stable shock waves and characteristic discontinuities [5,21]. It contains, in particular, the reduction to a fixed domain, using the "good unknown" of Alinhac [1], and a suitable Nash-Moser-type iteration scheme. A certain modification of such an approach is caused by the fact that the symbol associated to the free surface is not elliptic. This approach is still directly applicable to the relativistic version of our problem in the setting of special relativity and we briefly discuss its extension to general relativity.

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Well Posedness for the Motion of a Compressible Liquid with Free Surface Boundary

Cited by 104 publications

References 21 publications

Short-Time Structural Stability of Compressible Vortex Sheets with Surface Tension

Short-Time Structural Stability of Compressible Vortex Sheets with Surface Tension

Static self‐gravitating elastic bodies in Einstein gravity

Local existence for the free boundary problem for nonrelativistic and Relativistic compressible Euler equations with a vacuum boundary condition

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