Local Well-Posedness and Incompressible Limit of the Free-Boundary Problem in Compressible Elastodynamics

Zhang, Junyan

doi:10.1007/s00205-022-01774-4

Cited by 12 publications

(6 citation statements)

References 84 publications

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“…Later, the authors [46] proved the LWP without using Nash-Moser, but the energy functional in [46] is not uniform in Mach number, and thus we cannot derive the incompressible limit (see the next paragraph) while constructing the solution. The second author refined and simplified the techniques in [59,46] such that the LWP and the incompressible limit can be simultaneously proved in the study of compressible elastodynamics [71] which can be directly applied to Euler equations. However, the methods in these works do not apply to the case with nonzero surface tension.…”

Section: An Overview Of Previous Resultsmentioning

confidence: 99%

“…The first result was due to Lindblad and the first author [43] for the case of a bounded domain and zero surface tension. See also the first author's work [45] for compressible gravity water wave, the second author's work [71] for a simpler proof that works for both bounded and unbounded domain, and Disconzi and the first author [17] for the case σ > 0 in a bounded fluid domain.…”

Section: An Overview Of Previous Resultsmentioning

confidence: 99%

“…The compressible pressure q never converges to the incompressible pressure q in , because the former one is the solution to a quasilinear symmetric hyperbolic system but the latter one appears as a Lagrangian multiplier. Indeed, as was indicated by [43,45,71], it is the enthalpy h(ρ) := ρ 1 q ′ (r)/r dr of the compressible equations that converges to the incompressible pressure q in . On the other hand, the convergence of ρ λ,σ − 1 3 can be easily proved if we write the continuity equation to be D ϕ t (ρ − 1) = −ρ(∇ ϕ • v) and use Grönwall's inequality for its H 3 estimates.…”

Section: The Main Theoremsmentioning

confidence: 93%

“…For example, the approximation scheme in[46,71] works for Euler equations but may not work for compressible ideal MHD. An alternative way is to use Nash-Moser iteration as in[59,60,61], but that will introduce a big loss of regularity from initial data to solution.…”

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confidence: 99%

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Compressible Gravity-Capillary Water Waves with Vorticity: Local Well-Posedness, Incompressible and Zero-Surface-Tension Limits

Luo¹,

Zhang²

2022

Preprint

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We consider 3D compressible isentropic Euler equations describing the motion of a liquid in an unbounded initial domain with a moving boundary and a fixed flat bottom at finite depth. The liquid is under the influence of gravity and surface tension and it is not assumed to be irrotational. We prove the local well-posedness by introducing carefully-designed approximate equations which are asymptotically consistent with the a priori energy estimates. The energy estimates yield no regularity loss and are uniform in Mach number. Also, they are uniform in surface tension coefficient if the Rayleigh-Taylor sign condition holds initially. We can thus simultaneously obtain incompressible and vanishing-surface-tension limits. The method developed in this paper is a unified and robust hyperbolic approach to free-boundary problems in compressible Euler equations. It can be applied to some important complex fluid models as it relies on neither parabolic regularization nor irrotational assumption. This paper joined with our previous works [45,46] rigorously proves the local well-posedness and the incompressible limit for a compressible gravity water wave with or without surface tension.

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Section: An Overview Of Previous Resultsmentioning

confidence: 99%

Section: An Overview Of Previous Resultsmentioning

confidence: 99%

Section: The Main Theoremsmentioning

confidence: 93%

mentioning

confidence: 99%

See 2 more Smart Citations

Compressible Gravity-Capillary Water Waves with Vorticity: Local Well-Posedness, Incompressible and Zero-Surface-Tension Limits

Luo¹,

Zhang²

2022

Preprint

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“…, , u ρ τ be the solution to the linearized problem ( 24)-( 26) with ( 7) and ( 8) in [ ] 0,T Ω× satisfying the initial conditions (10), which are defined recursively by ( 24)- (26), and the compatibility conditions (11). Then the solution ( )…”

Section: Preliminaries and The Linearized Problemmentioning

confidence: 99%

Incompressible Limit of the Oldroyd-B Model with Density-Dependent Viscosity

Li¹,

Ren²

2023

JAMP

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This paper studies the existence and uniqueness of local strong solutions to an Oldroyd-B model with density-dependent viscosity in a bounded domain d Ω ⊂  , 2 d = or 3, via incompressible limit, in which the initial data is "well-prepared" and the velocity field enjoys the slip boundary conditions. The main idea is to derive the uniform energy estimates for nonlinear systems and corresponding incompressible limit.

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Low Mach number limit of nonisentropic inviscid Hookean elastodynamics

Wang

2023

Math Methods in App Sciences

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Local Well-Posedness and Incompressible Limit of the Free-Boundary Problem in Compressible Elastodynamics

Cited by 12 publications

References 84 publications

Compressible Gravity-Capillary Water Waves with Vorticity: Local Well-Posedness, Incompressible and Zero-Surface-Tension Limits

Compressible Gravity-Capillary Water Waves with Vorticity: Local Well-Posedness, Incompressible and Zero-Surface-Tension Limits

Incompressible Limit of the Oldroyd-B Model with Density-Dependent Viscosity

Low Mach number limit of nonisentropic inviscid Hookean elastodynamics

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