Well-posedness of the free boundary problem in incompressible elastodynamics under the mixed type stability condition

Gu, Xumin; Wang, Fan

doi:10.1016/j.jmaa.2019.123529

Cited by 10 publications

(7 citation statements)

References 43 publications

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“…For the incompressible case, Hao-Wang [36] proved the Christodoulou-Lindblad type a priori estimates under Rayleigh-Taylor sign condition (1.8). Gu-Wang [32] proved the LWP under a mixed stability condition. Li-Wang-Zhang [53,54] proved the LWP and elasto-vortex sheet problem under non-collinearity condition.…”

Section: History and Backgroundmentioning

confidence: 99%

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

Zhang

2021

Preprint

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We prove the local well-posedness in Sobolev spaces and incompressible limit for the motion of a compressible isentropic inviscid elastic fluid with moving physical vacuum boundary provided that Rayleigh-Taylor sign condition initially holds. The deformation tensor satisfies the neo-Hookean linear elasticity and is tangential to the free boundary. We apply the tangential smoothing method to set the approximation system. The key observation is that the structure of the wave equation of pressure together with Christodoulou-Lindblad [13] elliptic estimates reduces the energy estimates to the control of tangentiallydifferentiated wave equations in spite of a potential loss of derivative in the source term. Then we use the hyperbolic approach instead of fixed-point argument to solve the linearized system and Picard iteration to establish the well-posedness. Our energy estimate is of no regularity loss and uniform in sound speed which yields the incompressible limit: The solutions and energy functionals of the free-boundary compressible elastodynamic equations converge to the incompressible counterpart provided the convergence of initial datum. To the best of our knowledge, we first find out the suitable energy functionals without loss of regularity for free-boundary compressible elastodynamics system.It is worth emphasizing that our method is completely applicable to compressible Euler equations. Our observation also shows that it is not necessary to analyze the full time derivatives in boundary energy and higher order wave equations as in the previous works of compressible Euler equations even if we require the energy is uniform in sound speed. Moreover, the enhanced regularity for compressible Euler equations obtained in Lindblad-Luo [62] and Luo [64] can still be recovered for a slightly compressible elastic fluid by further delicate analysis which is completely different from Euler equations.

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Section: History and Backgroundmentioning

confidence: 99%

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

Zhang

2021

Preprint

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“…However, for the free boundary problem, the well-posedness theory, even the local one, seems far from being well understood. Only some a priori estimates and the local well-posedness were obtained under a special boundary condition (such as p = 0, F T n = 0), see [17,27,16,18,41]. To the best of our knowledge, the problem of the local existence under the force balance condition (1.2) 2 was left open.…”

Section: Introductionmentioning

confidence: 99%

Local Well-posedness of the Free Boundary Incompressible Elastodynamics with Surface Tension

Gu¹,

Lei²

2020

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In this paper, we consider a free boundary problem of the incompressible elatodynamics, a coupling system of the Euler equations for the fluid motion with a transport equation for the deformation tensor. Under a natural force balance law on the free boundary with the surface tension, we establish its well-posedness theory on a short time interval. Our method is the vanishing viscosity limit by establishing a uniform a priori estimates with respect to the viscosity. As a by-product, the inviscid limit of the incompressible viscoelasticity (the system coupling with the Navier-Stokes equations) is also justified. We point out that based on a crucial new observation of the inherent structure of the elastic term on the free boundary, the framework here is established solely in standard Sobolev spaces, but not the co-normal ones used in [33].

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“…The well-posedness theory of the free boundary problem for incompressible neo-Hookean elastodynamics is more complicated. A prior estimates and local well-posedness are obtained in [15][16][17]24] under some special boundary conditions, such as F T n = 0, p = 0 or p = 0, (F F T − I)n = (F T − I)n = 0. Recently, the first author and Lei in [14] established the local well-posedness result for free boundary problems being subject to the natural force balance law, which is consistent with (1.7) 5 in the compressible case here.…”

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confidence: 99%

Vanishing viscosity limits for the free boundary problem of compressible viscoelastic fluids with surface tension

Gu¹,

Yu²

2022

Preprint

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We consider the free boundary problem of compressible isentropic neo-Hookean viscoelastic fluid equations with surface tension. Under the physical kinetic and dynamic conditions proposed on the free boundary, we investigate regularities of classical solutions to viscoelastic fluid equations in Sobolev spaces which are uniform in viscosity and justify the corresponding vanishing viscosity limits. The key ingredient of our proof is that the deformation gradient tensor in Lagrangian coordinates can be represented as a parameter in terms of flow map so that the inherent structure of the elastic term improves the uniform regularity of normal derivatives in the limit of vanishing viscosity. This result indicates that the boundary layer does not appear in the free boundary problem of compressible viscoelastic fluids which is different to the case studied by the second author for the free boundary compressible Navier-Stokes system.

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Well-posedness of the free boundary problem in incompressible elastodynamics under the mixed type stability condition

Cited by 10 publications

References 43 publications

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

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

Local Well-posedness of the Free Boundary Incompressible Elastodynamics with Surface Tension

Vanishing viscosity limits for the free boundary problem of compressible viscoelastic fluids with surface tension

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