Well-posedness of the free boundary problem for incompressible elastodynamics

Hu, Xianpeng; Huang, Yongting

doi:10.1016/j.jde.2018.12.018

Cited by 11 publications

(5 citation statements)

References 30 publications

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“…Li-Wang-Zhang [53,54] proved the LWP and elasto-vortex sheet problem under non-collinearity condition. Hu-Huang [37] proved the LWP under Rayleigh-Taylor sign condition by generalizing Lindblad's [58,60] method and Nash-Moser iteration, and Zhang [98] gives an alternative proof with energy estimates without loss of regularity by generalizing Gu-Wang's [33] method in incompressible MHD. In the case of nonzero surface tension, Gu-Lei [31] proved the LWP by vanishing viscosity method.…”

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

Preprint

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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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“…Further discussions of these topics can be referred to previous studies. [9][10][11][12][13][14][15][16][17][18][19][20][21] A summary on some of the recent results can be found in a review paper. 22 We now revisit the system (1).…”

Section: Introductionmentioning

confidence: 99%

“…Yet, the global existence of weak solutions to the incompressible viscoelastic flows with large initial data is still an outstanding open question. Further discussions of these topics can be referred to previous studies . A summary on some of the recent results can be found in a review paper …”

Section: Introductionmentioning

confidence: 99%

On the Cauchy problems for certain regularized models to the incompressible viscoelastic flow

Qiu

Yao

2019

Math Methods in App Sciences

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In this paper, we study the Cauchy problems for certain regularized models to the incompressible viscoelastic flow in n space dimensions with n=2,3. Firstly, we establish a regularity condition for the solution under ∇u∈L1false(0,T;L∞false(double-struckRnfalse)false). Furthermore, we obtain a regularity condition to the smooth solution for the inviscid regularized models in two space dimensions. Finally, we prove a global existence result of classical solutions for a three‐dimensional incompressible Oldroyd‐α model with fractional diffusion.

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Well-posedness of the free boundary problem for incompressible elastodynamics

Cited by 11 publications

References 30 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

On the Cauchy problems for certain regularized models to the incompressible viscoelastic flow

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