[Paper Review] Well-posedness of the free boundary problem in compressible elastodynamics

Yuri Trakhinin|arXiv (Cornell University)|May 31, 2017

Navier-Stokes equation solutions30 references3 citations

TL;DR

This paper establishes the local-in-time well-posedness of the free boundary problem for compressible elastodynamics under two distinct conditions: either the deformation gradient columns are non-collinear at the initial boundary, or the classical Rayleigh-Taylor sign condition holds. It proves existence and uniqueness of smooth solutions and constructs a Hadamard-type ill-posedness example showing instability when both conditions fail, confirming the necessity of one of them for well-posedness.

ABSTRACT

We study the free boundary problem for the flow of a compressible isentropic inviscid elastic fluid. At the free boundary moving with the velocity of the fluid particles the columns of the deformation gradient are tangent to the boundary and the pressure vanishes outside the flow domain. We prove the local-in-time existence of a unique smooth solution of the free boundary problem provided that among three columns of the deformation gradient there are two which are non-collinear vectors at each point of the initial free boundary. If this non-collinearity condition fails, the local-in-time existence is proved under the classical Rayleigh-Taylor sign condition satisfied at the first moment. By constructing an Hadamard-type ill-posedness example for the frozen coefficients linearized problem we show that the simultaneous failure of the non-collinearity condition and the Rayleigh-Taylor sign condition leads to Rayleigh-Taylor instability.

Motivation & Objective

• To establish local-in-time existence and uniqueness of smooth solutions for the free boundary problem in compressible elastodynamics.
• To identify minimal structural conditions on the initial data that ensure well-posedness.
• To determine the sharpness of the non-collinearity and Rayleigh-Taylor sign conditions by constructing an ill-posedness example.
• To clarify the role of the deformation gradient's column structure in the stability of the free boundary problem.

Proposed method

• Formulates the compressible elastodynamic system in Eulerian coordinates using the symmetric hyperbolic system (4) with state variables (p, v, F₁, F₂, F₃).
• Imposes boundary conditions: pressure vanishes at the free boundary, and columns of the deformation gradient are tangent to it.
• Applies a Lagrangian-type transformation to reduce the free boundary problem to a fixed domain problem for analysis.
• Uses energy estimates and the Kreiss symmetrization framework to derive a priori estimates for the linearized problem.
• Performs a Fourier analysis on the frozen-coefficient linearized problem to derive a dispersion relation and identify unstable modes.
• Constructs a Hadamard-type ill-posedness example by analyzing the existence of exponentially growing modes when both non-collinearity and Rayleigh-Taylor conditions fail.

Experimental results

Research questions

• RQ1 Under what conditions on the initial deformation gradient does the free boundary problem for compressible elastodynamics admit a unique smooth solution locally in time?
• RQ2 Is the non-collinearity of two columns of the deformation gradient sufficient for well-posedness when the Rayleigh-Taylor sign condition fails?
• RQ3 What happens to the well-posedness when both the non-collinearity and Rayleigh-Taylor sign conditions are violated?
• RQ4 Can a Hadamard-type ill-posedness example be constructed to demonstrate instability under the simultaneous failure of both conditions?
• RQ5 What is the minimal structural assumption on the deformation gradient that still allows for a priori estimates and well-posedness?

Key findings

• Local-in-time existence and uniqueness of a smooth solution is proven if at least two columns of the deformation gradient are non-collinear at the initial free boundary.
• If the non-collinearity condition fails, well-posedness still holds under the classical Rayleigh-Taylor sign condition at the initial time.
• A Hadamard-type ill-posedness example is constructed for the frozen-coefficient linearized problem when both non-collinearity and Rayleigh-Taylor conditions fail, demonstrating Rayleigh-Taylor instability.
• The ill-posedness arises from the existence of exponentially growing modes in the Fourier analysis, specifically when the deformation gradient columns are collinear and the Rayleigh-Taylor condition is violated.
• The analysis confirms that the non-collinearity and Rayleigh-Taylor conditions are necessary in combination for stability, with neither being sufficient alone when the other fails.
• The results extend to the full nonlinear free boundary problem via a reduction to a fixed domain and a priori estimates, establishing well-posedness under the stated conditions.