Global Existence for 3d Incompressible Isotropic Elastodynamics
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چکیده
The behavior of elastic waves in a three-dimensional isotropic incompressible material is studied. Unlike compressible elastodynamics, where there are nonlinear interactions of shear and pressure waves, with incompressible elastodynamics the only waves present are shear waves. In an isotropic system, shear waves are linearly degenerate, and therefore global solutions to the perturbative incompressible equations can be expected via the generalized energy method. This article confirms this intuitive idea. Strong dispersive estimates are needed for long time stability of the solutions for incompressible elastodynamics. The generalized energy method based on the Lorentz invariance of the wave equation combines energy and decay estimates which together with a null condition provide global existence results (see [6],[7]). However, the equations of motion for elasticity are only Galilean invariant and the Lorentz rotations cannot be used. As Klainerman and Sideris observed in [8], however, Lorentz invariance is not necessary to obtain almost global existence in 3D for isotropic systems such as the equations of elasticity. Global small solutions to the equations for compressible elasticity were obtained in [12],[13] with the addition of a null condition for pressure waves, see also [1],[2]. With the observation that the null condition is inherently satisfied by shear waves the authors were able to show global existence for incompressible elasticity for small data as a limit of slightly compressible materials in [14]. The key step was that the pressure waves vanish in the limit and the shear waves are already null. For this work it was convenient to consider the equations of elasticity as a first order system in Eulerian coordinates. In this frame the singular term is linear. We find this same set of variables works well for the incompressible equations as the constraints are most naturally posed in that frame. The system can be viewed as an extension of the incompressible Euler equations where the inverse deformation gradient is coupled with the velocity and pressure. It is also shares common features with viscoelastic theories, in particular for the Oldroyd-B system for viscoelastic materials [3],[9],[10],[11]. The argument requires the use of weighted local decay estimates for the linearized incompressible equations. We obtain these as a special case of a new general result for certain isotropic symmetric hyperbolic systems presented in a separate paper [15]. This article contains further examples, including compressible elasticity. One must also handle the pressure which arises as a Lagrange multiplier which enforces the incompressibility constraint. Although it involves nonlocal operators,
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تاریخ انتشار 2005