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Stability of Multidimensional Thermoelastic Contact Discontinuities
Archive for Rational Mechanics and Analysis ( IF 2.6 ) Pub Date : 2020-05-22 , DOI: 10.1007/s00205-020-01531-5
Gui-Qiang G. Chen , Paolo Secchi , Tao Wang

We study the system of nonisentropic thermoelasticity describing the motion of thermoelastic nonconductors of heat in two and three spatial dimensions, where the frame-indifferent constitutive relation generalizes that for compressible neo-Hookean materials. Thermoelastic contact discontinuities are characteristic discontinuities for which the velocity is continuous across the discontinuity interface. Mathematically, this renders a nonlinear multidimensional hyperbolic problem with a characteristic free boundary. We identify a stability condition on the piecewise constant background states and establish the linear stability of thermoelastic contact discontinuities in the sense that the variable coefficient linearized problem satisfies a priori tame estimates in the usual Sobolev spaces under small perturbations. Our tame estimates for the linearized problem do not break down when the strength of thermoelastic contact discontinuities tends to zero. The missing normal derivatives are recovered from the estimates of several quantities relating to physical involutions. In the estimate of tangential derivatives, there is a significant new difficulty, namely the presence of characteristic variables in the boundary conditions. To overcome this difficulty, we explore an intrinsic cancellation effect, which reduces the boundary terms to an instant integral. Then we can absorb the instant integral into the instant tangential energy by means of the interpolation argument and an explicit estimate for the traces on the hyperplane.

中文翻译:

多维热弹性接触不连续性的稳定性

我们研究了非等熵热弹性系统,该系统描述了热弹性非导体在两个和三个空间维度上的运动,其中与框架无关的本构关系概括了可压缩新胡克材料的本构关系。热弹性接触不连续性是特征不连续性,其速度在不连续性界面上是连续的。在数学上,这呈现了一个具有特征自由边界的非线性多维双曲线问题。我们确定了分段恒定背景状态的稳定性条件,并建立了热弹性接触不连续性的线性稳定性,因为可变系数线性化问题在小扰动下满足通常的 Sobolev 空间中的先验估计。当热弹性接触不连续性的强度趋于零时,我们对线性化问题的温和估计不会失效。从与物理对合相关的几个数量的估计中恢复丢失的正态导数。在切向导数的估计中,存在一个重要的新困难,即边界条件中存在特征变量。为了克服这个困难,我们探索了一种内在的抵消效应,它将边界项减少到一个即时积分。然后我们可以通过插值参数和对超平面上的迹线的显式估计将瞬时积分吸收到瞬时切向能量中。从与物理对合相关的几个数量的估计中恢复丢失的正态导数。在切向导数的估计中,存在一个重要的新困难,即边界条件中存在特征变量。为了克服这个困难,我们探索了一种内在的抵消效应,它将边界项减少到一个即时积分。然后我们可以通过插值参数和对超平面上的迹线的显式估计将瞬时积分吸收到瞬时切向能量中。从与物理对合相关的几个数量的估计中恢复丢失的正态导数。在切向导数的估计中,存在一个重要的新困难,即边界条件中存在特征变量。为了克服这个困难,我们探索了一种内在的抵消效应,它将边界项减少到一个即时积分。然后我们可以通过插值参数和对超平面上的迹线的显式估计将瞬时积分吸收到瞬时切向能量中。我们探索了一种内在的抵消效应,它将边界项减少到一个即时积分。然后我们可以通过插值参数和对超平面上的迹线的显式估计将瞬时积分吸收到瞬时切向能量中。我们探索了一种内在的抵消效应,它将边界项减少到一个即时积分。然后我们可以通过插值参数和对超平面上的迹线的显式估计将瞬时积分吸收到瞬时切向能量中。
更新日期:2020-05-22
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