Linear stability of compressible vortex sheets in 2D elastodynamics: variable coefficients
Linear stability of compressible vortex sheets in 2D elastodynamics: variable coefficients
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DOI：
10.1007/s00208-018-01798-w
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发表时间：
2019-01
影响因子：
1.4
通讯作者：
R. Chen;Jilong Hu;Dehua Wang
中科院分区：
文献类型：
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作者：
R. Chen;Jilong Hu;Dehua Wang
研究了二维可压缩弹性流动涡片的变系数线性稳定性。与我们早期的工作一样(Chen等人)。在ADV Math 311：18-60,2007b)中，常系数线性稳定性问题有一个特征的自由边界，而且Kreiss-Lopatinskii条件也不一致满足。另外，拟线性化系统的Lopatinskii行列式的根可能与系统的极点重合。这种新的折叠现象在应用Coulombel(SIAM J Math Anal 34(1)：142-172,2002；Ann Inst H Poincaréanal non Linéaire 21(4)：401-443,2004)和Coulombel and Secchi(印第安纳大学数学J 53(4)：941-1012,2004)的双特征延拓方法时造成了严重的困难。受我们在常系数情形(ADV Math 311：18-60,2007b)中介绍的方法的启发，我们对拟线性化系统执行上三角化，以将出射模式分离为闭合形式，其中出射模式只出现在前导阶处。这一过程导致了出射模式的规律性增益，这允许我们克服极点特征分量的规律性损失，从而关闭所有能量估计。我们发现，与常系数情形类似，弹性能产生显著的稳定化效果，并且与等熵欧拉流相比，存在额外的稳定亚音速区域。此外，由于我们的方法不依赖于特征曲线的构造，因此它也可以应用于其他流体模型，如非等熵欧拉方程和MHD方程。
The linear stability with variable coefficients of the vortex sheets for the two-dimensional compressible elastic flows is studied. As in our earlier work (Chen et al. in Adv Math 311:18–60, 2017b) on the linear stability with constant coefficients, the problem has a free boundary which is characteristic, and also the Kreiss–Lopatinskii condition is not uniformly satisfied. In addition, the roots of the Lopatinskii determinant of the para-linearized system may coincide with the poles of the system. Such a new collapsing phenomenon causes serious difficulties when applying the bicharacteristic extension method of Coulombel (SIAM J Math Anal 34(1):142–172, 2002; Ann Inst H Poincaré Anal Non Linéaire 21(4):401–443, 2004) and Coulombel and Secchi (Indiana Univ Math J 53(4):941–1012, 2004). Motivated by our method introduced in the constant-coefficient case (Adv Math 311:18–60, 2017b), we perform an upper triangularization to the para-linearized system to separate the outgoing mode into a closed form where the outgoing mode only appears at the leading order. This procedure results in a gain of regularity for the outgoing mode, which allows us to overcome the loss of regularity of the characteristic components at the poles and hence to close all the energy estimates. We find that, analogous to the constant-coefficient case, elasticity generates notable stabilization effects, and there are additional stable subsonic regions compared with the isentropic Euler flows. Moreover, since our method does not rely on the construction of the characterisic curves, it can also be applied to other fluid models such as the non-isentropic Euler equations and the MHD equations.