本文研究二维不可压缩各向同性弹性动力学系统最高阶能量的时间增长。Lei [31]首先得到了平衡点附近光滑解的整体适定性，其中最高阶广义能量可以随时间有一定的增长。我们改进了上述结果，证明了最高阶广义能量始终一致有界。二维不可压弹性动力学是一个非局部拟线性波动方程组，其中未知量按$\langle t\rangle^{-\frac12}$衰减。这表明这个问题是超临界的，因为衰变率远非可积的。令人惊讶的是，我们发现，在最高阶的能量估计，时间衰减可以强烈增强到亚临界$\langle t \rangle^{-\frac54}$。分析基于Alinhac [5]的鬼权能量方法、Lei [31]的固有强零结构和系统的固有div-curl结构。
英文摘要：
This paper concerns the time growth of the highest-order energy of the systems of incompressible isotropic elastodynamics in two space dimensions. The global well-posedness of smooth solutions near equilibrium was first obtained by Lei [31] where the highest-order generalized energy may have certain growth in time. We improve above result and show that the highest-order generalized energy is uniformly bounded for all the time. The two dimensional incompressible elastodynamics is a system of nonlocal quasilinear wave equations where the unknowns decay as $\langle t\rangle^{-\frac12}$. This suggests the problem is supercritical in the sense that the decay rate is far from integrable. Surprisingly, we showed that in the highest-order energy estimate, the temporal decay can be strongly enhanced to be subcritical $\langle t \rangle^{-\frac54}$. The analysis is based on the ghost weight energy method by Alinhac [5], the inherent strong null structure by Lei [31] and the inherent div-curl structure of the system.
Finite-Time Splash in Free Boundary Problem of 3D Neo-Hookean Elastodynamics
Hefei University（合肥大学）；Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院）；University of Chinese Academy of Sciences（中国科学院大学）；Xiamen University（厦门大学）
