论文标题
2D不可压缩弹性动力学的最高级能量的统一结合
Uniform Bound of the Highest-order Energy of the 2D Incompressible Elastodynamics
论文作者
论文摘要
本文涉及两个空间维度中不可压缩的各向同性弹性动力学系统的最高阶段的时间增长。 Lei [31]首先获得了平衡近均衡的平滑溶液的全球良好性，其中最高阶的广义能量可能会在时间上具有一定的增长。我们高于结果，并表明最高阶的广义能量一直均匀地界定。二维不可压缩的弹性动力学是一个非局部准线性波方程的系统，其中未知数为$ \ langle t \ rangle^{ - \ frac12} $。这表明问题是超临界的，因为衰减速率远非可以整合。令人惊讶的是，我们表明，在最高阶的能量估计中，可以强烈增强时间衰减，以成为亚临界$ \ langle t \ rangle^{ - \ frac54} $。该分析基于Alinhac [5]的幽灵重量能法，Lei [31]的固有的强零结构以及系统的固有DIV-Curl结构。
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.
