[论文解读] On the Motion of a Self-Gravitating Incompressible Fluid with Free Boundary and Constant Vorticity: An Appendix

Lydia Bieri, Shuang Miao+2 more|arXiv (Cornell University)|Nov 23, 2015

Advanced Mathematical Physics Problems参考文献 1被引用 3

一句话总结

本文将先前对具有自由边界自引力不可压缩流体的分析扩展至包含恒定涡度的情形，证明了对平衡态的 $ \epsilon $-阶扰动，解的存在时间下界仍为 $ T \gtrsim \epsilon^{-2} $。作者将无旋情形下的非线性坐标变换与能量法加以改进，表明恒定涡度仅引入有界线性项，可通过进一步的线性变换消除，从而在变换后的系统中保持无二次非线性项。

ABSTRACT

In a recent work [1] the authors studied the dynamics of the interface separating a vacuum from an inviscid incompressible fluid, subject to the self-gravitational force and neglecting surface tension, in two space dimensions. The fluid is additionally assumed to be irrotational, and we proved that for data which are size $ε$ perturbations of an equilibrium state, the lifespan $T$ of solutions satisfies $T \gtrsim ε^{-2}$. The key to the proof is to find a nonlinear transformation of the unknown function and a coordinate change, such that the equation for the new unknown in the new coordinate system has no quadratic nonlinear terms. For the related irrotational gravity water wave equation with constant gravity the analogous transformation was carried out by the last author in [3]. While our approach is inspired by the last author's work [3], the self-gravity in the present problem is a new nonlinearity which needs separate investigation. Upon completing [1] we learned of the work of Ifrim and Tataru [2] where the gravity water wave equation with constant gravity and constant vorticity is studied and a similar estimate on the lifespan of the solution is obtained. In this short note we demonstrate that our transformations in [1] can be easily modified to allow for nonzero constant vorticity, and a similar energy method as in [1] gives an estimate $T\gtrsimε^{-2}$ for the lifespan $T$ of solutions with data which are size $ε$ perturbations of the equilibrium. In particular, the effect of the constant vorticity is an extra linear term with constant coefficient in the transformed equation, which can be further transformed away by a bounded linear transformation. This note serves as an appendix to the aforementioned work of the authors.

研究动机与目标

• 将自引力不可压缩流体在自由边界条件下的寿命估计 $ T \gtrsim \epsilon^{-2} $ 扩展至非零恒定涡度的情形。
• 证明先前工作[1]中发展的非线性变换与能量法可适用于恒定涡度情形，通过表明涡度项仅引入有界线性扰动。
• 证明变换后的系统仍不包含二次非线性项，从而确保与无旋情形相同的能量估计与寿命下界成立。
• 在假设 $ \omega_0^2 < \pi $ 的条件下，证明泰勒符号条件依然成立，该条件是局部适定性的必要条件。

提出的方法

• 将无旋情形下的非线性变换与坐标变换方法推广，以在速度场中引入恒定涡度项 $ 2\omega_0 $。
• 将速度分解为 $ \mathbf{v} = \mathbf{v}_0 + \mathfrak{v} $，其中 $ \mathbf{v}_0 = \omega_0(y, -x) $，以实现涡度贡献的解耦。
• 采用复变量形式，利用希尔伯特变换 $ H $，证明在边界上 $ (I - \overline{H})(z_t + i\omega_0 z) = 0 $。
• 通过引入依赖于时间的变换 $ \tilde{\delta} = e^{-i\omega_0 t} \delta $，定义新未知量 $ \tilde{\delta} $，从而从方程中消除振荡相位。
• 推导出形式为 $ (\partial_t^2 + ia\partial_\alpha - (\pi - \omega_0^2))\tilde{\delta} = \widetilde{\mathcal{M}}_1 $ 的变换系统，且系统中无二次非线性项。
• 验证能量估计中所有非线性项均为三次或更高阶，从而确保相同的 $ \epsilon^{-2} $ 寿命下界成立。

实验结果

研究问题

• RQ1 能否将无旋情形下使用的非线性变换与能量法推广至包含恒定涡度的自引力不可压缩流体自由边界问题？
• RQ2 恒定涡度的引入是否会带来二次非线性项，从而破坏 $ \epsilon^{-2} $ 的寿命估计？
• RQ3 涡度项如何影响边界方程的结构以及希尔伯特变换框架的适用性？
• RQ4 当 $ \omega_0^2 < \pi $ 时，泰勒符号条件是否仍然成立，从而保证局部适定性？
• RQ5 能否通过有界线性变换消除线性涡度项，而不会引入新的非线性项？

主要发现

• 对于自引力不可压缩流体在自由边界与恒定涡度条件下的解，其存在时间下界满足 $ T \gtrsim \epsilon^{-2} $，其中 $ \epsilon $ 为平衡态的扰动大小。
• 恒定涡度项在变换后的方程中仅贡献有界线性项，可通过有界线性变换消除。
• 变换后系统的非线性结构仍保持无二次非线性项，从而保留了无旋情形下的能量法框架。
• 变换后系统的能量估计形式与无旋情形完全相同，确保了相同的定量寿命下界。
• 能量估计中所有非线性贡献均为三次或更高阶，证实了在引入恒定涡度后 $ \epsilon^{-2} $ 寿命下界依然稳健。
• 假设 $ \omega_0^2 < \pi $ 足够保证泰勒符号条件成立，而该条件是局部适定性的必要条件。