论文标题
重力stokes流动的长时间接口动力学
Long time interface dynamics for gravity Stokes flow
论文作者
论文摘要
我们研究了两种不可压缩的粘性液体在Stokes状态中填充2D水平周期性条的界面的动力学。流体受重力的影响，密度差会诱导界面的动力学。我们通过Stokeslet的$ X_1 $周期版本来得出此问题的轮廓动力学公式。使用这个新系统，当没有自行体的曲线和$ c^{1+γ} $hölder规律性，$ 0 <γ<1 $时，我们显示了本地时间供应良好性。无论物理系统的瑞利 - 泰勒稳定性如何，这种良好的结果都会成立。此外，在雷利 - 泰勒稳定制度中证明了全球及时的存在和衰减对平坦的固定状态，以用于小型初始数据。最后，在雷利 - 泰勒不稳定的政权中，我们建立了一个广泛的平滑解决方案家族，其时间增长为任意的大间隔。值得注意的是，导致这种指数增长的初始数据可能缺乏任何对称性。
We study the dynamics of the interface given by two incompressible viscous fluids in the Stokes regime filling a 2D horizontally periodic strip. The fluids are subject to the gravity force and the density difference induces the dynamics of the interface. We derive the contour dynamics formulation for this problem through a $x_1$-periodic version of the Stokeslet. Using this new system, we show local-in-time well-posedness when the initial interface is described by a curve with no self-intersections and $C^{1+γ}$ Hölder regularity, $0<γ<1$. This well-posedness result holds regardless of the Rayleigh-Taylor stability of the physical system. In addition, global-in-time existence and decay to the flat stationary state is proved in the Rayleigh-Taylor stable regime for small initial data. Finally, in the Rayleigh-Taylor unstable regime, we construct a wide family of smooth solutions with exponential in time growth for an arbitrary large interval of existence. Remarkably, the initial data leading to this exponential growth possibly lack any symmetry.
