论文标题
一阶段麝香问题的行进波解决方案：存在与稳定性
Traveling wave solutions to the one-phase Muskat problem: existence and stability
论文作者
论文摘要
我们研究了一个任意维度的一种流体的麝香问题，在下面由平坦床及以上的自由边界界定为图。除了固定的均匀重力场外，流体还由散装中的通用力场和自由边界的外部压力作用，两者都以波动波形为单位。我们证明，对于Sobolev空间中的足够小的力和压力数据，在Sobolev型空间中存在局部独特的行驶波解决方案。波动波解决方案的自由边界在空间无穷大处是周期性或渐近的平坦。此外，我们证明，仅由外部压力诱导的小型周期性行驶波解决方案是渐近稳定的。这些结果为问题提供了第一类非平凡稳定解决方案。
We study the Muskat problem for one fluid in arbitrary dimension, bounded below by a flat bed and above by a free boundary given as a graph. In addition to a fixed uniform gravitational field, the fluid is acted upon by a generic force field in the bulk and an external pressure on the free boundary, both of which are posited to be in traveling wave form. We prove that for sufficiently small force and pressure data in Sobolev spaces, there exists a locally unique traveling wave solution in Sobolev-type spaces. The free boundary of the traveling wave solutions is either periodic or asymptotically flat at spatial infinity. Moreover, we prove that small periodic traveling wave solutions induced by external pressure only are asymptotically stable. These results provide the first class of nontrivial stable solutions for the problem.
