当前位置: X-MOL 学术J. Math. Fluid Mech. › 论文详情
Our official English website, www.x-mol.net, welcomes your feedback! (Note: you will need to create a separate account there.)
Well-Posedness and Stability Results for Some Periodic Muskat Problems
Journal of Mathematical Fluid Mechanics ( IF 1.3 ) Pub Date : 2020-06-06 , DOI: 10.1007/s00021-020-00494-7
Bogdan-Vasile Matioc

We study the two-dimensional Muskat problem in a horizontally periodic setting and for fluids with arbitrary densities and viscosities. We show that in the presence of surface tension effects the Muskat problem is a quasilinear parabolic problem which is well-posed in the Sobolev space \(H^r({\mathbb {S}})\) for each \(r\in (2,3)\). When neglecting surface tension effects, the Muskat problem is a fully nonlinear evolution equation and of parabolic type in the regime where the Rayleigh–Taylor condition is satisfied. We then establish the well-posedness of the Muskat problem in the open subset of \(H^2({\mathbb {S}})\) defined by the Rayleigh–Taylor condition. Besides, we identify all equilibrium solutions and study the stability properties of trivial and of small finger-shaped equilibria. Also other qualitative properties of solutions such as parabolic smoothing, blow-up behavior, and criteria for global existence are outlined.

中文翻译:

一些周期性Muskat问题的适定性和稳定性结果

我们研究了水平周期周期以及具有任意密度和粘度的流体的二维Muskat问题。我们表明,在存在表面张力效应的情况下,Muskat问题是一个拟线性抛物线问题,它在Sobolev空间\(H ^ r({\ mathbb {S}})\)中对于每个\(r \ in (2,3)\)。当忽略表面张力效应时,Muskat问题是一个完全非线性的演化方程,在满足瑞利-泰勒条件的状态下是抛物线型的。然后,我们在\(H ^ 2({\ mathbb {S}})\)的开放子集中建立Muskat问题的适定性由瑞利-泰勒条件定义。此外,我们确定了所有平衡解,并研究了琐碎的和小指状平衡的稳定性。还概述了解决方案的其他定性性质,例如抛物线平滑,爆炸行为和整体存在准则。
更新日期:2020-06-06
down
wechat
bug