Gebhard Björn, Kolumbán József J, Székelyhidi László
Mathematisches Institut, Universität Leipzig, Augustusplatz 10, 04109 Leipzig, Germany.
Arch Ration Mech Anal. 2021;241(3):1243-1280. doi: 10.1007/s00205-021-01672-1. Epub 2021 Jun 12.
In this article we consider the inhomogeneous incompressible Euler equations describing two fluids with different constant densities under the influence of gravity as a differential inclusion. By considering the relaxation of the constitutive laws we formulate a general criterion for the existence of infinitely many weak solutions which reflect the turbulent mixing of the two fluids. Our criterion can be verified in the case that initially the fluids are at rest and separated by a flat interface with the heavier one being above the lighter one-the classical configuration giving rise to the Rayleigh-Taylor instability. We construct specific examples when the Atwood number is in the ultra high range, for which the zone in which the mixing occurs grows quadratically in time.
在本文中，我们将描述在重力影响下具有不同恒定密度的两种流体的非齐次不可压缩欧拉方程视为一个微分包含。通过考虑本构定律的松弛，我们制定了一个关于存在无限多个弱解的一般准则，这些弱解反映了两种流体的湍流混合。在初始时流体静止且由一个平坦界面分隔，较重的流体在较轻的流体上方（这种经典构型会引发瑞利 - 泰勒不稳定性）的情况下，我们的准则可以得到验证。当阿特伍德数处于超高范围时，我们构造了具体的例子，对于这些例子，发生混合的区域随时间呈二次方增长。
