论文标题
粘度跳跃的2D不可压缩的多孔介质方程的H原理
H-principle for the 2D incompressible porous media equation with viscosity jump
论文作者
论文摘要
在这项工作中，我们将结果扩展到2D IPM系统的[6,32]中，具有持续的粘度（Atwood Number $a_μ= 0 $），以粘度跳跃（$ |a_μ| <1 $）。我们证明了H-Principle，该H-Principle（无限的许多）弱解决方案在$ C_TL_ {W^*}^{\ infty} $中，每当提供一个subsolution时，都会通过凸集成恢复。作为第一个例子，获得了随着时间的推移提供紧凑支持的非平凡弱解。其次，我们通过初始平面界面构建了混合解决方案，以解决不稳定的Muskat问题。作为副产品，我们检查了SzékelyHidi以$a_μ= 0 $建立的连接，在订阅和lagrangian宽松解决方案之间，也以$ |a_μ| <1 $持有。对于不同的粘度，我们显示弛豫中的捏合奇异性如何阻止两种流体在界面上既没有雷利 - 泰勒和涡度的地方混合。
In this work we extend the results in [6,32] on the 2D IPM system with constant viscosity (Atwood number $A_μ=0$) to the case of viscosity jump ($|A_μ|<1$). We prove a h-principle whereby (infinitely many) weak solutions in $C_tL_{w^*}^{\infty}$ are recovered via convex integration whenever a subsolution is provided. As a first example, non-trivial weak solutions with compact support in time are obtained. Secondly, we construct mixing solutions to the unstable Muskat problem with initial flat interface. As a byproduct, we check that the connection, established by Székelyhidi for $A_μ=0$, between the subsolution and the Lagrangian relaxed solution of Otto, holds for $|A_μ|<1$ too. For different viscosities, we show how a pinch singularity in the relaxation prevents the two fluids from mixing wherever there is neither Rayleigh-Taylor nor vorticity at the interface.
