Linear Inviscid Damping for Monotone Shear Flows in a Finite Periodic Channel, Boundary Effects, Blow-up and Critical Sobolev Regularity

In a previous article (Zillinger, Linear inviscid damping for monotone shear flows, 2014), we have established linear inviscid damping for a large class of monotone shear flows in a finite periodic channel and have further shown that boundary effects asymptotically lead to the formation of singularities of derivatives of the solution as . As the main results of this article, we provide a detailed description of the singularity formation and establish stability in all sub-critical fractional Sobolev spaces and blow-up in all super-critical spaces. Furthermore, we discuss the implications of the blow-up to the problem of nonlinear inviscid damping in a finite periodic channel, where high regularity would be essential to control nonlinear effects.