Linear Inviscid Damping for Monotone Shear Flows, Boundary Effects and Sharp Sobolev Regularity

Abstract

In this thesis, we study the linear stability and long-time asymptotic behavior of solutions to the 2D incompressible Euler equations, which model the evolution of an inviscid, incompressible fluid. <br /> The Euler equations possess many conserved quantities, among them the kinetic energy, the enstrophy and entropy, and in particular exhibit neither dissipation nor entropy increase. As shown by Arnold, they even possess the structure of an infinite-dimensional Hamiltonian system on the Lie algebra of smooth volume-preserving diffeomorphisms. It was thus a very surprising observation of Kelvin and later Orr that small perturbations to Couette flow, i.e. the linear shear v(t,x,y)=(y,0), are damped back to a (possibly different) shear flow. As the linearized Euler equations around Couette flow can be solved explicitly, direct computations show that under the linear dynamics perturbations to the velocity field decay with algebraic rates. This phenomenon is commonly called linear inviscid damping in analogy to Landau damping in plasma physics. <br /> Going beyond these classic results, which due to the explicit solution are in a sense trivial, has, however, remained mostly open until recently. As the main result of this thesis, we prove that linear inviscid damping holds for a large class of regular monotone shear flows. There, we consider both the common setting of an infinite periodic channel as well as the physically relevant finite periodic channel with impermeable walls. In the latter setting, boundary effects are shown to qualitatively change the behavior of solutions and that, in general, asymptotically (logarithmic) singularities develop on the boundary. In particular, regularity results with respect to y are thus limited to the critical fractional Sobolev spaces. <br /> We further discuss the implications of our (in)stability results for the problem of nonlinear inviscid damping, where high regularity would be essential to control nonlinear effects. In particular, we show that the stability results for the finite periodic channel and the associated instability in supercritical Sobolev norms provide an upper bound on the Sobolev regularity that can be controlled in the nonlinear setting.