No abstract

In this paper, we study the generation of eigenvalues of a stable monotonic shear flow under perturbations in C s with s < 2. More precisely, we study the Rayleigh operator LU m,γ = Um,γ ∂x − U ′′ m,γ ∂x∆ −1 associated with perturbed shear flow (Um,γ (y), 0) in a finite channel T2π × [−1, 1] where Um,γ (y) = U (y) + mγ 2 Γ(y/γ) with U (y) being a stable monotonic shear flow and {mγ 2 Γ(y/γ)} m≥0 being a family of perturbations parameterized by m. We prove that there exists m * such that for 0 ≤ m < m * , the Rayleigh operator has no eigenvalue or embedded eigenvalue, therefore linear inviscid damping holds. Otherwise, instability occurs when m ≥ m * . Moreover, at the nonlinear level, we show that asymptotic instability holds for m near m * and a growing mode exists for m > m * which also leads to instability.

We present a large-amplitude existence theory for two-dimensional solitary waves propagating through a two layer body of water. The domain of the fluid is bounded below by an impermeable flat ocean floor and above by a free boundary at constant pressure. For any piecewise smooth upstream density distribution and laminar background current, we construct a global curve of solutions. This curve bifurcates from the background current and, following along the curve, we find waves that are arbitrarily close to having horizontal stagnation points.The small-amplitude waves are constructed using a center manifold reduction technique. The large-amplitude theory is obtained through analytical global bifurcation together with refined qualitative properties of the waves.

The present work concerns two mathematical problems on wave in a stratified body of water governed by the incompressible Euler equations. In the first part, we present a large-amplitude existence theory for two-dimensional solitary waves by means of a global bifurcation theoretic approach. That is for any piece-wise smooth upstream density distribution and laminar background current, we construct a global curve of solutions. This curve bifurcates from the background current and, following along the curve, we find waves that are arbitrarily close to having horizontal stagnation points. The second part of the work tackles the problem on orbital stability of solitary waves under the presence of surface tension and constant vorticities. Using a spatial dynamics technique, we established the existence of such waves. Following that, we proved an orbital stability result. It is accomplished via a variant of Grillakis-Shatah-Strauss method.