**Le****cturer****：Zeng Chongchun** GIT

**Time：June 6 ****2016 ****15:00**

**Place：6th classroom**

Abstract：In this talk, we start with the mathematical theory of wind-generated water waves in the framework of the interface problem between two incompressible inviscid fluids under the influence of gravity. This entails the careful study of the stability of the shear flow solutions to the interface problem of the two-phase Euler equation. Based on a rigorous derivation of the linearized equations about shear flow solutions, we obtained rigorously the linear instability criterion of Miles due to the presence of the critical layer in the steady shear flows. Our analysis is valid even in the presence of surface tension and a vortex sheet (discontinuity in the tangential velocity across the air--sea interface). We are thus able to give a unified equation including the Kelvin--Helmholtz and quasi-laminar models of wave generation put forward by Miles. While the rigorous nonlinear instability proof is still missing for this problem, we are aiming at a stronger statement -- constructing local unstable manifolds of the full nonlinear system of the interface problem of the Euler equation.If time permits, in the second half of the talk, we discuss the unstable manifolds of steady states of the Euler equation on fixed domains. Suppose the linearized equation at a steady state $v_*$ has an exponential dichotomy with a finite dimensional unstable subspace.By rewriting the Euler equation as an ODE on an infinite dimensional manifold in $H^k$, $k>\frac n2 +1$, the unstable manifold of $v_*$ is constructed under certain conditions on the Lyapunov exponents of the vector field $v_*$. In particular, this leads to the desired nonlinear instability of $v_*$ in the sense that small $H^k$ perturbations can lead to $L^2$ derivation of the solutions.