Lecture Notes Ii: on Local and Global Theory for the Kdv Equation

The second part of the notes are written jointly with my collaborator from University of Illinois, M. B. Erdogan. We developed the material with two goals in mind. First to prove existence and uniqueness results in the case of dispersive PDE evolving from initial data that are periodic in the space variable. Secondly we develop new tools to address the problem of wellposedness of solutions, in the case that the nonlinearity is not a polynomial and may for example contain derivatives (of order less that the order of the linear PDE). An example of such an equation is the Korteweg de Vries (KdV) equation. 1. The Korteweg de Vries equation. In this section we prove existence and uniqueness of smooth solutions for the KdV equation. We follow the proof in [6]. Consider the initial value problem: ut + uxxx + uux = 0 with initial data u(0, x) = u0(x) ∈ H(R) with s being a sufficiently large integer. In this section, we say u is a classical solution of KdV in H if u ∈ C([−δ, δ];H) ∩ C([−δ, δ];Hs−3) and if u satisfies KdV for each x and t. We start with the following energy inequality: if u is a smooth solution of KdV, then there exists T0 = T0(‖u0‖Hs) such that on [0, T0], ‖u‖Hs ≤ 2‖u0‖Hs . Indeed, ∂t‖∂ xu‖L2 = 2 ∫ ∂ xut∂ s xudx = −2 ∫ ∂ x u∂ s xudx− 2 ∫ ∂ x(uux)∂ s xudx. The first term is zero, the highest order contribution of the second term is −2 ∫ u∂ x u∂ s xudx = ∫ ux(∂ s xu) dx. Thus, we obtain for s > 3/2 ∂t‖u‖Hs . ‖ux‖L∞‖u‖Hs . ‖u‖Hs . Integrating in time implies that ‖u(T )‖Hs ≤ ‖u0‖Hs + ∫ T