Low Regularity Solutions of Korteweg-de Vries and Chern-Simons-Schrodinger Equations

The aim of this thesis is to understand the locall wellposedness theory for some nonlinear dispersive equations at low regularity.;The Korteweg-de Vries equation has sharp wellposedness at H-34 if we are concerned about the Lipschitz dependence of solutions on the initial data. For lower regularity, one might still have a weaker form of wellposedness only with continuous dependence on data. Here we prove that the smooth solutions satisfy a-priori local in time Hs bound in terms of the Hs size of the initial data for s ≥ --⅘. Together with the bounds we obtained on the nonlinearity, the result here ensures that the equation is satisfied in the sense of distributions even for weak limits.;The Chern-Simons-Schrodinger equation is a planar gauged Schrodinger equation which has some similarity to the derivative formulation of the Schrodinger map problem. We work on to prove local wellposedness in the full subcritical range Hs ( R2 ), s > 0.;One important idea in working on these problems is to find a suitable space to characterize the solution. We use Xs,b spaces introduced by Bourgain, and U 2, V2 spaces introduced by Koch and Tataru. For the Chern-Simons-Schrodinger equation, we also need to fix a suitable gauge to make the problem well-posed. The heat gauge is a variation of Coulomb gauge, and it serves as a good candidate for this problem.