A study on certain periodic Schrodinger equations

In the first part of this thesis we consider the cubic Schrodinger equation.;[special characters omitted].;T is the time of existence of the solutions and [special characters omitted] is the irrational torus given by [special characters omitted] and theta1/theta 2 irrational. Our main result is an improvement of the Strichartz estimates on irrational tori using a counting argument by Huxley [43], which estimates the number of lattice points on ellipsoids. With this Strichartz estimate, we obtain a local well-posedness result in Hs for s > [special characters omitted] . We also use energy type estimates to control the Hs norm of the solution and obtain improved growth bounds for higher order Sobolev norms.;In the second and the third parts of this thesis, we study the Cauchy problem for the 1d periodic fractional Schrodinger equation:;[special characters omitted].;where alpha ∈ (1/2; 1). First, we prove a Strichartz type estimate for this equation. Using the arguments from Chapter 3, this estimate implies local well-posedness in Hs for [special characters omitted] . However, we prove local wellposedness using direct Xs,b estimates. In addition, we show the existence of global-in-time infinite energy solutions. We also show that the nonlinear evolution of the equation is smoother than the initial data. As an important consequence of this smoothing estimate, we prove that there is global well-posedness in Hs for [special characters omitted] . Finally, for the fractional Schrodinger equation, we define an invariant probability measure on Hs for [special characters omitted], called a Gibbs measure. We define mu so that for any epsilon=epsilon > 0 there is a set Hs such that [special characters omitted] and the equation is globally well-posed for initial data in O. We achieve this by showing that for the initial data in O, the Hs norms of the solutions stay finite for all times. This fills the gap between the local well-posedness and the global well-posedness range in almost sure sense for [special characters omitted].