The Well-posedness Of The Stochastic Nonlinear Schr(?)dinger Equations

In this paper,we discuss two types of stochastic nonlinear Schr(?)dinger equations with random terms.First of all,we consider L2-critical stochastic nonlinear Schr(?)dinger equation with white noise dispersion in one-dimensional torus:(?)The main conclusion is the global well-posedness of the equation in space H1(T).We get Strichartz type estimate of the equation,and then get the result by using nonlinear analysis,compression mapping principle and other methods.Secondly,we study the derivative nonlinear Schr(?)dinger equation with additive random nonlinear term in the torus:(?)The main conclusion is the local well-posedness in space H2(T).We first employ Yosida-type regularization and construct approximate solutions.Next,we use the tools such as Gagliaro-Nirenberg inequality,Burkholder-Davis-Gundy inequality and martingale properties of random integrals in order to obtain H2 estimate of the solution.Because of the existence of the random term,the charge M(u(t))and the energy E(u(t))are no longer conserved,which is the main difficulty for us.