This paper considers the existence and uniqueness of the solution to the initial boundary value problem for a class of generalized Zakharov equations in $(2+1)$ dimensions, and proves the global existence of the solution to the problem by a priori integral estimates and the Galerkin method.

We prove the existence and the invariance of a Gibbs measure associated to the defocusing sub-quintic Nonlinear Schrödinger equations on the disc of the plane ${ℝ}^2$. We also prove an estimate giving some intuition to what may happen in $3$ dimensions.

We construct an invariant weighted Wiener measure associated to the periodic derivative nonlinear Schrödinger equation in one dimension and establish global well-posedness for data living in its support. In particular almost surely for data in a Fourier–Lebesgue space $ℱL^{s,r}\left(T\right)$ with $s\ge \frac{1}{2},2<r<4,(s-1)r<-1$ and scaling like $H^{\frac{1}{2}-ϵ}\left(𝕋\right)$, for small $ϵ>0$. We also show the invariance of this measure.