The nonlinear Schrödinger equation (NLS) on is a prototypical dispersive equation, i.e. it is characterized by different frequencies travelling at different velocities and by the lack of a smoothing effect over time. Furthermore, NLS is a prototypical infinite-dimensional Hamiltonian system. Constructing an invariant measure for the NLS flow is a natural, albeit very difficult problem. It requires showing local well-posedness in low regularity spaces, in an appropriate probabilistic sense. We prove the probabilistic local well-posedness of \[ (i\partial_{t}+\Delta)u=\pm |u|^{2}u \text{ on } [0,T)\times \mathbb{R}^{d}, \] with initial data being a unit-scale Wiener randomization of a given function \(f\in H^{S}_{x}(\mathbb{R}^{d})\). When \(d=3\) we obtain the full range \(S>0\). The solutions are constructed as a sum of an explicit multilinear expansion of the flow in terms of the random initial data and of an additional smoother remainder term with deterministically subcritical regularity. We show how directional behavior of solutions can be used to control the (probabilistic) multilinear approximations of the solution and the remainder term. We obtain improved bilinear probabilistic-deterministic Strichartz estimates, and we shed light on NLS in dimensions \(d>3\), and potentially with other power nonlinearities.

In this talk I will show that every solution of a KdV-Burgers-Sivashinsky type equation blows up in the energy space, backward in time, provided the solution does not belong to the global attractor. This is a phenomenon contrast to the backward behavior of the 2D Navier-Stokes equations, subject to periodic boundary condition, studied by Constantin, Foias, Kukavica and Majda, but analogous to the backward behavior of the Kuramoto-Sivashinsky equation discovered by Kukavica and Malcok. I will also discuss the backward behavior of solutions to the damped driven nonlinear Schroedinger equation, the complex Ginzburg-Landau equation, and the hyperviscous Navier-Stokes equations. In addition, I will provide some physical interpretation of various backward behaviors of several perturbations of the KdV equation by studying explicit cnoidal wave solutions. Furthermore, I will discuss the connection between the backward behavior and the energy spectra of the solutions. The study of backward behavior of dissipative evolution equations is motivated by a conjecture of Bardos and Tartar which states that the solution operator of the two-dimensional Navier-Stokes equations maps the phase space into a dense subset in this space. This is a joint work with Yanqui Guo.

TBA