Randomness and Nonlinear Evolution Equations

In this paper we survey some results on existence, and when possible also uniqueness, of solutions to certain evolution equations obtained by injecting randomness either on the set of initial data or as a perturbative term.     

Sharp global well-posedness for KdV and modified KdV on

The initial value problems for the Korteweg-de Vries (KdV) and modified KdV (mKdV) equations under periodic and decaying boundary conditions are considered. These initial value problems are shown to be globally well-posed in all -based Sobolev spaces where local well-posedness is presently known, apart from the endpoint for mKdV and the endpoint for KdV. The result for KdV relies on a new method for constructing almost conserved quantities using multilinear harmonic analysis and the available local-in-time theory. Miura's transformation is used to show that global well-posedness of modified KdV is implied by global well-posedness of the standard KdV equation.

     

Bilinear estimates and applications to 2d NLS

The three bilinearities for functions are sharply estimated in function spaces associated to the Schrödinger operator . These bilinear estimates imply local wellposedness results for Schrödinger equations with quadratic nonlinearity. Improved bounds on the growth of spatial Sobolev norms of finite energy global-in-time and blow-up solutions of the cubic nonlinear Schrödinger equation (and certain generalizations) are also obtained.

     

Local well-posedness for higher order nonlinear dispersive systems

We study nonlinear dispersive systems of the form

Semilinear Schrödinger flows on hyperbolic spaces: scattering in <Emphasis Type="Italic">H</Emphasis><Superscript>1</Superscript>

We prove global well-posedness and scattering in H ¹ for the defocusing nonlinear Schrödinger equations

$\left\{\begin{array}{ll}(i\partial_t+\Delta_g)u=u|u|^{2\sigma};\\u(0)=\phi,\end{array}\right.$

on the hyperbolic spaces \({\mathbb{H}^d}\), d ≥ 2, for exponents \({\sigma \in (0, 2/(d-2))}\). The main unexpected conclusion is scattering to linear solutions in the case of small exponents σ; for comparison, on Euclidean spaces scattering in H ¹ is not known for any exponent \({\sigma \in (1/d, 2/d]}\) and is known to fail for \({\sigma \in (0, 1/d]}\). Our main ingredients are certain noneuclidean global in time Strichartz estimates and noneuclidean Morawetz inequalities.

     

Randomization and the Gross–Pitaevskii Hierarchy

Archive for Rational Mechanics and Analysis - We study the Gross–Pitaevskii hierarchy on the spatial domain $${\mathbb{T}^3}$$ . By using an appropriate randomization of the Fourier...     

On the uniqueness of solutions to the periodic 3D Gross–Pitaevskii hierarchy

In this paper, we present a uniqueness result for solutions to the Gross–Pitaevskii hierarchy on the three-dimensional torus, under the assumption of an a priori spacetime bound. We show that this a priori bound is satisfied for factorized solutions to the hierarchy which come from solutions of the nonlinear Schrödinger equation. In this way, we obtain a periodic analogue of the uniqueness result on R³${\mathbb{R}}^3$ previously proved by Klainerman and Machedon [75], except that, in the periodic setting, we need to assume additional regularity. In particular, we need to work in the Sobolev class H^α$H^{\alpha }$ for α>1$\alpha >1$. By constructing a specific counterexample, we show that, on T³${\mathbb{T}}^3$, the existing techniques from the work of Klainerman and Machedon approach do not apply in the endpoint case α=1$\alpha =1$. This is in contrast to the known results in the non-periodic setting, where these techniques are known to hold for all α?1$\alpha ?1$. In our analysis, we give a detailed study of the crucial spacetime estimate associated to the free evolution operator. In this step of the proof, our methods rely on lattice point counting techniques based on the concept of the determinant of a lattice. This method allows us to obtain improved bounds on the number of lattice points which lie in the intersection of a plane and a set of radius R, depending on the number-theoretic properties of the normal vector to the plane. We are hence able to obtain a sharp range of admissible Sobolev exponents for which the spacetime estimate holds.