Kirchhoff equations in generalized Gevrey spaces: local existence, global existence, uniqueness

In this note we present some recent results for Kirchhoff equations in generalized Gevrey spaces. We show that these spaces are the natural framework where classical results can be unified and extended. In particular we focus on existence and uniqueness results for initial data whose regularity depends on the continuity modulus of the nonlinear term, both in the strictly hyperbolic case, and in the degenerate hyperbolic case.Comment: 20 pages, 4 tables, conference paper (7th ISAAC congress, London 2009

Randomization and the Gross-Pitaevskii hierarchy

We study the Gross-Pitaevskii hierarchy on the spatial domain $\mathbb{T}^3$. By using an appropriate randomization of the Fourier coefficients in the collision operator, we prove an averaged form of the main estimate which is used in order to contract the Duhamel terms that occur in the study of the hierarchy. In the averaged estimate, we do not need to integrate in the time variable. An averaged spacetime estimate for this range of regularity exponents then follows as a direct corollary. The range of regularity exponents that we obtain is $\alpha>\frac{3}{4}$. It was shown in our previous joint work with Gressman that the range $\alpha>1$ is sharp in the corresponding deterministic spacetime estimate. This is in contrast to the non-periodic setting, which was studied by Klainerman and Machedon, in which the spacetime estimate is known to hold whenever $\alpha \geq 1$. The goal of our paper is to extend the range of $\alpha$ in this class of estimates in a \emph{probabilistic sense}. We use the new estimate and the ideas from its proof in order to study randomized forms of the Gross-Pitaevskii hierarchy. More precisely, we consider hierarchies similar to the Gross-Pitaevskii hierarchy, but in which the collision operator has been randomized. For these hierarchies, we show convergence to zero in low regularity Sobolev spaces of Duhamel expansions of fixed deterministic density matrices. We believe that the study of the randomized collision operators could be the first step in the understanding of a nonlinear form of randomization.Comment: 51 pages. Revised versio

On the one-dimensional cubic nonlinear Schrodinger equation below L^2

In this paper, we review several recent results concerning well-posedness of the one-dimensional, cubic Nonlinear Schrodinger equation (NLS) on the real line R and on the circle T for solutions below the L^2-threshold. We point out common results for NLS on R and the so-called "Wick ordered NLS" (WNLS) on T, suggesting that WNLS may be an appropriate model for the study of solutions below L^2(T). In particular, in contrast with a recent result of Molinet who proved that the solution map for the periodic cubic NLS equation is not weakly continuous from L^2(T) to the space of distributions, we show that this is not the case for WNLS.Comment: 14 pages, additional reference

Degenerate dispersive equations arising in the study of magma dynamics

An outstanding problem in Earth science is understanding the method of transport of magma in the Earth's mantle. Models for this process, transport in a viscously deformable porous media, give rise to scalar degenerate, dispersive, nonlinear wave equations. We establish a general local well-posedness for a physical class of data (roughly $H^1$) via fixed point methods. The strategy requires positive lower bounds on the solution. This is extended to global existence for a subset of possible nonlinearities by making use of certain conservation laws associated with the equations. Furthermore, we construct a Lyapunov energy functional, which is locally convex about the uniform state, and prove (global in time) nonlinear dynamic stability of the uniform state for any choice of nonlinearity. We compare the dynamics to that of other problems and discuss open questions concerning a larger range of nonlinearities, for which we conjecture global existence.Comment: 27 Pages, 7 figures are not present in this version. See http://www.columbia.edu/~grs2103/ for a PDF with figures. Submitted to Nonlinearit

Unconditional Uniqueness of the cubic Gross-Pitaevskii Hierarchy with Low Regularity

In this paper, we establish the unconditional uniqueness of solutions to the cubic Gross-Pitaevskii hierarchy on $\mathbb{R}^d$ in a low regularity Sobolev type space. More precisely, we reduce the regularity $s$ down to the currently known regularity requirement for unconditional uniqueness of solutions to the cubic nonlinear Schr\"odinger equation ($s\ge\frac{d}{6}$ if $d=1,2$ and $s>s_c=\frac{d-2}{2}$ if $d\ge 3$). In such a way, we extend the recent work of Chen-Hainzl-Pavlovi\'c-Seiringer.Comment: 26 pages, 1 figur