相关论文: On well-posedness and uniqueness for general hiera…
In this paper, we study solutions to the three-dimensional quintic Gross-Pitaevskii hierarchy. We prove unconditional uniqueness among all small solutions in the critical space $\mathfrak{H}^1$ (which corresponds to $H^1$ on the NLS level).…
In this article, we consider Hartree equations generalised to $2p+1$ order nonlinearities. These equations arise in the study of the mean-field limits of Bose gases with $p$-body interactions. We study their well-posedness properties in…
Consider a system of $N$ bosons on the three dimensional unit torus interacting via a pair potential $N^2V(N(x_i-x_j))$, where $\bx=(x_1, ..., x_N)$ denotes the positions of the particles. Suppose that the initial data $\psi_{N,0}$…
We consider the Cauchy problem for the Gross-Pitaevskii infinite linear hierarchy of equations on $\mathbb{R}^n.$ By introducing a (F)-norm in certain Sobolev type spaces of sequences of marginal density matrices, we establish local…
The Cauchy problem for the Gross-Pitaevskii equation in three space dimensions is shown to have an unconditionally unique global solution for data of the form 1 + H^s for 5/6 < s < 1, which do not have necessarily finite energy. The proof…
The purpose of this paper is to investigate the Cauchy problem for the Gross-Pitaevskii infinite linear hierarchy of equations on $\mathbb{R}^n,$ $n \geq 1.$ We prove local existence and uniqueness of solutions in certain Sobolev type…
Our main goal is the comparative study of singularities of solutions to the systems of first order quasilinear PDEs and their perturbations containing higher derivatives. The study is focused on the subclass of Hamiltonian PDEs with one…
We review $H^{1}$-well-posedness for initial value problems of ordinary differential equations with state-dependent right-hand side. We streamline known approaches to infer existence and uniqueness of solutions for small times given a…
In this work, we study the Gross-Pitaevskii hierarchy on general --rational and irrational-- rectangular tori of dimension two and three. This is a system of infinitely many linear partial differential equations which arises in the rigorous…
We give a new proof of uniqueness of solutions to the Gross-Pitaevskii hierarchy, first established by Erdos, Schlein and Yau, in a different space, based on space-time estimates.
In this paper, we study the local-in-time existence of solutions to randomized forms of the Gross-Pitaevskii hierarchy on periodic domains. In particular, we study the independently randomized Gross-Pitaevskii hierarchy and the dependently…
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…
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…
We consider a system of PDEs of Monge-Kantorovich type that, in the isotropic case, describes the stationary configurations of two-layers models in granular matter theory with a general source and a general boundary data. We propose a new…
We consider the cubic and quintic Gross-Pitaevskii (GP) hierarchies in $d$ dimensions, for focusing and defocusing interactions. We introduce new higher order energy functionals and prove that they are conserved for solutions of energy…
We consider the dynamical Gross-Pitaevskii (GP) hierarchy on $\R^d$, $d\geq1$, for cubic, quintic, focusing and defocusing interactions. For both the focusing and defocusing case, and any $d\geq1$, we prove local existence and uniqueness of…
We introduce a new theory of generalised solutions which applies to fully nonlinear PDE systems of any order and allows for merely measurable maps as solutions. This approach bypasses the standard problems arising by the application of…
This course explains how the usual mean field evolution partial differential equations (PDEs) in Statistical Physics - such as the Vlasov-Poisson system, the vorticity formulation of the two-dimensional Euler equation for incompressible…
We study the generalized Hartree equation, which is a nonlinear Schr\"odinger-type equation with a nonlocal potential $iu_t + \Delta u + (|x|^{-b} \ast |u|^p)|u|^{p-2}u=0, x \in \mathbb{R}^N$.We establish the local well-posedness at the…
We investigate the properties of the one-dimensional Bose gas at zero temperature, for which exact results exist for some model systems. We treat the interactions between particles in the gas with an approximate form of the many-body…