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We extend the results of the 2019 paper by the third and fourth author globally in time. More precisely, we prove uniform in $N$ estimates for the solutions $\phi$, $\Lambda$ and $\Gamma$ of a coupled system of Hartree-Fock-Bogoliubov type with interaction potential $V_N(x-y)=N^{3 \beta}v(N^{\beta}(x-y))$ with $\beta<1$. The potential satisfies some technical conditions, but is not small. The initial conditions have finite energy and the "pair correlation" part satisfies a smallness condition, but are otherwise general functions in suitable Sobolev spaces, and the expected correlations in $\Lambda$ develop dynamically in time. The estimates are expected to improve the Fock space bounds from the 2021 paper of the first and fifth author. This will be addressed in a different paper., Comment: 53 pages. Comments are welcome!

We provide the analysis of a physically inspired model for a trapped dilute Bose gas with repulsive pairwise atomic interactions at zero temperature. Our goal is to describe aspects of the excited many-body quantum states by accounting for the scattering of atoms in pairs from the macroscopic state (condensate). We formally construct a many-body Hamiltonian, $\mathcal{H}_{\text{app}}$, that is quadratic in the Boson field operators for noncondensate atoms. This $\mathcal{H}_{\text{app}}$ conserves the total number of atoms. Inspired by Wu (J. Math. Phys., 2:105-123, 1961), we apply a non-unitary transformation to $\mathcal{H}_{\text{app}}$. Key in this non-Hermitian view is the pair-excitation kernel, which in operator form obeys a Riccati equation. In the stationary case, we develop an existence theory for solutions to this operator equation by a variational approach. We connect this theory to the one-particle excitation wave functions heuristically derived by Fetter (Ann. Phys., 70:67-101, 1972). These functions solve an eigenvalue problem for a $J$-self-adjoint operator. From the non-Hermitian Hamiltonian, we derive a one-particle nonlocal equation for low-lying excitations, describe its solutions, and recover Fetter's excitation spectrum. Our approach leads to a description of the excited eigenstates of the reduced Hamiltonian in the $N$-particle sector of Fock space., Comment: 38 pages

We prove that certain Sobolev-type norms, slightly stronger than those given by energy conservation, stay bounded uniformly in time and $N$. This allows one to extend the local existence results of the second and third author globally in time. The proof is based on interaction Morawetz-type estimates and Strichartz estimates (including some new end-point results) for the equation $\{ \frac{1}{i}\partial_t-\Delta_{x}-\Delta_{y}+\frac{1}{N}V_N(x-y) \}\Lambda(t, x, y) =F$ in mixed coordinates such as $L^p(dt) L^q(dx) L^2(dy)$, $L^p(dt) L^q(dy) L^2(dx)$, $L^p(dt) L^q(d(x-y)) L^2(d(x+y))$. The main new technical ingredient is a dispersive estimate in mixed coordinates, which may be of interest in its own right.

We establish both a local and a global well-posedness theories for the nonlinear Hartree-Fock equations and its reduced analog in the setting of the modulation spaces on $\mathbb R^d$. In addition, we prove similar results when a harmonic potential is added to the equations. In the process, we prove the boundedeness of certain multilinear operators on products of the modulation spaces which may be of independent interest., Comment: 28 pages

We prove local in time, uniform in $N$, estimates for the solutions $\phi$, $\Lambda$ and $\Gamma$ of a coupled system of Hartree-Fock-Bogoliubov type with interaction potential $v_N(x-y) =N^{3 \beta} v(N^{\beta}(x-y))$, with $\beta <1$ and $v$ a Schwartz function (satisfying additional technical requirements). The initial conditions are general functions in a Sobolev-type space, and the expected correlations in $\Lambda$ develop dynamically in time. As shown in our previous work, as well as the work of J. Chong, (both in the case $\beta<2/3$), using the conserved quantities of the system of equations, this type of local in time estimates can be extended globally. Also, they can be used to derive Fock space estimates for the approximation of the exact evolution of a Bosonic system by states of the form $e^{\sqrt N A(\phi)}e^{B(k)} \Omega$. This will be addressed in detail in future work., Comment: Revised version, to appear in Comm. PDE

This article addresses the large data global regularity for the equivariant case of the $2+1$-dimensional Faddeev model and shows that it holds true for initial data in $H^s\times H^{s-1}(\mathbb{R}^2)$ with $s>3$., Comment: 35 pages. arXiv admin note: text overlap with arXiv:1707.02917

This article is concerned with the large data global regularity for the equivariant case of the classical Skyrme model and proves that this is valid for initial data in $H^s\times H^{s-1}(\mathbb{R}^3)$ with $s>7/2$., Comment: 37 pages

We consider a large number of Bosons with interaction potential $v_N(x)=N^{3 \beta}v(N^{\beta}x)$. In earlier papers we considered a set of equations for the condensate $\phi$ and pair excitation function $k$ and proved that they provide a Fock space approximation to the exact evolution of the condensate for $\beta <\frac{1}{3}$. This result was extended to the case $\beta<\frac{1}{2}$ by E. Kuz, where it was also argued informally that the equations of our earlier work do not provide an approximation for $\beta>\frac{1}{2}$. In 2013, we introduced a coupled refinement of our original equations and conjectured that they provide a Fock space approximation in the range $0 \le \beta < 1$. In the current paper we prove that this is indeed the case for $\beta < \frac{2}{3}$, at least locally in time. In order to do that, we re-formulate the equations of \cite{GMM} in a way reminiscent of BBGKY and apply harmonic analysis techniques in the spirit of X. Chen and J. Holmer to prove the necessary estimates. In turn, these estimates provide bounds for the pair excitation function $k$.

In our previous work \cite{GMM1},\cite{GMM2} we introduced a correction to the mean field approximation of interacting Bosons. This correction describes the evolution of pairs of particles that leave the condensate and subsequently evolve on a background formed by the condensate. In \cite{GMM2} we carried out the analysis assuming that the interactions are independent of the number of particles $N$. Here we consider the case of stronger interactions. We offer a new transparent derivation for the evolution of pair excitations. Indeed, we obtain a pair of linear equations describing their evolution. Furthermore, we obtain apriory estimates independent of the number of particles and use these to compare the exact with the approximate dynamics.

We study the evolution of a N-body weakly interacting system of Bosons. Our work forms an extension of our previous paper I, in which we derived a second-order correction to a mean-field evolution law for coherent states in the presence of small interaction potential. Here, we remove the assumption of smallness of the interaction potential and prove global existence of solutions to the equation for the second-order correction. This implies an improved Fock-space estimate for our approximation of the N-body state.

We present a unified approach for obtaining global a priori estimates for solutions of nonlinear defocusing Schr\"odinger equations with defocusing nonlinearities. The estimates are produced by contracting the local momentum conservation law with appropriate vector fields. The corresponding law is written for defocusing equations of tensored solutions. In particular, we obtain a new estimate in two dimensions. We bound the restricted $L_t^4L_{\gamma}^4$ Strichartz norm of the solution on any curve $\gamma$ in $\mathbb R^2$. For the specific case of a straight line we upgrade this estimate to a weighted Strichartz estimate valid in the full plane.

Inspired by the works of Rodnianski and Schlein and Wu, we derive a new nonlinear Schr\"odinger equation that describes a second-order correction to the usual tensor product (mean-field) approximation for the Hamiltonian evolution of a many-particle system in Bose-Einstein condensation. We show that our new equation, if it has solutions with appropriate smoothness and decay properties, implies a new Fock space estimate. We also show that for an interaction potential $v(x)= \epsilon \chi(x) |x|^{-1}$, where $\epsilon$ is sufficiently small and $\chi \in C_0^{\infty}$, our program can be easily implemented locally in time. We leave global in time issues, more singular potentials and sophisticated estimates for a subsequent part (part II) of this paper.

We prove new interaction Morawetz type (correlation) estimates in one and two dimensions. In dimension two the estimate corresponds to the nonlinear diagonal analogue of Bourgain's bilinear refinement of Strichartz. For the 2d case we provide a proof in two different ways. First, we follow the original approach of Lin and Strauss but applied to tensor products of solutions. We then demonstrate the proof using commutator vector operators acting on the conservation laws of the equation. This method can be generalized to obtain correlation estimates in all dimensions. In one dimension we use the Gauss-Weierstrass summability method acting on the conservation laws. We then apply the 2d estimate to nonlinear Schr\"odinger equations and derive a direct proof of Nakanishi's $H^1$ scattering result for every $L^{2}$-supercritical nonlinearity. We also prove scattering below the energy space for a certain class of $L^{2}$-supercritical equations., Comment: 45 pages

We study the evolution of a many-particle system whose wave function obeys the N-body Schroedinger equation under Bose symmetry. The system Hamiltonian describes pairwise particle interactions in the absence of an external potential. We derive apriori dispersive estimates that express the overall repulsive nature of the particle interactions. These estimates hold for a wide class of two-body interaction potentials which are independent of the particle number, N. We discuss applications of these estimates to the BBGKY hierarchy for reduced density matrices analyzed by Elgart, Erdos, Schlein and Yau., Comment: 25 pages, 2 figures

We prove global well-posedness for low regularity data for the $L^2-critical$ defocusing nonlinear Schr\"odinger equation (NLS) in 2d. More precisely we show that a global solution exists for initial data in the Sobolev space $H^{s}(\mathbb R^2)$ and any $s>{2/5}$. This improves the previous result of Fang and Grillakis where global well-posedness was established for any $s \geq {1/2}$. We use the $I$-method to take advantage of the conservation laws of the equation. The new ingredient is an interaction Morawetz estimate similar to one that has been used to obtain global well-posedness and scattering for the cubic NLS in 3d. The derivation of the estimate in our case is technical since the smoothed out version of the solution $Iu$ introduces error terms in the interaction Morawetz inequality. A byproduct of the method is that the $H^{s}$ norm of the solution obeys polynomial-in-time bounds., Comment: 21 pages