[Paper Review] Analysis of Hartree equation with an interaction growing at the spatial infinity

Satoshi Masaki|arXiv (Cornell University)|Mar 16, 2011

Advanced Mathematical Physics Problems6 references3 citations

TL;DR

This paper establishes global well-posedness for the nonlinear Schrödinger equation with a Hartree-type nonlinearity where the interaction potential grows quadratically at spatial infinity (i.e., $\nu = -2$). By exploiting mass conservation and a decomposition of the nonlinearity, the authors prove existence and uniqueness of global solutions in a weighted energy space $\Sigma^{1,\gamma/2}$, with explicit solutions when $\gamma = 2$, and show that momentum conservation is essential for faster-than-linear growth rates.

ABSTRACT

We consider nonlinear Schrödinger equation with a Hartree-type nonlocal nonlinearity. The case where a nonlinear interaction potential grows at the spatial infinity is studied. By virtue of an effective decomposition of the nonlinearity based on conservation of mass, this kind of growing nonlinear interaction is known to contain an effect like a linear potential. In this paper, a well-posedness result is obtained in a suitable energy space for a class of interaction potential growing at the spatial infinity in at most the quadratic order. If the growth rate of interaction potential is faster than the linear order, a priori information on the center of mass plays a crucial role. When the interaction potential is exactly in the quadratic order, solution are written explicitly.

Motivation & Objective

• To establish global well-posedness for the Hartree equation with a nonlocal nonlinearity whose potential grows at spatial infinity, particularly for $\nu \in [-2, -1)$.
• To analyze the role of the center of mass and momentum conservation in controlling solutions when the interaction potential grows faster than linearly.
• To provide explicit solutions in the case of quadratic growth ($\nu = -2$) and extend well-posedness results beyond previous works.
• To develop a framework based on energy estimates and effective decomposition of the nonlinearity to handle singular and growing potentials.
• To prove unconditional uniqueness in the case of sub-quadratic growth ($\gamma \in (0,1]$) and conditional uniqueness for $\gamma \in (1,2)$, requiring momentum conservation.

Proposed method

• Use of a weighted energy space $\Sigma^{1,\gamma/2}$ to control both $L^2$-norm of the gradient and the weighted $L^2$-norm of the solution.
• Decomposition of the nonlinearity via mass conservation to treat the growing potential as if it behaves like a linear potential.
• Application of abstract existence theory from Section 3, adapted to the case where the potential $V(x) = \lambda |x|^\gamma \chi(|x|)$ is truncated and analyzed via regularity and decay estimates.
• Use of energy conservation and a priori $L^2$-bounds on $\nabla u(t)$ and $\langle x \rangle^{\gamma/2} u(t)$ to prevent finite-time blow-up.
• Establishment of uniqueness under momentum conservation for $\gamma \in (1,2)$, and unconditional uniqueness for $\gamma \in (0,1]$ via $\zeta = \infty$ in the abstract framework.
• Explicit solution derivation when $\gamma = 2$, using the structure of the equation and the form of the potential $|x|^2$.

Experimental results

Research questions

• RQ1 What is the well-posedness status of the Hartree equation when the interaction potential grows at spatial infinity, particularly for $\nu \in [-2, -1)$?
• RQ2 How does the center of mass and momentum conservation influence the global existence and uniqueness of solutions in the presence of growing nonlocal nonlinearities?
• RQ3 Can explicit solutions be constructed when the interaction potential grows quadratically?
• RQ4 Under what conditions does the solution remain bounded in the weighted energy space $\Sigma^{1,\gamma/2}$ for growing potentials?
• RQ5 What is the role of the abstract framework (Assumptions (V1)-(V4)) in proving global well-posedness for non-Lipschitz growing potentials?

Key findings

• For $\gamma \in (1,2)$, the Hartree equation is globally well-posed in $\Sigma^{1,\gamma/2}$, with solutions conserving mass, energy, and momentum.
• Uniqueness holds conditionally in $C(\mathbb{R}; \Sigma^{1,\gamma/2})$ only when momentum is conserved, indicating that momentum information is essential for uniqueness in this regime.
• When $\lambda > 0$, the $L^2$-norm of the gradient $\|\nabla u(t)\|_{L^2}$ grows at most like $\langle t \rangle^{\gamma/(2-\gamma)}$, while $\|\langle x \rangle^{\gamma/2} u(t)\|_{L^2}$ grows similarly.
• For $\lambda < 0$, both $\|\nabla u(t)\|_{L^2}$ and $\|\langle x \rangle^{\gamma/2} u(t)\|_{L^2}$ remain uniformly bounded in time.
• When $\gamma = 2$, the equation admits an explicit solution, confirming the integrability of the system under quadratic growth.
• For $\gamma \in (0,1]$, global well-posedness holds unconditionally in $\Sigma^{1,\gamma/2}$, without requiring momentum conservation, due to stronger regularity in the abstract framework.