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The well-posedness and convergence of higher-order Hartree equations in critical Sobolev spaces on $\mathbb{T}^3$

Ryan L. Acosta Babb, Andrew Rout

TL;DR

The paper develops a rigorous theory for higher-order Hartree equations on the periodic domain $\mathbb{T}^3$, establishing local and global well-posedness in the scaling-critical Sobolev spaces $H^{s_c}$ with $s_c=\frac{3}{2}-\frac{1}{p}$, and proving convergence to the corresponding nonlinear Schrödinger equations as the interaction potential concentrates to a delta. It provides sharp energy and multilinear estimates in the periodic setting, enabling a fixed-point argument and energy conservation to yield global well-posedness for small data and perturbative global results for mixed nonlinearities. A key contribution is the demonstration that Hartree dynamics with $p$-body interactions converge to local NLS in $H^{s_c}$, and that mixed nonlocal nonlinearities can be treated as perturbations of higher-order Hartree or quintic NLS, with robust convergence results. These results advance the mathematical understanding of mean-field limits and the rigorous connection between nonlocal Hartree dynamics and local NLS on a torus, with implications for Bose gas models and periodic dispersive PDE theory.

Abstract

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 $H^{s_c}(\mathbb{T}^3)$, where $\mathbb{T}^3$ is the three dimensional torus and $s_c = 3/2 - 1/p$ is the scaling-critical regularity. The convergence of solutions of the Hartree equation to solutions of the nonlinear Schrödinger equation is proved. We also consider the case of mixed nonlinearities, proving local well-posedness in $s_c$ by considering the problem as a perturbation of the higher-order Hartree equation. In the particular case of the (defocusing) quintic-cubic Hartree equation, we also prove global well-posedness for all initial conditions in $H^1(\mathbb{T}^3)$. This is done by viewing it as a perturbation of the local quintic NLS.

The well-posedness and convergence of higher-order Hartree equations in critical Sobolev spaces on $\mathbb{T}^3$

TL;DR

The paper develops a rigorous theory for higher-order Hartree equations on the periodic domain , establishing local and global well-posedness in the scaling-critical Sobolev spaces with , and proving convergence to the corresponding nonlinear Schrödinger equations as the interaction potential concentrates to a delta. It provides sharp energy and multilinear estimates in the periodic setting, enabling a fixed-point argument and energy conservation to yield global well-posedness for small data and perturbative global results for mixed nonlinearities. A key contribution is the demonstration that Hartree dynamics with -body interactions converge to local NLS in , and that mixed nonlocal nonlinearities can be treated as perturbations of higher-order Hartree or quintic NLS, with robust convergence results. These results advance the mathematical understanding of mean-field limits and the rigorous connection between nonlocal Hartree dynamics and local NLS on a torus, with implications for Bose gas models and periodic dispersive PDE theory.

Abstract

In this article, we consider Hartree equations generalised to order nonlinearities. These equations arise in the study of the mean-field limits of Bose gases with -body interactions. We study their well-posedness properties in , where is the three dimensional torus and is the scaling-critical regularity. The convergence of solutions of the Hartree equation to solutions of the nonlinear Schrödinger equation is proved. We also consider the case of mixed nonlinearities, proving local well-posedness in by considering the problem as a perturbation of the higher-order Hartree equation. In the particular case of the (defocusing) quintic-cubic Hartree equation, we also prove global well-posedness for all initial conditions in . This is done by viewing it as a perturbation of the local quintic NLS.

Paper Structure

This paper contains 19 sections, 19 theorems, 154 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Statement of results
  4. Hartree equation with mixed nonlinearity
  5. Previously known results
  6. Outline of the paper
  7. Well-posedness of the Hartree equation
  8. Energy estimates
  9. Multilinear estimates
  10. Well-posedness arguments for \ref{['higher_order_Hartree']}
  11. Proof of convergence
  12. Mixed Hartree nonlinearities
  13. Energy and multilinear estimates
  14. Well-posedness of mixed Hartree-equation for general $p$
  15. Global well-posedness for all data for perturbations of the quintic NLS
  16. ...and 4 more sections

Key Result

Proposition 2.1

Fix an integer $p \geq 2$. Suppose that $u_0 \in H^{s_c}(\mathbf{X})$ with $s_c$ as in critical_sobolev and $V \in L^1(\mathbf{X}^{p})$. Then there exist a $T > 0$ --- depending on $\|u\|_{H^{s_c}}$, $\|V\|_{L^1}$, and $p$ --- and a unique $u \in C_t((-T,T);H^{s_c}(\mathbf{X})) \cap X^{s_c}((-T,T))$

Theorems & Definitions (57)

  • Remark 1.1
  • Remark 1.2
  • Proposition 2.1: Local well-posedness of \ref{['higher_order_Hartree']}
  • Proposition 2.2: Small data global well-posedness of \ref{['higher_order_Hartree']} for $p=2$
  • Theorem 2.3: Convergence of the Hartree equation to the NLS
  • Remark 2.4
  • Remark 2.5
  • Remark 2.6
  • Remark 2.7
  • Remark 2.8
  • ...and 47 more