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Existence and Large Time Behavior for a Dissipative Variant of the Rotational NLS Equation

Paolo Antonelli, Boris Shakarov

TL;DR

We analyze a dissipative variant of the rotational Gross-Pitaevskii equation with a mass-conserving nonlocal chemical potential $\mu[\psi]$. The paper establishes local and global well-posedness across physically relevant regimes, derives Strichartz-based estimates for the associated semigroup, and characterizes long-time behavior: in the linear case, solutions converge to the lowest-energy eigenspace of the rotating Hamiltonian $H_\Omega$, while in the nonlinear case solutions converge to stationary states and, under a conjectured spectral gap, strongly to the ground state up to a phase. Ground states exist variationally at fixed mass, and their potential orbital stability is discussed in the context of the conjecture. Overall, the results provide a rigorous foundation for dissipative, rotating Bose-Einstein condensate dynamics with mass conservation and offer insight into vortex formation and relaxation phenomena in these systems.

Abstract

We study a dissipative variant of the Gross-Pitaevskii equation with rotation. The model contains a nonlocal, nonlinear term that forces the conservation of $L^2$-norm of solutions. We are motivated by several physical experiments and numerical simulations studying the formation of vortices in Bose-Einstein condensates. We show local and global well-posedness of this model and investigate the asymptotic behavior of its solutions. In the linear case, the solution asymptotically tends to the eigenspace associated with the smallest eigenvalue in the decomposition of the initial datum. In the nonlinear case, we obtain weak convergence to a stationary state. Moreover, for initial energies in a specific range, we prove strong asymptotic stability of ground state solutions.

Existence and Large Time Behavior for a Dissipative Variant of the Rotational NLS Equation

TL;DR

We analyze a dissipative variant of the rotational Gross-Pitaevskii equation with a mass-conserving nonlocal chemical potential . The paper establishes local and global well-posedness across physically relevant regimes, derives Strichartz-based estimates for the associated semigroup, and characterizes long-time behavior: in the linear case, solutions converge to the lowest-energy eigenspace of the rotating Hamiltonian , while in the nonlinear case solutions converge to stationary states and, under a conjectured spectral gap, strongly to the ground state up to a phase. Ground states exist variationally at fixed mass, and their potential orbital stability is discussed in the context of the conjecture. Overall, the results provide a rigorous foundation for dissipative, rotating Bose-Einstein condensate dynamics with mass conservation and offer insight into vortex formation and relaxation phenomena in these systems.

Abstract

We study a dissipative variant of the Gross-Pitaevskii equation with rotation. The model contains a nonlocal, nonlinear term that forces the conservation of -norm of solutions. We are motivated by several physical experiments and numerical simulations studying the formation of vortices in Bose-Einstein condensates. We show local and global well-posedness of this model and investigate the asymptotic behavior of its solutions. In the linear case, the solution asymptotically tends to the eigenspace associated with the smallest eigenvalue in the decomposition of the initial datum. In the nonlinear case, we obtain weak convergence to a stationary state. Moreover, for initial energies in a specific range, we prove strong asymptotic stability of ground state solutions.
Paper Structure (12 sections, 25 theorems, 228 equations)

This paper contains 12 sections, 25 theorems, 228 equations.

Table of Contents

  1. Introduction
  2. Physical background and motivations
  3. Mathematical Setting and Main Results
  4. Preliminaries
  5. Properties of the semigroup
  6. Spectral properties of the linear Hamiltonian
  7. Existence of Solutions
  8. Local well-posedness for $\sigma \geq \frac{1}{2}$
  9. Local well-posedness for $\sigma<\frac{1}{2}$
  10. Global Well-posedness
  11. Asymptotic Behavior of the Linear Equation
  12. Asymptotic Behavior in the Nonlinear Case

Key Result

Theorem 1.1

Let $\frac{1}{2} \leq \sigma < \frac{2}{(d-2)^+}$, or let $g=0$. Then for any $\psi_0 \in \Sigma(\mathbb R^d)$ there exist a maximal existence time $T_{max}>0$ and a unique local solution $\psi \in C([0,T_{max}), \Sigma(\mathbb R^d))$ to equation eq:gpe. Moreover, either $T_{max} = \infty$, or $T_{m

Theorems & Definitions (48)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Conjecture 1.6: Fundamental gap conjecture
  • Theorem 1.7
  • Proposition 2.1
  • proof
  • Proposition 2.2
  • ...and 38 more