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Navier-Stokes Equations on Quantum Euclidean Spaces

Deyu Chen, Guixiang Hong, Liang Wang, Wenhua Wang

TL;DR

This work develops a comprehensive real-analysis framework for the Navier–Stokes equations on quantum Euclidean spaces ${\mathbb{R}^d_{\theta}}$, using noncommutative $L_p$-spaces, Besov spaces, and a transference principle to transfer harmonic-analytic estimates from the classical setting. It proves quantum analogues of Ladyzhenskaya and Kato results: global well-posedness in 2D for the symmetric nonlinearity and local well-posedness in higher dimensions for the same, with uniform-in-$\theta$ estimates and smoothing properties. A key novelty is the theta-independent approach to time–space estimates and the semiclassical limit, showing convergence to the classical Navier–Stokes equations as $\|\theta\|\to0$. The results establish a robust framework for quantum PDEs, enabling sharp $L_p$–type estimates, transference of Fourier multipliers, and Besov-space analysis, with potential applications to other quantum-fluid models and semiclassical analysis.

Abstract

We investigate in the present paper the Navier-Stokes equations on quantum Euclidean spaces $\mathbb{R}^d_θ$ with $θ$ being a $d\times d$ antisymmetric matrix, which is a standard example of non-compact noncommutative manifolds. The quantum analogues of Ladyzhenskaya and Kato's results are established, that is, we obtain the global well-posedness in the 2D case and the local well-posedness with solution in $L_d(\mathbb{R}^d)$ in higher dimensions. To achieve these optimal results, we develop the related theory of harmonic analysis and function spaces on $\mathbb{R}^d_θ$, and apply the sharp estimates around noncommutative $L_p$-spaces to quantum Navier-Stokes equations. Moreover, our techniques, which are independent of the deformed parameter $θ$, allow us to conclude some results on the semiclassical limits. This is the first instance of systematical applications to the theory of quantum partial differential equations of the powerful real analysis techniques around noncommutative $L_p$-spaces, which date back to the seminal work \cite{PiXu97} in 1997 on noncommutative martingale inequalities. As in classical case, one may expect numerous similar applications in the future.

Navier-Stokes Equations on Quantum Euclidean Spaces

TL;DR

This work develops a comprehensive real-analysis framework for the Navier–Stokes equations on quantum Euclidean spaces , using noncommutative -spaces, Besov spaces, and a transference principle to transfer harmonic-analytic estimates from the classical setting. It proves quantum analogues of Ladyzhenskaya and Kato results: global well-posedness in 2D for the symmetric nonlinearity and local well-posedness in higher dimensions for the same, with uniform-in- estimates and smoothing properties. A key novelty is the theta-independent approach to time–space estimates and the semiclassical limit, showing convergence to the classical Navier–Stokes equations as . The results establish a robust framework for quantum PDEs, enabling sharp –type estimates, transference of Fourier multipliers, and Besov-space analysis, with potential applications to other quantum-fluid models and semiclassical analysis.

Abstract

We investigate in the present paper the Navier-Stokes equations on quantum Euclidean spaces with being a antisymmetric matrix, which is a standard example of non-compact noncommutative manifolds. The quantum analogues of Ladyzhenskaya and Kato's results are established, that is, we obtain the global well-posedness in the 2D case and the local well-posedness with solution in in higher dimensions. To achieve these optimal results, we develop the related theory of harmonic analysis and function spaces on , and apply the sharp estimates around noncommutative -spaces to quantum Navier-Stokes equations. Moreover, our techniques, which are independent of the deformed parameter , allow us to conclude some results on the semiclassical limits. This is the first instance of systematical applications to the theory of quantum partial differential equations of the powerful real analysis techniques around noncommutative -spaces, which date back to the seminal work \cite{PiXu97} in 1997 on noncommutative martingale inequalities. As in classical case, one may expect numerous similar applications in the future.

Paper Structure

This paper contains 21 sections, 37 theorems, 351 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Noncommutative $L_p$-spaces, Lorentz spaces
  4. Quantum Euclidean spaces
  5. Sobolev spaces on quantum Euclidean spaces
  6. Navier-Stokes equations on quantum Euclidean spaces
  7. The definition of Navier-Stokes equations on $\mathbb{R}^d_{\theta}$
  8. An equivalent form of Navier-Stokes equations on $\mathbb{R}^d_{\theta}$
  9. The equivalence between differential and integral forms of Navier-Stokes equations
  10. Critical spaces for Navier-Stokes equations on $\mathbb{R}^d_{\theta}$
  11. A transference technique on quantum Euclidean spaces
  12. Besov spaces on quantum Euclidean spaces
  13. The interpolation and some embeddings on $\mathbb{R}^d_{\theta}$
  14. A product estimate
  15. The sharp time-space estimates
  16. ...and 6 more sections

Key Result

Theorem 1.1

Let $d\geq2$ and $u_0\in [L_d({\mathbb R}_{\theta}^d)]^d_0$. Then we have the following conclusions:

Theorems & Definitions (84)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Remark 2.1
  • Lemma 2.2
  • Definition 2.3
  • Remark 2.4
  • Remark 2.5
  • ...and 74 more