Navier-Stokes Equations on Quantum Euclidean Spaces

TL;DR

This paper extends classical Navier-Stokes analysis to quantum Euclidean spaces, establishing well-posedness results and developing harmonic analysis tools for noncommutative PDEs, with implications for semiclassical limits.

Contribution

It introduces a systematic approach to quantum Navier-Stokes equations using noncommutative harmonic analysis, achieving well-posedness results and independent techniques from the deformation parameter.

Findings

Global well-posedness in 2D quantum spaces

Local well-posedness in higher dimensions

Development of harmonic analysis on quantum Euclidean spaces

Abstract

We investigate in the present paper the Navier-Stokes equations on quantum Euclidean spaces $\mathbb{R}^d_{\theta}$ with $\theta$ 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 $\mathbb{R}^d_{\theta}$, 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 $\theta$, 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.