[Paper Review] On the global regularity of axisymmetric Navier-Stokes-Boussinesq system

Hamadi Abidi, Taoufik Hmidi+1 more|ArXiv.org|Aug 6, 2009

Advanced Mathematical Physics Problems12 references3 citations

TL;DR

This paper establishes global well-posedness for the three-dimensional axisymmetric Navier-Stokes-Boussinesq system with initial data in critical regularity spaces, proving existence and uniqueness of global strong solutions when the initial velocity is axisymmetric without swirl and the initial density is bounded. The proof relies on vorticity structure, weighted energy estimates, and a priori bounds in anisotropic function spaces, extending classical results to the coupled system with density transport.

ABSTRACT

In this paper we prove a global well-posedness result for tridimensional Navier-Stokes-Boussinesq system with axisymmetric initial data. This system couples Navier-Stokes equations with a transport equation governing the density.

Motivation & Objective

• To establish global well-posedness for the 3D axisymmetric Navier-Stokes-Boussinesq system with initial data in critical regularity spaces.
• To extend the classical global regularity results for axisymmetric Navier-Stokes to the case where density is transported and acts as a source term in the vertical direction.
• To analyze the role of axisymmetry and the absence of swirl in preventing finite-time singularity formation despite the presence of density coupling.
• To prove uniform a priori estimates in anisotropic Sobolev and Besov spaces that control the growth of the solution over time.

Proposed method

• Utilizes the axisymmetric structure of the velocity field and the absence of swirl to exploit cancellation in the vortex-stretching term.
• Introduces the quantity $ \Gamma = \omega_\theta / r $, which satisfies a transport-diffusion equation without source terms, enabling $ L^p $-norm conservation.
• Employs weighted energy estimates and anisotropic Sobolev embeddings to control the growth of the velocity and density in critical spaces.
• Applies a vanishing viscosity approximation and regularization via convolution to construct approximate solutions with uniform bounds.
• Uses Gronwall-type inequalities in conjunction with Besov space norms to control the difference between two solutions and prove uniqueness.
• Relies on the Beale-Kato-Majda criterion and the conservation of $ \| \Gamma \|_{L^p} $ to prevent blow-up in finite time.

Experimental results

Research questions

• RQ1 Can global regularity be established for the 3D axisymmetric Navier-Stokes-Boussinesq system when the initial data are not small in critical norms but possess special geometric structure?
• RQ2 How does the presence of a transported density term $ \rho e_z $ affect the vorticity dynamics and singularity formation in axisymmetric flows?
• RQ3 To what extent do the conservation laws of $ \Gamma = \omega_\theta / r $ persist and control the solution in the presence of density coupling?
• RQ4 Can the uniqueness and existence of global strong solutions be proven in critical regularity spaces for the full Navier-Stokes-Boussinesq system with axisymmetry?
• RQ5 What is the role of the Besov space norm $ \| \nabla v \|_{L^1_t B^{3/p}_{p,1}} $ in the uniqueness argument for the system?

Key findings

• The system admits a unique global strong solution for initial data $ v_0 $ axisymmetric without swirl and $ \rho_0 \in L^\infty \cap L^2 $, with $ v_0 \in H^1 $ and $ \omega_0, \omega_0/r \in L^2 \cap L^\infty $.
• The quantity $ \Gamma = \omega_\theta / r $ satisfies a conservation law in $ L^p $-norms for all $ p \in [1, \infty] $, which is crucial for preventing finite-time blow-up.
• A priori estimates in $ L^\infty_T L^2 \cap L^2_T H^1 $ for velocity and $ L^\infty_T H^{-1} $ for density are uniformly bounded, ensuring global existence.
• Uniqueness is established via Gronwall's inequality applied to the difference of two solutions, using the bound $ \| \nabla v^2 \|_{L^1_t B^{3/p}_{p,1}} $ and the $ H^{-1} $-norm of density difference.
• The solution remains axisymmetric and without swirl for all time, preserving the initial geometric structure.
• The method extends previous results on axisymmetric Navier-Stokes to the Boussinesq system, even for large initial data, by leveraging the cancellation in the vortex-stretching term.