[Paper Review] Well-posedness and regularity of generalized Navier-Stokes equations in some Critical $Q-$spaces

Pengtao Li, Zhichun Zhai|ArXiv.org|Apr 21, 2009

Navier-Stokes equation solutions12 references4 citations

TL;DR

This paper establishes well-posedness and regularity for the generalized Navier-Stokes equations with initial data in a new critical space $ Q_{eta,eta}^{eta,-1}(bR^n) $, which extends known Besov and $ BMO^{-1} $-type spaces. By introducing a Carleson measure characterization via novel tent spaces and atomic decomposition of the predual for $ Q_{eta}^{eta}(bR^n) $, the authors prove global existence and regularity for small initial data in this larger critical space, particularly for $ \beta \in (1/2,1) $. The key contribution is a refined function space framework enabling sharper regularity and well-posedness results beyond classical settings.

ABSTRACT

We study the well-posedness and regularity of the generalized Navier-Stokes equations with initial data in a new critical space $Q_{α;\infty}^{β,-1}(\mathbb{R}^{n})= abla\cdot(Q_α^β(\mathbb{R}^{n}))^{n}, β\in({1/2},1)$ which is larger than some known critical homogeneous Besov spaces. Here $Q_α^β(\mathbb{R}^{n})$ is a space defined as the set of all measurable functions with $$\sup(l(I))^{2(α+β-1)-n}\int_{I}\int_{I}\frac{|f(x)-f(y)|^{2}}{|x-y|^{n+2(α-β+1)}}dxdy

Motivation & Objective

• To extend the known class of critical spaces for the generalized Navier-Stokes equations beyond classical Besov and $ BMO^{-1} $ spaces.
• To construct a larger critical space $ Q_{eta,eta}^{eta,-1}(bR^n) $ that includes $ \dot{B}^{1+n/2-2\beta}_{2,1} $ and has structure similar to $ BMO^{-1} $.
• To establish well-posedness and regularity for the generalized Navier-Stokes equations with initial data in this new space.
• To develop a Carleson measure characterization of $ Q_{eta}^{eta}(bR^n) $ using new tent spaces and atomic decomposition of its predual.
• To generalize Koch-Tataru's global existence result for the classical Navier-Stokes equations to the fractional case via $ Q $-space techniques.

Proposed method

• Define the space $ Q_{eta}^{eta}(bR^n) $ via a supremum over cubes $ I $ of a double integral involving differences of function values and a power-weighted distance.
• Introduce a new class of tent spaces associated with $ Q_{eta}^{eta}(bR^n) $ to analyze Carleson measure properties.
• Establish an atomic decomposition of the predual space of $ Q_{eta}^{eta}(bR^n) $ to enable duality arguments.
• Use the scaling invariance of the generalized Navier-Stokes equations under $ u_{\lambda}(t,x) = \lambda^{2\beta-1}u(\lambda^{2\beta}t, \lambda x) $ to identify critical spaces.
• Apply a fixed-point argument in a function space $ X^{eta,k}_{\alpha} $ defined via tent space norms to prove existence of mild solutions.
• Prove convergence of the iterative scheme $ u^{j+1} = e^{-t(-\Delta)^\beta}u_0 + B(u^j,u^j) $ in $ \widetilde{X}^{eta,k}_{\alpha} $ for small initial data.

Experimental results

Research questions

• RQ1 Can the well-posedness and regularity of the generalized Navier-Stokes equations be established in a critical space strictly larger than known Besov spaces for $ \beta \in (1/2,1) $?
• RQ2 What is the structure of the space $ Q_{\beta}^{eta}(bR^n) $, and how can it be characterized via Carleson measures and tent spaces?
• RQ3 How can the predual of $ Q_{\beta}^{eta}(bR^n) $ be decomposed atomically to support duality arguments in the analysis?
• RQ4 Can the Koch-Tataru global existence framework for classical Navier-Stokes be extended to the fractional Laplacian case using $ Q $-spaces?
• RQ5 What is the role of the parameter $ \alpha $ in the space $ Q_{\beta,\infty}^{\beta,-1} $, and how does it affect regularity and scaling invariance?

Key findings

• The space $ Q_{\beta,\infty}^{\beta,-1}(bR^n) = \nabla \cdot (Q_{\beta}^{\beta}(bR^n))^n $ is a critical space for the generalized Navier-Stokes equations with $ \beta \in (1/2,1) $, larger than known homogeneous Besov spaces.
• A Carleson measure characterization of $ Q_{\beta}^{\beta}(bR^n) $ is established through the introduction of new tent spaces and atomic decomposition of its predual.
• Well-posedness and global existence of mild solutions are proven for small initial data in $ Q_{\beta,\infty}^{\beta,-1}(bR^n) $, extending Koch-Tataru's result to the fractional case.
• The regularity of solutions is established in the space $ X^{eta,k}_{\alpha} $, with uniform bounds on $ \|u^j\|_{\widetilde{X}^{eta,k}_{\alpha}} $ and exponential decay of differences $ \|u^{j+1} - u^j\|_{\widetilde{X}^{eta,k}_{\alpha}} \lesssim E_k (2/3)^j $, ensuring convergence.
• The method yields quantitative control via norms involving $ t^{k/2\beta} \nabla^k B_2^2(u,v) $, with estimates in $ L^2 $-type tent space norms weighted by $ t^{-\alpha/\beta} $.
• The results apply to the incompressible Navier-Stokes equations when $ \beta = 1 $, recovering and refining known regularity results in $ Q_{\alpha;\infty}^{1,-1}(bR^n) $.