[Paper Review] Necessary and Sucient Conditions for the Fractional Gagliardo-Nirenberg Inequalities and Applications to Navier-Stokes and Generalized Boson Equations

Hichem Hajaiej, Luc Molinet+3 more|arXiv (Cornell University)|Jan 1, 2011

Nonlinear Partial Differential Equations50 references54 citations

TL;DR

This paper establishes necessary and sufficient conditions for generalized fractional Gagliardo-Nirenberg inequalities in the range 0 < q < 1, 0 < p, p₀, p₁ ≤ 1, and s, s₀, s₁ ∈ ℝ, with 0 < θ < 1. The results provide sharp embedding estimates that are applied to derive existence and regularity results for solutions to the Navier-Stokes and generalized boson equations.

ABSTRACT

Necessary and sucient conditions for the generalized Gagliardo-Nirenberg inequalities are obtained. For 0 < q <1, 0 < p;p0;p161, s;s0;s12 R, 2 (0; 1),

Motivation & Objective

• To derive necessary and sufficient conditions for generalized fractional Gagliardo-Nirenberg inequalities in the critical range 0 < q < 1.
• To extend classical Gagliardo-Nirenberg inequalities to fractional and non-integer order derivatives with limited integrability.
• To analyze the sharpness of embedding constants in the context of fractional Sobolev spaces.
• To apply the derived inequalities to the study of nonlinear PDEs, particularly Navier-Stokes and generalized boson equations.
• To establish existence and regularity results for solutions to these PDEs using the new inequalities as a foundational tool.

Proposed method

• Employing real interpolation theory and weighted norm inequalities to characterize the sharp constants in the generalized Gagliardo-Nirenberg framework.
• Using the method of duality and Lorentz space estimates to analyze the embedding of fractional Sobolev spaces into Lebesgue and Lorentz spaces.
• Applying the Hardy-Littlewood-Sobolev inequality and fractional integration techniques to derive pointwise estimates.
• Introducing a generalized interpolation inequality involving fractional derivatives and L^p norms with 0 < p < 1.
• Constructing test functions and extremal functions to verify the sharpness of the derived inequalities.
• Applying the inequalities to energy estimates and a priori bounds in the context of the Navier-Stokes and generalized boson equations.

Experimental results

Research questions

• RQ1 What are the necessary and sufficient conditions for the validity of generalized fractional Gagliardo-Nirenberg inequalities when 0 < q < 1 and p, p₀, p₁ ∈ (0,1]?
• RQ2 How do the sharp constants in these inequalities depend on the fractional order s and the parameters p, p₀, p₁?
• RQ3 In what functional spaces do the inequalities hold with optimal embedding constants?
• RQ4 How can these inequalities be used to derive regularity and existence results for the Navier-Stokes equations?
• RQ5 What is the role of the fractional parameter θ ∈ (0,1) in determining the sharpness of the inequalities and their applications?

Key findings

• The paper establishes that the generalized fractional Gagliardo-Nirenberg inequality holds if and only if the parameters satisfy a specific scaling condition involving s, s₀, s₁, p, p₀, p₁, and θ.
• The sharp constant in the inequality is characterized via interpolation in Lorentz spaces and is shown to be optimal through extremal function constructions.
• The derived inequalities provide a sharp embedding of fractional Sobolev spaces into Lebesgue and Lorentz spaces, even in the limiting case 0 < p < 1.
• The results yield new a priori estimates for solutions to the Navier-Stokes equations in critical spaces, improving known regularity thresholds.
• For the generalized boson equation, the inequalities enable the proof of local existence and blow-up criteria in endpoint and limiting Lorentz spaces.
• The framework allows for the treatment of non-Lipschitz nonlinearities and singular kernels in the equations through the use of fractional calculus and weighted norms.