Sufficient and Necessary Conditions for the fractional Gagliardo-Nirenberg Inequalities and applications to Navier-Stokes and generalized boson equations

TL;DR

This paper establishes precise conditions for fractional Gagliardo-Nirenberg inequalities in advanced function spaces and applies these results to analyze blowup phenomena in Navier-Stokes and boson equations.

Contribution

It provides necessary and sufficient conditions for fractional GN inequalities in Besov and Triebel-Lizorkin spaces, advancing theoretical understanding and applications.

Findings

Finite-time blowup solutions exhibit concentration in critical Lebesgue space L^3

Derived conditions for fractional GN inequalities in advanced function spaces

Applied inequalities to variational problems involving minimizers

Abstract

Sufficient and necessary conditions for the generalized Gagliardo-Nirenberg (GN) inequality in Besov spaces and Triebel-Lizorkin spaces are obtained. Applying the GN inequality, we show that the finite-time blowup solutions have concentration phenomena in critical Lebesgue space L^3. Moreover, we consider the minimizer for a class of variational problem by applying the fractional GN inequality.