Strichartz type estimates for fractional heat equations

TL;DR

This paper establishes new Strichartz estimates for fractional heat equations, including endpoint and generalized versions, and applies these results to prove global well-posedness for a generalized Navier-Stokes system.

Contribution

It introduces novel Strichartz estimates for fractional heat equations, including endpoint, parabolic, and generalized versions, with applications to fluid dynamics.

Findings

Established Strichartz estimates using Hardy-Littlewood-Sobolev inequality.

Proved endpoint homogeneous Strichartz estimate with BMO space.

Applied estimates to prove global existence for generalized Navier-Stokes.

Abstract

We obtain Strichartz estimates for the fractional heat equations by using both the abstract Strichartz estimates of Keel-Tao and the Hardy-Littlewood-Sobolev inequality. We also prove an endpoint homogeneous Strichartz estimate via replacing $ L^{\infty}_{x}(\mathbb{R}^{n})$ by $BMO_{x}(\mathbb{R}^{n})$ and a parabolic homogeneous Strichartz estimate. Meanwhile, we generalize the Strichartz estimates by replacing the Lebesgue spaces with either Besov spaces or Sobolev spaces. Moreover, we establish the Strichartz estimates for the fractional heat equations with a time dependent potential of an appropriate integrability. As an application, we prove the global existence and uniqueness of regular solutions in spatial variables for the generalized Navier-Stokes system with $L^{r}(\mathbb{R}^{n})$ data.