Gevrey regularity solution for initial data in Triebel-Lizorkin-Lorentz spaces via single norm defined by nonlinearity of frequency

The properties of solutions to Navier-Stokes equations, including well-posedness and Gevrey regularity, are a class of highly interesting problems. We are inspired by the location result on Triebel-Lizorkin-Lorentz space of Hobus and Saal in 2019. In order to overcome the difficulties they encountered when dealing with global well-posedness, we introduce the single norm iterative space ${^{m'}_{m} \dot{F}}^{\frac{n}{p}-1,q}_{p,r}$ and utilize tools such as the Fefferman-Stein inequality to investigate the properties of our iterative spaces. As a result, we establish the global well-posedness of Navier-Stokes equations in critical Triebel-Lizorkin-Lorentz space and obtain the Gevrey regularity of the mild solution. Regarding that there're many regularity studies focused on Besov spaces, such as Bae-Biswas-Tadmor(2012) and Liu-Zhang (2024), our Triebel-Lizorkin-Lorentz spaces contain more general initial value spaces, including part of Besov spaces and all of Triebel-Lizorkin spaces, etc.. Furthermore, compared with Germain-Pavlovi\'c-Staffilani (2007), our Gevrey estimation also implies spatial analyticity and is more convenient to unify the estimates of gradient of any order.