Global well-posedness and Gevrey regularity of Navier-Stokes equations in critical Triebel-Lizorkin-Lorentz spaces

The properties of solutions to Navier-Stokes equations, including well-posedness and Gevrey regularity, are a class of highly interesting problems. Inspired by the property of Lorentz type spaces that they reflect the distribution of large value points, we establish the global well-posedness of Navier-Stokes equations in critical Triebel-Lizorkin-Lorentz space. Based on this, we obtained the Gevrey regularity of the mild solution. Compared with Germain-Pavlović-Staffilani (2007), the Gevrey regularity we studied is stronger than analyticity. Furthermore, regarding that previous regularity studies mostly focused on Besov spaces, such as 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..