Joint space-time analyticity of mild solutions to the Navier-Stokes equations

In this paper, we show the optimal decay rate estimates of space-time derivatives and joint analyticity solutions to Navier-Stokes equations. As it is known from Hartogs's theorem, for a complex function with two variables, respect variables can be derived combining each variable. However, as real space time, equations cannot directly obtained combination time analyticity. Our result seems first quantitative equations, proof only involves variable methods. Moreover, also yield bounds on growth (in time) radius analyticity, solutions.