Long-time existence of Gevrey-2 solutions to the 3D Prandtl boundary layer equations

For the three dimensional Prandtl boundary layer equations, we will show that for arbitrary $M$ and sufficiently small $\epsilon$, the lifespan of the Gevrey-2 solution is at least of size ${\epsilon }^{-M}$ if the initial data lies in suitable Gevrey-2 spaces with size of $\epsilon$.