Second derivatives of solutions to the 3D incompressible Navier-Stokes equation in Lebesgue spaces
Second derivatives of solutions to the 3D incompressible Navier-Stokes equation in Lebesgue spaces
We obtain new controls for the Leray solutions $u$ of the incompressible Navier-Stokes equation in $\mathbb{R}^3$. Specifically, we estimate $u$, $\nabla u$, and $\nabla^2 u$ in suitable Lebesgue spaces $L^{\tilde r}_TL^r$, $r <+ \infty$ with some constraints on $\tilde r>0$. Our method is based on a Duhamel formula around a …