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 $u$, $\mathrm{\nabla }u$, and ${\mathrm{\nabla }}^{2}u$ in suitable Lebesgue spaces $L_T^{\tilde{r}}{L}^r$, $r<+\mathrm{\infty }$ with some constraints on $\tilde{r}>0$. Our method is based on a Duhamel formula around a perturbed heat equation, allowing to thoroughly exploit the well-known energy estimates which balances the potential singularities. We also perform a new Bihari-LaSalle argument in this context. Eventually, we adapt our strategy to prove that ${\sup }_{t\in [0,T]}{\int }_0^t(t-s{)}^{-\theta }\mathrm{\parallel }{\mathrm{\nabla }}^{k}u(s,\cdot ){\mathrm{\parallel }}_{L^r}ds<+\mathrm{\infty }$, for all $\theta <\frac{3-kr}{2r}$, $k\in [0,2]$, and $1<r<\frac{3}{k}$.