Last Update: 16:52 - Wednesday - October 3^{rd} 2018.

In this work we consider the Navier-Stokes problem in $\mathbb{R}^N$:
$$
\begin{array}{l}
u_t=\Delta u - \nabla \pi + f(t) - (u\cdot \nabla) u, \quad x\in
\Omega\\
\hbox{div}(u)=0,\quad x\in \Omega \\
u=0, \quad x\in \partial \Omega\\
u(0,x)=u_0(x),
\end{array}
$$
where $u_0\in L^N(\Omega)^N$ and $\Omega$ is a bounded open subset of $\mathbb{R}^N$ with smooth boundary.
We prove that this problem is locally well posed and provide conditions to show that these solutions are defined for all $t\geq 0$.
We offer an interpretation for the problem of the Clay Mathematics Institute concerning the Navier-Stokes equations.

**Room:** Room 202 - Maths Department
**Date:** Thursday - October 18^{th}, 2018
**Time:** 14:00 to 15:00