\(\newcommand{\R}{\mathbb{R}}\)
[stextbox id=”Theorem” caption=”Navier-Stokes equation in \(d=2,3\)”]
\[ \begin{aligned}
u &: [0,\infty) \times \Omega \to \R^d &\text{velocity (unknown)} & \text{ length / time } \\
p &: [0,\infty) \times \Omega \to \R^d &\text{pressure (unknown)} & \text{ force / area } \\
f &: [0,\infty) \times \Omega \to \R^d &\text{external volume force (data)} & \text{ force / volume } \\
\varrho &\in [0,\infty) &\text{mass density (data)} & \text{ mass / volume } \\
\mu&\in (0,\infty) &\text{dynamic viscosity (data)} & \text{ pressure \(\cdot\) time }
\end{aligned}\]
\[\begin{aligned}
\varrho \left( \partial_t u + u \cdot \nabla u \right) – \mu \Delta u + \nabla p &= f &&\text{in \((0,\infty)\times \Omega\)},\\
\nabla \cdot u &= 0 &&\text{in \((0,\infty)\times \Omega\)},\\
u &= 0 &&\text{on \((0,\infty)\times \partial \Omega\)}
\end{aligned}\]
and initial conditions \(u(t=0,\cdot)=u_0(\cdot)\).
[/stextbox]
Claim: The pressure \(p\) can be interpreted as Lagarange multiplicator that comes from incompressibility constraint \(\nabla \cdot u=0\).
Basis: Lagrange multiplicator are associated to variational problem.
Problem: Navier-Stokes equation has no known variational structure.
Alternative: Consider special cases: Stokes and Euler equation, which have a variational structure.