\(\newcommand{\R}{\mathbb{R}}\)

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 \(u(t=0,\cdot)=u_0(\cdot)\).

**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 variational structure.

**Alternative:** Consider special cases: *Stokes* and *Euler equation*, which have a variational structure.

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