# Pressure as Lagrange multiplicator in Stokes and Euler equation

$$\newcommand{\R}{\mathbb{R}}$$

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.

# weak L¹-convergence


# Classic results for porous medium equation

### Selfsimilar solutions


# Fokker Planck equation

### Description
