Pressure as Lagrange multiplicator in Stokes and Euler equation

Navier-Stokes equation in $d=2,3$

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.

Claim 1 (Stokes equation)

The Stokes equation is the limit equation coming from the Navier-Stokes equation for $\varrho\to 0$

Then the Pressure $p$ is the Lagrange multiplicator coming from the incompressibility constraint.

Claim 2 (Euler equation)

The Euler equation is the limit equation coming from the Navier-Stokes equation for $\mu\to 0$ and $f\equiv 0$

Then the Pressure $p$ is the Lagrange multiplicator coming from the incompressibility constraint.

To proof both claims, we first appeal to the following auxilliary result:

Auxilliary Lemma (Dirichlet boundary conditions)

Suppose $g :\Omega \to \R^d$ satisfies

Then there exists $p:\Omega \to \R$ such that $g= - \nabla p$.

Proof of Claim 1

Variational principle behind the Stokes equation: The stokes equation is the Euler-Lagrange equation of the minimazation problem

where the minimum is taken among all

Let us calculate the first variation of the functional in the minimization problem. Therefore take $v:\bar \Omega\to \R^d$ incompressible with zero Dirichlet boundary values, then we have

results By the previous Lemma, there exists $p:\Omega\to \R$ such that

whch is nothing else than the Stokes equation.

For the proof of Claim 2, we use Arnolds interpretation of the Euler equation. For simplicity we restrict to the case $\Omega=\R^d$.

Arnolds interpretation of the Euler equation

$u$ is a solution of the Euler equation if and only if the associated flow $\Phi : (0,T)\times \R^d \to \R^d$ given by

is a stationary point of the action functional

subject to $\Phi(t=0,\cdot)=\text{Id}$, $\Phi(t=T,\cdot)=\Phi_T(\cdot)$ and

For the proof we again an auxillairy result, but with different boundary conditions.

Auxilliary Lemma (Neumann boundary conditions)

Suppose $g :\Omega \to \R^d$ satisfies

Then there exists $p:\Omega \to \R$ such that $g= - \nabla p$.

Proof of Claim 2

Step 1: Weak form of the euler equation. We reformulate the Euler equation by starting from the auxilliary Lemma with Neumann boundary conditions. With the same reasoning as in the proof of the Claim for the Stokes equation, we obtain that $u$ satisfies the Euler equation if and only if

Furthermore, this characterization is equivalent to the time-integrated one

The boundary conditions allow for easy integratation by parts. Let us first investigate the nonlinear term

where we used the incompressibility condition. Hence, together with the term from the time derivative we arrive at

This we call the weak form of the Euler equation.

Step 2: From Arnold to Euler. Let $v: [0,T]\times \Omega\to \R^d$ be an admissible test vectorfield, i.e. incompressible, Neumann boundary and vanishing for $t=0,T$. Therewith, we want to construct a variation $\tilde \Phi(s,t,y)$ of $\Phi(t,y)$. Therefore, we first define the flow $\Psi(s,t,z)$ generated by $v(t,z)$ as usual

Therewith we set

Let us check, that this is an admissible variation

$\tilde \Phi(0,t,y)= Phi(t,y)$ follows from $\Psi(0,t,z)=z$.
$\tilde \Phi(s,t=\{0,T\},y)= \Phi(t=\{0,T\},y)$ is a consequence of $\Psi(s,t=\{0,T\},z)=z$ following from $v(t=\{0,T\},\cdot)=0$.
$\det D\tilde \Phi(s,t,y)=1$ is a consequence of $D\Psi(s,t,z)=1$ following from $\nabla \cdot v=0$.

Hence $\tilde \Phi$ is an admissible variation and therefore by assumption that $\Phi$ is a stationary point, we obtain

The first observation is that $\partial_t \tilde\Phi(0,t,y)=\partial_t \Phi(t,y)=u(t,\Phi(t,y))$. Let us calculate the second factor in the scalar product