# 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.

# Classic results for porous medium equation

### Selfsimilar solutions

 Let us seek for solutions of  satisfying the scaling hypothesis

where  and  are reparametrization of time and . The prefactor  ensures that  conserves mass, i.e.

 The Fokker Planck equation has the form
where  is a smooth function,  some parameter and  a probability density on . The partial differential equation is in divergence form and conserves mass. Hence, also  is a probability density on . In the case, where  has some growth at , the equilibrium solutions  are characterized by