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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.

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Fokker Planck equation


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

leading to the solution

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