# Solutions of the Navier-Stokes equations

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The Navier-Stokes equations can be geometrized in the following form:

$$\dot{u} + \nabla_u u =\nu \Delta (u)+ df^*$$

$$d^* u=0$$

$\nabla_X Y$ is the connection $dY(X)$. If we define $u=\dot{\gamma}$, we recognize the equation of geodesics:

$$\nabla_{\dot{\gamma}}\dot{\gamma}=0$$

Can we solve the Navier-Stokes equations with help of a lagrangian formalism?

edited Nov 10

No. The Lagrangian formalism is just a tool to set up equations, not to solve them.

Moreover, the Navier-Stokes equations are dissipative, while the Lagrangian formalism produces conservative equations. Thus there is no useful Lagrnagian formulation of the Navier-Stokes equation.

This gives a second order equation for $\gamma$, hence with your substitution a first order equation for $u$. There is no natural way to get a dissipative equation from a Lagrangian. (One can get one - like for a damped harmonic oscillator - by doubling the fields, but then the resulting Hamiltonian has nothing to do with the energy.)

As noted, the Navier-Stokes equations contain something like the geodesic equations. With the help of a dissipation functional the NS equations can be reframed in a variational formalism, but I don't see how this gives new information about the solutions.

The NS equations are dissipative, but the energy isn't lost, simply it is transformed in heat. So, I propose to introduce a temperature parameter $T$ and to make a variational approach with a Lagrangian decomposed in a term depending on $u$ and another depending on the temperature $T$ which is a function of $u$.

You'd need to propose an actual Lagrangian that reproduces the Navier-Stokes equation. Just speculating that such a thing might exist is not enough for a discussion.

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