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Hello,

The Navier-Stokes equations can be written on a riemannian manifold: $$\nabla_u \dot{u}+\nabla_u u+ \Delta u=(df)^*$$ $$d^* u=0$$ where $\nabla$ is the Levi-Civita connection, $u$ is a vector fields, $\Delta$ is the laplacian, $df$ is the differential of $f$, $(df)^*$ is the dual of $df$ by the metric, $d^*u$ is the divergence of $u$.

The problem is due to Antoine Balan.

Do you have references ?

Thanks in advance.

1

# Navier-Stokes equations in riemannian geometry

Hello,

The Navier-Stokes equations can be written on a riemannian manifold: $$\nabla_u u+ \Delta u=(df)^*$$ $$d^* u=0$$ where $\nabla$ is the Levi-Civita connection, $u$ is a vector fields, $\Delta$ is the laplacian, $df$ is the differential of $f$, $(df)^*$ is the dual of $df$ by the metric, $d^*u$ is the divergence of $u$.

The problem is due to Antoine Balan.

Do you have references ?

Thanks in advance.