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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. |
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Navier-Stokes equations in riemannian geometryHello, 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.
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