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Navier-Stokes equations on Riemannian manifolds

  • Technology and innovations

Master of Science Maryam Samavaki’s thesis presents some new solutions regarding the Navier-Stokes equations on Riemannian manifolds. The used methods are based on linearization.

Navier-Stokes equations describe flows, such as the flow of liquid in a pipe, flow of air around an airplane wing, ocean currents, and weather. The equations express how the movement of a single particle varies depending on the pressure and internal viscous forces of the fluid. The Clay Mathematics Institute has declared a reward of one million dollars regarding a question about Navier-Stokes equations and turbulence.

Riemannian manifold is a mathematically more general setting to study flows than a physical surface or space. In a Riemannian manifold, the concepts are independent of coordinates and the distance function (metric) can be more general. A Killing vector field is a mapping defined on a Riemannian manifold, which preserves distances. Ricci tensor describes, roughly, how much the geometry on the manifold differs from Euclidean space.

In the thesis, linearizing is used to provide many new results and then to present more solutions with the Lie bracket of the Killing fields for linearized Navier-Stokes equations. The equations for the Killing and conformal Killing vector fields which are overdetermined systems of PDE can be formulated as an eigenvalue problem. Moreover, analyzing several classes of Riemannian manifolds which are described by first imposing certain conditions on the Ricci tensor can be interpreted as overdetermined PDE systems whose unknowns are the Riemannian metric components.

The doctoral dissertation of Master of Science Maryam Samavaki, entitled Navier-Stokes equations on Riemannian manifolds will be examined at the Faculty of Science and Forestry on the 29th of May (online). The opponent in the public examination will be Professor Mikko Salo, University of Jyväskylä, Finland, and the custos will be Professor Jukka Tuomela, University of Eastern Finland. The public examination will be held in English.

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