In the thesis we investigate two problems on Partial Differential Equations (PDEs) in differential geometry and fluid mechanics. First, we prove the weak L<sup> p</sup> continuity of the Gauss-Codazzi-Ricci (GCR) equations, which serve as a compatibility condition for the isometric immersions of Riemannian and semi-Riemannian manifolds. Our arguments, based on the generalised compensated compactness theorems established via functional and micro-local analytic methods, are intrinsic and global. Second, we prove the vanishing viscosity limit of an incompressible fluid in three-dimensional smooth, curved domains, with the kinematic and Navier boundary conditions. It is shown that the strong solution of the Navier-Stokes equation in H<sup> r+1</sup> (r > 5/2) converges to the strong solution of the Euler equation with the kinematic boundary condition in H<sup> r</sup>, as the viscosity tends to zero. For the proof, we derive energy estimates using the special geometric structure of the Navier boundary conditions; in particular, the second fundamental form of the fluid boundary and the vorticity thereon play a crucial role. In these projects we emphasise the linkages between the techniques in differential geometry and mathematical hydrodynamics.
| Identifer | oai:union.ndltd.org:bl.uk/oai:ethos.bl.uk:729067 |
| Date | January 2017 |
| Creators | Li, Siran |
| Contributors | Chen, Gui-Qiang G. |
| Publisher | University of Oxford |
| Source Sets | Ethos UK |
| Detected Language | English |
| Type | Electronic Thesis or Dissertation |
| Source | https://ora.ox.ac.uk/objects/uuid:20866cbb-e5ab-4a6b-b9dc-88a247d15572 |
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