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Title: Analysis of several non-linear PDEs in fluid mechanics and differential geometry
Author: Li, Siran
ISNI:       0000 0004 6498 5566
Awarding Body: University of Oxford
Current Institution: University of Oxford
Date of Award: 2017
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In the thesis we investigate two problems on Partial Differential Equations (PDEs) in differential geometry and fluid mechanics. First, we prove the weak L p 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 r+1 (r > 5/2) converges to the strong solution of the Euler equation with the kinematic boundary condition in H r, 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.
Supervisor: Chen, Gui-Qiang G. Sponsor: Not available
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral
EThOS ID:  DOI: Not available
Keywords: Mathematics ; Euler Equations ; Gauss--Codazzi--Ricci Equations ; Isometric Immersions ; Differential Geometry ; Navier--Stokes Equations ; Compensated Compactness ; Weak Continuity ; Vanishing Viscosity Limits ; Partial Differential Equations (PDEs) ; Fluid Mechanics