Use this URL to cite or link to this record in EThOS: | https://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.729067 |
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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
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Awarding Body: | University of Oxford | ||||||
Current Institution: | University of Oxford | ||||||
Date of Award: | 2017 | ||||||
Availability of Full Text: |
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Abstract: | |||||||
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.
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Supervisor: | Chen, Gui-Qiang G. | Sponsor: | Not available | ||||
Qualification Name: | Thesis (Ph.D.) | Qualification Level: | Doctoral | ||||
EThOS ID: | uk.bl.ethos.729067 | 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 | ||||||
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