Analysis of several non-linear PDEs in fluid mechanics and differential geometry

Thesis

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.

Authors: Li, S

• DOI: 10.5287/ora-7qeze1v8g
• Type of award: DPhil
• Level of award: Doctoral
• Awarding institution: University of Oxford
• Language: English
• Keywords: Isometric Immersions; Partial Differential Equations (PDEs); Navier--Stokes Equations; Gauss--Codazzi--Ricci Equations; Euler Equations; Differential Geometry; Fluid Mechanics; Weak Continuity; Compensated Compactness; Vanishing Viscosity Limits
• Subjects: Mathematics