Abstract
This thesis investigates the behavior of solutions to nonlinear partial differential equations (PDEs) in the presence of boundary conditions. In Chapter 1, some background knowledge of mathematical modes we considered is presented.Chapter 2 focuses on the incompressible magnetohydrodynamics (MHD) equations within a general three-dimensional bounded domain with a curved boundary. We employ general Navier-Slip boundary conditions for the velocity field and perfect conducting conditions for the magnetic field. By analyzing the anisotropic regularity of tangential and normal directions, we establish uniform estimates for conormal Sobolev norms and Lipschitz regularity. These results enable the rigorous justification of the vanishing dissipation limit as both the viscosity and magnetic diffusion coefficients approach zero.
Chapter 3 investigates the uniform regularity of solutions to the compressible MHD system in the half-space in small Alfvén and Mach numbers. We focus on the case where the magnetic field is tangential to the physical boundary, satisfying the perfect conducting boundary condition, while the velocity field adheres to a Navier-slip boundary condition. Under a critical condition that bounds the normal derivatives of the velocity and magnetic field uniformly in ε (the small Alfvén and Mach numbers ), we establish uniform estimates for the solutions in high-order conormal Sobolev norms. This condition ensures the absence of strong boundary layers as ε → 0.
In Chapter 4, we analyze the compressible Navier-Stokes-Poisson equations in a three-dimensional exterior domain with a non-uniform doping profile. By considering general Navier-slip boundary conditions, we establish the global existence of strong solutions near a steady-state configuration. This work extends previous results by incorporating non-flat doping profiles and more flexible boundary constraints.
| Date of Award | 19 Aug 2025 |
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| Original language | English |
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| Supervisor | Tao LUO (Supervisor) |