Introduction -- Stability to the global large solutions of the Navier-Stokes equations -- Global Smooth Solutions to the 2-D inhomogeneous Navier-Stokes Equations with Variable Viscosity -- On the decay and stability to global solutions of the 3-D inhomogeneous Navier-Stokes equations.

This thesis contains results of Dr. Guilong Gui during his PhD period with the aim to understand incompressible Navier-Stokes equations. It is devoted to the study of the stability to the incompressible Navier-Stokes equations. There is great potential for further theoretical and numerical research in this field. The techniques developed in carrying out this work are expected to be useful for other physical model equations. It is also hopeful that the thesis could serve as a valuable reference on current developments in research topics related to the incompressible Navier-Stokes equations. It was nominated by the Graduate University of Chinese Academy of Sciences as an outstanding Ph.D. thesis.