Hao Jia

This group has two research objectives that require precise numerical calculations:

• The study of hydrodynamic stability of two and three dimensional flows. The main goal is to develop rigorous analytic methods to prove asymptotic stability of physically relevant incompressible flows, such as shear flows, vortices, vortex columns, Blasius boundary layers, and vortex filaments. The problem is challenging since the linearized operators are very complicated - they involve continuous spectrum, are not self-adjoint, and when viscosity is present, also involve strong singular perturbations. Numerical computations are essential in some of the more complicated problems, to calculate eigenvalues (which would then be rigorously validated using perturbation analysis and computer assited proof arguments).
• Well-posedness and regularity of Navier Stokes equations. This project has two fundamental goals: to determine the critical space in which the equation is well-posed, and to investigate scenarios where solution may become unbounded in a finite time. Numerical computations are essential in both problems, since these are large data problems and classical perturbation arguments are not enough to solve them. There are concrete calculation problems in both goals.