Well-Posedness and Long Time Behavior of Some Nonlinear Partial Differential Equations

The proposed projects aim to solve important problems for fundamental nonlinear partial differential equations from areas of mathematical physics, including the Navier Stokes equations and the nonlinear wave equations, which describe the flow of incompressible fluid and a wide variety of wave phenomenon respectively. In practice, the questions about solutions to these equations that we are interested in can be simple, such as what is the drag force of a boat with a given shape, or when nonlinear interference between different electro-magnetic waves becomes too serious. The answers to these simple questions can however be quite complicated, and usually depend on deep understanding of the underlying mathematical equations. These equations are nonlinear, for which our current understanding is still fundamentally incomplete. As is often the case, relatively few ``global quantities' are already sufficient to provide satisfactory control on the most interesting aspects of the complicated solutions. It is an ultimate goal in many theoretic studies of nonlinear partial differential equations to find these quantities and the mechanism through which they control the solutions. These quantities can play an essential role in guiding practical applications of these fundamental equations, such as in the design of numerical schemes to calculate the solutions, by allowing us to focus on relatively few important parameters, while ignoring large amount of other non-essential parameters.

The projects on Navier Stokes focus on the following problems. 1. The regularity of axi-symmetric solutions with small initial swirl component. Since the other components can still be large, compactness arguments are not sufficient to obtain regularity and suitable dynamical control on the solution is needed. 2. The spectral assumption related to large scale invariant solutions. Such spectral assumption appears naturally in the study of large scale invariant solutions in the non-perturbative regime, and has profound applications in the uniqueness problem of Leray-Hopf weak solutions. 3. Large distance asymptotics of steady state solutions. For such problems, it is well documented that nonlinearity is important even for small solutions. The projects on energy critical nonlinear wave equations are centered around the soliton resolution conjecture for various models. For the focusing energy critical wave equation and energy critical wave map equations, the PI aims to prove the soliton resolution conjecture along a sequence of times in the non-radial case, based on recent partial results. The PI also aims to establish full soliton resolution under certain additional conditions, such as in the case of one bubble concentration, or in the radial case. Another model the PI plans to study is the defocusing energy critical wave equation with a trapping potential, in the non-radial case. In this case, basic questions such as the ``ground state conjecture' are still open. More interestingly, it appears that one can now describe rigorously the generic and non-generic behavior of solutions in a non-perturbative regime. The PI plans to address some of these questions.