Aspects of well-possedeness and long time behavior for non-linear PDEs

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Mathematical description of fluid flows is mostly based on partial differential equations. These equations express well-known physical laws in the context of fluids and describe how quantities characterizing the flow, such as the velocity and the pressure will change in time and space. It turns out the equations are difficult to solve, even with the help of large computers. One reason for the difficulties comes from the highly non-trivial behavior exhibited by the solutions, which includes the emergence of complicated small-scale structures and fast oscillations in time. This is a result of the non-linearity in the equations, which can transfer energy between various scales. The fluid motion can be complicated, but the practical questions are in some sense simple: will a tornado form? At which speed will a plane stall? Open theoretical questions about the equations can also be formulated in a relatively simple language: do the equations give a self-consistent description of the fluid evolution, in the sense that they can uniquely predict the future state of the fluid based on a known current state? This is one of the well-known open mathematical problems surrounding the equations, closely related to the possible development of singularities in the solutions. Our mathematical understanding of the equations is currently incomplete. The research at both theoretical and practical aspects of the equation has ultimately the same goal: to find some relatively simple set of parameters which control the solutions. One hopes that by identifying the right quantities, one will be able to give a good description of the flow and sufficiently characterize its important features. The main effort of this research project is aimed at several open mathematical problems surrounding these issues.At a more technical level, the proposed topics include:1. Well-posedness, ill-posedness and uniqueness for the Navier-Stokes equation and related equations, questions such as: Is the Navier-Stokes equation well-posed in the natural energy space? Based on recent work concerning scale-invariant solutions, the PI expects that the answer to this questions is negative, but significant work is still needed to confirm this. (The question is also related to the open problem of uniqueness of the Leray-Hopf weak solutions with initial date of finite energy, but not necessarily smooth.) The proposed methods should also work for the surface quasi-geostrophic equation, where similar questions are open. The Euler equation also presents a number of open problem concerning well-posedness and stability, some of which will be addressed. Many issues are not clear even at the linearized level. These are in some sense more subtle than in the Navier-Stokes case (due to the strong role of continuous spectra), but should provide valuable insights into low-viscosity flows.2. Singularities and possible non-uniqueness for the complex Ginzburg-Landau equation. This equation has the same energy estimates and the same scaling symmetry as the Navier-Stokes equation. There is strong evidence that solution can develop singularities even when starting from smooth initial conditions. One can develop a theory of global weak solutions, but it remains open whether these uniquely predict the behavior of the system. The question of uniqueness is important for assessing the predictive power of the equation.3. Long-time behavior of solutions for the 2d Euler equation and PDE problems associated with models used in that connection. The long-time behavior of 2d flows (relevant for example for modelling of meteorological phenomena and making predictions concerning climate) exhibits some striking features whose mathematical understanding remains incomplete. There are strong connections to Statistical Mechanics and other infinite-dimensional Hamiltonian PDEs. The research will address some of the open PDE problems arising in this context, such as the properties of invariant measures in flows with stochastic forcing and properties of steady-states arising from statistical theories. Other Hamiltonian PDEs which can serve as good models for these questions will also be studied.