The Navier-Stokes equations have been around for more than 150 years. Physicists use them to model ocean currents, weather patterns and turbulent flows behind a commercial jet or ship. Mathematicians and physicists believe that an explanation for and the prediction of both the breeze and the turbulence can be found through an understanding of solutions to Navier-Stokes equations. Although most physicists and engineers believe that the smooth solutions of the Navier-Stokes equations will not break down without external forcing, currently there is no theoretical guarantee that this is indeed the case. Our recent study indicates that the Euler equations, a special case of the Navier-Stokes equations with zero viscosity, could develop a catastrophic behavior if one starts with a highly symmetric but perfectly smooth flow. Such a scenario corresponds to a perfect storm in which all things that could potentially go wrong indeed go wrong. Potentially singular behavior of the Navier-Stokes equations could post tremendous damage to our environment, affect the safety of our planes and ships, and our ability to do accurate weather forecasting. This award will investigate under what conditions the Euler or Navier-Stokes equations may develop singular behavior. The ultimate goal of this research is to develop effective analytical and computational tools that would enhance our ability to model and predict various complex phenomena in nature so that we can have more confidence in the safety of commercial jets and ships, and weather forecasting. Additional impact of this project will be the involvement of graduate students. The interdisciplinary training they receive in this project will be important for their future careers in mathematics and science.
The award seeks to understand whether the incompressible 3D Euler and Navier-Stokes equations could develop a finite-time singularity from a smooth initial condition with finite energy. A unique aspect of the research is the integration of highly resolved numerical simulations and rigorous mathematical analysis. Our strategy is to reformulate the problem of proving finite time self-similar singularity into the problem of establishing the nonlinear stability of an approximate self-similar profile using the dynamic rescaling equation. We first construct a highly accurate self-similar profile using a high order numerical method. We then use the energy method with carefully chosen singular weight functions and take into account cancellation among various nonlinear terms to extract the inviscid damping effect from the linearized operator around the approximate self-similar profile. Our stability result enables us to prove that the dynamic rescaling solution converges to the steady state self-similar solution exponentially fast in time. Moreover, by introducing a cut-off to the self-similar profile, we obtain a smooth initial condition with finite energy that develops a self-similar blowup in finite time. The novel approach of investigating finite-time singularity formation by studying the stability of spatial profiles in the potential singular solutions also forms the basis of a novel analytical framework for other nonlinear nonlocal systems of partial differential equations and has the potential to be applied to study a larger class of nonlinear dynamic problems.
This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.
