Relaxation time, Damping, and Blowup of Solutions to an Equation from Fluid Dynamics

The incompressible Navier-Stokes equations govern the evolution of fluids under the influence of viscosity. Their inviscid counterpart, the incompressible Euler equations, are used when the fluid in question has negligible viscosity. The fundamental problem of whether solutions of either set of equations with smooth initial data cease to be smooth at some fine time, known as the finite-time blowup problem, is currently open and of great interest in mathematical physics and engineering, among other areas. Additionally, the ability of the viscosity coefficient in the Navier-Stokes equa tions to suppress or even delay a potential Euler blowup is also unknown. In this presentation, we compare special solutions of the Euler equations with and without damping. The special structure of the solutions considered reduce a system of partial differential equations (PDE’s) to a single PDE which we can be solved explicitly along Lagrangian trajectories. Using this solution formula, we show that a reciprocal relationship exists between the damping coefficient, the undamped blowup time, and the relaxation time of the system which determines whether finite-time blowup occurs in the associated damped solution.