Mathematics Colloquia and Seminars

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I will begin with a short introduction to the Navier-Stokes equations for incompressible fluid motion and discuss well-posedness results for classical and weak solutions. In turbulent regimes, it is reasonable to study a statistical theory of turbulence and the heart of my talk will be on the anisotropic Lagrangian averaged Navier-Stokes equations recently developed by Marsden and Shkoller. These equations are a coupled system of PDEs for the mean velocity field and Lagrangian covariance tensor designed to capture the dynamics of the Navier-Stokes equations at length scales larger than a parameter \alpha, while averaging the motion at scales smaller than \alpha. I will provide a short introduction to the anisotropic Lagrangian averaged Navier-Stokes equation, briefly review previous analytical results, and outline the proof of local-in-time well-posedness of solutions to these equations when the viscosity term is of a certain form. In addition, with time remaining, I will present numerical solutions to the equations assuming steady channel flow in the two cases of no-slip and inhomogeneous boundary conditions.