Navier-Stokes evolutions as self-dual variational problems

PDE Seminar
Tuesday, October 7, 2008 - 15:15
1.5 hours (actually 80 minutes)
Skiles 255
University of British Columbia, Canada
We describe how several nonlinear PDEs and evolutions ­including stationary and dynamic Navier-Stokes equations­ can be formulated and resolved variationally by minimizing energy functionalsof the form I(u) = L(u, -\Lambda u) + \langle \Lambda u, u\rangle and I(u) = \Int^T_0 [L(t, u(t), -\dot u(t) - \Lambda u(t)) + \langle\Lambda u(t), u(t)\rangle]dt + \ell (u(0) - u(T) \frac{u(T) + u(0)}{2} where L is a time-dependent "selfdual Lagrangian" on state space, is another selfdual "boundary Lagrangian", and is a nonlinear operator (such as \Lambda u = div(u \otimes u) in the Navier-Stokes case). However, just like the selfdual Yang-Mills equations, the equations are not obtained via Euler-Lagrange theory, but from the fact that a natural infimum is attained. In dimension 2, we recover the well known solutions for the corresponding initial-value problem as well as periodic and anti-periodic ones, while in dimension 3 we get Leray solutions for the initial-value problems, but also solutions satisfying u(0) = \alpha u(T ) for any given in (-1, 1). It is worth noting that our variational principles translate into Leray's energy identity in dimension 2 (resp., inequality in dimension 3). Our approach is quite general and does apply to many other situations.