RESOLVENT ESTIMATES FOR A TWO-DIMENSIONAL NON-SELF-ADJOINT OPERATOR

We consider a two-dimensional non-self-adjoint differential operator, originated from a stability problem in the two-dimensional Navier-Stokes equation, given by L-alpha = -Delta + vertical bar x vertical bar(2) + alpha sigma(vertical bar x vertical bar)partial derivative(theta), where sigma(r) = r-2(1 - e(-r2)), partial derivative(theta) = x(1)partial derivative(2) - x(2)partial derivative(1) and alpha is a positive parameter tending to + infinity. We give a complete study of the resolvent of L-alpha along the imaginary axis in the fast rotation limit alpha -> + infinity and we prove sup(lambda is an element of R) parallel to(L-alpha - i lambda)(-1)parallel to (L((L) over tilde2 (R2))) <= C alpha(-1/3), which is an optimal estimate. Our proof is based on a multiplier method, metrics on the phase space and localization techniques.