Local and global existence for the Lagrangian Averaged Navier-Stokes equations in Besov spaces

Through the use of a non-standard Leibntiz rule estimate, we prove the existence of unique short time solutions to the incompressible, isotropic Lagrangian Averaged Navier-Stokes equation with initial data in the Besov space B ^r _p,q(ℝ ⁿ), r > 0, for p > n and n ≥ 3. When p = 2, we obtain unique local solutions with initial data in the Besov space B ^n/2-1 _2,q (ℝ ⁿ), again with n ≥ 3, which recovers the optimal regularity available by these methods for the Navier-Stokes equation. Also, when p = 2 and n = 3, the local solution can be extended to a global solution for all 1 ≤ q ≤ ∞. For p = 2 and n = 4, the local solution can be extended to a global solution for 2 ≤ q ≤ ∞. Since B ^s _2,2(ℝ ⁿ) can be identified with the Sobolev space H ^s(ℝ ⁿ), this improves previous Sobolev space results, which only held for initial data in H ^3/4(ℝ ³).