Electronic Journal of Differential Equations,
Vol. 2012 (2012), No. 89, pp. 1-19.
Title: Local and global existence for the Lagrangian Averaged Navier-Stokes
equations in Besov spaces
Author: Nathan Pennington (Kansas State Univ., Manhattan, KS, USA)
Abstract:
Through the use of a non-standard Leibntiz rule estimate, we prove the
existence of unique short time solutions to the incompressible,
iso\-tropic Lagrangian Averaged Navier-Stokes equation with initial
data in the Besov space $B^{r}_{p,q}(\mathbb{R}^n)$, $r>0$,
for $p>n$ and $n\geq 3$. When $p=2$, we obtain unique local
solutions with initial data in the Besov space $B^{n/2-1}_{2,q}(\mathbb{R}^n)$,
again with $n\geq 3$, 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
$1\leq q\leq \infty$. For $p=2$ and $n=4$, the local solution can be extended
to a global solution for $2\leq q\leq \infty$.
Since $B^s_{2,2}(\mathbb{R}^n)$ can be identified with the Sobolev space
$H^s(\mathbb{R}^n)$, this improves previous Sobolev space results,
which only held for initial data in $H^{3/4}(\mathbb{R}^3)$.
Submitted February 22, 2012. Published June 05, 2012.
Math Subject Classifications: 76D05, 35A02, 35K58.
Key Words: Navier-Stokes; Lagrangian averaging; global existence;
Besov spaces.