Electron. J. Diff. Equ., Vol. 2012 (2012), No. 89, pp. 1-19.

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

Nathan Pennington

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 5, 2012.
Math Subject Classifications: 76D05, 35A02, 35K58.
Key Words: Navier-Stokes; Lagrangian averaging; global existence; Besov spaces.

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Nathan Pennington
Department of Mathematics
Kansas State University
138 Cardwell Hall
Manhattan, KS 66506, USA
email: npenning@math.ksu.edu

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