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 , , for and . When , we obtain unique local solutions with initial data in the Besov space , again with , which recovers the optimal regularity available by these methods for the Navier-Stokes equation. Also, when and , the local solution can be extended to a global solution for all . For and , the local solution can be extended to a global solution for . Since can be identified with the Sobolev space , this improves previous Sobolev space results, which only held for initial data in .
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
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