### Abstract

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}).

Original language | English |
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Journal | Electronic Journal of Differential Equations |

Volume | 2012 |

State | Published - Jun 5 2012 |

Externally published | Yes |

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### All Science Journal Classification (ASJC) codes

- Analysis

### Cite this

**Local and global existence for the Lagrangian Averaged Navier-Stokes equations in Besov spaces.** / Pennington, Nathan.

Research output: Contribution to journal › Article

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TY - JOUR

T1 - Local and global existence for the Lagrangian Averaged Navier-Stokes equations in Besov spaces

AU - Pennington, Nathan

PY - 2012/6/5

Y1 - 2012/6/5

N2 - 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).

AB - 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).

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M3 - Article

VL - 2012

JO - Electronic Journal of Differential Equations

JF - Electronic Journal of Differential Equations

SN - 1072-6691

ER -