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

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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 languageEnglish
JournalElectronic Journal of Differential Equations
Volume2012
StatePublished - Jun 5 2012
Externally publishedYes

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Local Existence
Local Solution
Besov Spaces
Global Existence
Navier-Stokes Equations
Global Solution
Sobolev Spaces
Regularity
Estimate

All Science Journal Classification (ASJC) codes

  • Analysis

Cite this

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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).",
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AU - Pennington, Nathan

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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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