### Abstract

It has recently become common to study approximating equations for the Navier-Stokes equation. One of these is the Leray-α equation, which regularizes the Navier-Stokes equation by replacing (in most locations) the solution u with (1 -α^{2}Δ)u. Another is the generalized Navier-Stokes equation, which replaces the Laplacian with a Fourier multiplier with symbol of the form -|ξ|^{γ} (γ = 2 is the standard Navier-Stokes equation), and recently in [16] Tao also considered multipliers of the form -|ξ|^{γ} /g(|ξ|), where g is (essentially) alogarithm. The generalized Leray-a equation combines these two modifications by incorporating the regularizing term and replacing the Laplacians with more general Fourier multipliers, including allowing for g terms similar to those used in [16]. Our goal in this paper is to obtain existence and uniqueness results with low regularity and/or non-L^{2} initial data. We will also use energy estimates to extend some of these local existence results to global existence results.

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

Volume | 2015 |

State | Published - Jun 18 2015 |

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### All Science Journal Classification (ASJC) codes

- Analysis

### Cite this

**Local and global low-regularity solutions to generalized Leray-alpha equations.** / Pennington, Nathan.

Research output: Contribution to journal › Article

}

TY - JOUR

T1 - Local and global low-regularity solutions to generalized Leray-alpha equations

AU - Pennington, Nathan

PY - 2015/6/18

Y1 - 2015/6/18

N2 - It has recently become common to study approximating equations for the Navier-Stokes equation. One of these is the Leray-α equation, which regularizes the Navier-Stokes equation by replacing (in most locations) the solution u with (1 -α2Δ)u. Another is the generalized Navier-Stokes equation, which replaces the Laplacian with a Fourier multiplier with symbol of the form -|ξ|γ (γ = 2 is the standard Navier-Stokes equation), and recently in [16] Tao also considered multipliers of the form -|ξ|γ /g(|ξ|), where g is (essentially) alogarithm. The generalized Leray-a equation combines these two modifications by incorporating the regularizing term and replacing the Laplacians with more general Fourier multipliers, including allowing for g terms similar to those used in [16]. Our goal in this paper is to obtain existence and uniqueness results with low regularity and/or non-L2 initial data. We will also use energy estimates to extend some of these local existence results to global existence results.

AB - It has recently become common to study approximating equations for the Navier-Stokes equation. One of these is the Leray-α equation, which regularizes the Navier-Stokes equation by replacing (in most locations) the solution u with (1 -α2Δ)u. Another is the generalized Navier-Stokes equation, which replaces the Laplacian with a Fourier multiplier with symbol of the form -|ξ|γ (γ = 2 is the standard Navier-Stokes equation), and recently in [16] Tao also considered multipliers of the form -|ξ|γ /g(|ξ|), where g is (essentially) alogarithm. The generalized Leray-a equation combines these two modifications by incorporating the regularizing term and replacing the Laplacians with more general Fourier multipliers, including allowing for g terms similar to those used in [16]. Our goal in this paper is to obtain existence and uniqueness results with low regularity and/or non-L2 initial data. We will also use energy estimates to extend some of these local existence results to global existence results.

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M3 - Article

VL - 2015

JO - Electronic Journal of Differential Equations

JF - Electronic Journal of Differential Equations

SN - 1072-6691

ER -