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 -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.
| Original language | English |
|---|---|
| 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.
In: Electronic Journal of Differential Equations, Vol. 2015, 18.06.2015.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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UR - http://www.scopus.com/inward/citedby.url?scp=84934987768&partnerID=8YFLogxK
M3 - Article
AN - SCOPUS:84934987768
VL - 2015
JO - Electronic Journal of Differential Equations
JF - Electronic Journal of Differential Equations
SN - 1072-6691
ER -