### Abstract

Due to the intractability of the Navier-Stokes equation, it is common to study approximating equations. Two of the most common of these are the Leray-α equation (which replaces the solution u with (1-^{α2} _{L1})u for a Fourier Multiplier L) and the generalized Navier-Stokes equation (which replaces the viscosity term νδ with ν_{L2}). In this paper we consider the combination of these two equations, called the generalized Leray-α equation. We provide a brief outline of the typical strategies used to solve such equations, and prove, with initial data in a low-regularity ^{Lp}(^{Rn}) based Sobolev space, the existence of a unique local solution with _{γ1}+_{γ2}>n/p+1. In the p=2 case, the local solution is extended to a global solution, improving on previously known results.

Original language | English |
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Pages (from-to) | 102-116 |

Number of pages | 15 |

Journal | Nonlinear Analysis, Theory, Methods and Applications |

Volume | 136 |

DOIs | |

State | Published - May 1 2016 |

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### All Science Journal Classification (ASJC) codes

- Analysis
- Applied Mathematics

### Cite this

**Global solutions to the generalized Leray-alpha equation with mixed dissipation terms.** / Pennington, Nathan.

Research output: Contribution to journal › Article

}

TY - JOUR

T1 - Global solutions to the generalized Leray-alpha equation with mixed dissipation terms

AU - Pennington, Nathan

PY - 2016/5/1

Y1 - 2016/5/1

N2 - Due to the intractability of the Navier-Stokes equation, it is common to study approximating equations. Two of the most common of these are the Leray-α equation (which replaces the solution u with (1-α2 L1)u for a Fourier Multiplier L) and the generalized Navier-Stokes equation (which replaces the viscosity term νδ with νL2). In this paper we consider the combination of these two equations, called the generalized Leray-α equation. We provide a brief outline of the typical strategies used to solve such equations, and prove, with initial data in a low-regularity Lp(Rn) based Sobolev space, the existence of a unique local solution with γ1+γ2>n/p+1. In the p=2 case, the local solution is extended to a global solution, improving on previously known results.

AB - Due to the intractability of the Navier-Stokes equation, it is common to study approximating equations. Two of the most common of these are the Leray-α equation (which replaces the solution u with (1-α2 L1)u for a Fourier Multiplier L) and the generalized Navier-Stokes equation (which replaces the viscosity term νδ with νL2). In this paper we consider the combination of these two equations, called the generalized Leray-α equation. We provide a brief outline of the typical strategies used to solve such equations, and prove, with initial data in a low-regularity Lp(Rn) based Sobolev space, the existence of a unique local solution with γ1+γ2>n/p+1. In the p=2 case, the local solution is extended to a global solution, improving on previously known results.

UR - http://www.scopus.com/inward/record.url?scp=84959525624&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84959525624&partnerID=8YFLogxK

U2 - 10.1016/j.na.2016.02.006

DO - 10.1016/j.na.2016.02.006

M3 - Article

VL - 136

SP - 102

EP - 116

JO - Nonlinear Analysis, Theory, Methods and Applications

JF - Nonlinear Analysis, Theory, Methods and Applications

SN - 0362-546X

ER -