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

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4 Citations (Scopus)

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 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.

Original languageEnglish
Pages (from-to)102-116
Number of pages15
JournalNonlinear Analysis, Theory, Methods and Applications
Volume136
DOIs
StatePublished - May 1 2016

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Global Solution
Navier Stokes equations
Dissipation
Sobolev spaces
Term
Local Solution
Viscosity
Navier-Stokes Equations
Fourier multipliers
Generalized Equation
Sobolev Spaces
Regularity

All Science Journal Classification (ASJC) codes

  • Analysis
  • Applied Mathematics

Cite this

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title = "Global solutions to the generalized Leray-alpha equation with mixed dissipation terms",
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 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.",
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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.

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