TY - JOUR

T1 - Asymptotic integration of Navier-Stokes equations with potential forces. II. An explicit Poincaré-Dulac normal form

AU - Foias, Ciprian

AU - Hoang, Luan

AU - Saut, Jean Claude

N1 - Funding Information:
L.H. acknowledges the support by NSF Grant DMS-0908177.

PY - 2011/5/15

Y1 - 2011/5/15

N2 - We study the incompressible Navier-Stokes equations with potential body forces on the three-dimensional torus. We show that the normalization introduced in the paper [C. Foias, J.-C. Saut, Linearization and normal form of the Navier-Stokes equations with potential forces, Ann. Inst. H. Poincaré Anal. Non Linéaire 4 (1) (1987) 1-47], produces a Poincaré-Dulac normal form which is obtained by an explicit change of variable. This change is the formal power series expansion of the inverse of the normalization map. Each homogeneous term of a finite degree in the series is proved to be well-defined in appropriate Sobolev spaces and is estimated recursively by using a family of homogeneous gauges which is suitable for estimating homogeneous polynomials in infinite dimensional spaces.

AB - We study the incompressible Navier-Stokes equations with potential body forces on the three-dimensional torus. We show that the normalization introduced in the paper [C. Foias, J.-C. Saut, Linearization and normal form of the Navier-Stokes equations with potential forces, Ann. Inst. H. Poincaré Anal. Non Linéaire 4 (1) (1987) 1-47], produces a Poincaré-Dulac normal form which is obtained by an explicit change of variable. This change is the formal power series expansion of the inverse of the normalization map. Each homogeneous term of a finite degree in the series is proved to be well-defined in appropriate Sobolev spaces and is estimated recursively by using a family of homogeneous gauges which is suitable for estimating homogeneous polynomials in infinite dimensional spaces.

KW - Homogeneous gauge

KW - Navier-Stokes equations

KW - Nonlinear dynamics

KW - Poincaré-Dulac normal form

UR - http://www.scopus.com/inward/record.url?scp=79952105412&partnerID=8YFLogxK

U2 - 10.1016/j.jfa.2011.02.005

DO - 10.1016/j.jfa.2011.02.005

M3 - Article

AN - SCOPUS:79952105412

VL - 260

SP - 3007

EP - 3035

JO - Journal of Functional Analysis

JF - Journal of Functional Analysis

SN - 0022-1236

IS - 10

ER -