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 -