Invariant Manifolds and the Long-Time Asymptotics of the Navier-Stokes and Vorticity Equations on R2

Abstract We construct finite-dimensional invariant manifolds in the phase space of the Navier-Stokes equation on R2 and show that these manifolds control the long-time behavior of the solutions. This gives geometric insight into the existing results on the asymptotics of such solutions and also allows us to extend those results in a number of ways.

[1]  J. Cooper.  SINGULAR INTEGRALS AND DIFFERENTIABILITY PROPERTIES OF FUNCTIONS , 1973 .

[2]  Maria E. Schonbek,et al.  L2 decay for weak solutions of the Navier-Stokes equations , 1985 .

[3]  J. Eckmann,et al.  Geometric Stability Analysis for Periodic Solutions of the Swift-Hohenberg Equation , 1996, patt-sol/9612002.

[4]  Y. Giga,et al.  Large time behavior of the vorticity of two-dimensional viscous flow and its application to vortex formation , 1988 .

[5]  Tosio Kato.  The Navier-Stokes equation for an incompressible fluid in ${\bf R}^2$ with a measure as the initial vorticity , 1994, Differential and Integral Equations.

[6]  乔花玲,et al.  关于Semigroups of Linear Operators and Applications to Partial Differential Equations的两个注解 , 2003 .

[7]  T. Miyakawa,et al.  On optimal decay rates for weak solutions to the Navier-Stokes equations in $R^n$ , 2001 .

[8]  Yoshikazu Giga,et al.  Two-dimensional Navier-Stokes flow with measures as initial vorticity , 1988 .

[9]  Daniel B. Henry.  Geometric Theory of Semilinear Parabolic Equations , 1989 .

[10]  E. Titi,et al.  Remark on the Rate of Decay of Higher Order Derivatives for Solutions to the Navier–Stokes Equations in Rn☆ , 2000 .

[11]  H. Brezis.  Remarks on the preceding paper by M. Ben-Artzi “Global solutions of two-dimensional Navier-Stokes and Euler equations” , 1994 .

[12]  E. Stein.  Singular Integrals and Di?erentiability Properties of Functions , 1971 .

[13]  M. Cannone,et al.  Self-similar solutions for navier-stokes equations in , 1996 .

[14]  Thierry Gallay,et al.  A center-stable manifold theorem for differential equations in Banach spaces , 1993 .

[15]  P. Lax.  Remarks on the preceding paper , 1957 .

[16]  A. Carpio.  Asymptotic behavior for the vorticity equations in dimensions two and three , 1994 .

[17]  Ana Rodríguez.  Large-time behavior in incompressible Navier-Stokes equations , 1996 .

[18]  M. Ben-Artzi.  Global solutions of two-dimensional Navier-Stokes and euler equations , 1994 .

[19]  C. Eugene Wayne,et al.  Invariant Manifolds for Parabolic Partial Differential Equations on Unbounded Domains , 1997 .

[20]  Geneviève Raugel,et al.  Scaling Variables and Asymptotic Expansions in Damped Wave Equations , 1998 .

[21]  K. Masuda.  Weak solutions of Navier-Stokes equations , 1984 .

[22]  T. Miyakawa,et al.  Asymptotic profiles of nonstationary incompressible Navier–Stokes flows in the half-space , 2001 .

[23]  Long-time asymptotics of the Navier-Stokes and vorticity equations on R^3 , 2001 .

[24]  Tetsuro Miyakawa,et al.  OnL2 decay of weak solutions of the Navier-Stokes equations inRn , 1986 .

[25]  H. Beirao da Veiga,et al.  Existence and Asymptotic Behavior for Strong Solutions of Navier-Stokes Equations in the Whole Space , 1985 .

[26]  Takashi Kato,et al.  StrongLp-solutions of the Navier-Stokes equation inRm, with applications to weak solutions , 1984 .

[27]  M. Cannone.  Self-Similar Solutions for Navier-Stokes Equations in R 3 , 2022 .

[28]  Non-Linear Stability Analysis of Higher Order Dissipative Partial Differential Equations , 1998, math-ph/9805014.

[29]  Michael Wiegner,et al.  Decay Results for Weak Solutions of the Navier–Stokes Equations on Rn , 1987 .

[30]  Maria E. Schonbek,et al.  Lower bounds of rates of decay for solutions to the Navier-Stokes equations , 1991 .

[31]  Daniel J. Amit,et al.  Asymptotic behavior of the , 1983 .