TY - JOUR

T1 - A determining form for the two-dimensional Navier-Stokes equations

T2 - The Fourier modes case

AU - Foias, Ciprian

AU - Jolly, Michael S.

AU - Kravchenko, Rostyslav

AU - Titi, Edriss S.

N1 - Funding Information:
The work of C.F. is supported in part by National Science Foundation (NSF) Grant No. DMS-1109784, that of M.J. by DMS-1008661 and DMS-1109638, that of R. K. by the ERC starting Grant No. GA 257110 RaWG, and that of E.S.T. by DMS-1009950, DMS-1109640, and DMS-1109645, as well as the Minerva Stiftung/Foundation. The authors would like to thank the referee for several suggestions on how to rearrange the paper.

PY - 2012/11/27

Y1 - 2012/11/27

N2 - The determining modes for the two-dimensional incompressible Navier-Stokes equations (NSE) are shown to satisfy an ordinary differential equation (ODE) of the form dv/dt = F(υ), in the Banach space, X, of all bounded continuous functions of the variable ε ℝ with values in certain finite-dimensional linear space. This new evolution ODE, named determining form, induces an infinite-dimensional dynamical system in the space X which is noteworthy for two reasons. One is that F is globally Lipschitz from X into itself. The other is that the long-term dynamics of the determining form contains that of the NSE; the traveling wave solutions of the determining form, i.e., those of the form υ(t, s) = υ0(t + s), correspond exactly to initial data v0 that are projections of solutions of the global attractor of the NSE onto the determining modes. The determining form is also shown to be dissipative; an estimate for the radius of an absorbing ball is derived in terms of the number of determining modes and the Grashof number (a dimensionless physical parameter).

AB - The determining modes for the two-dimensional incompressible Navier-Stokes equations (NSE) are shown to satisfy an ordinary differential equation (ODE) of the form dv/dt = F(υ), in the Banach space, X, of all bounded continuous functions of the variable ε ℝ with values in certain finite-dimensional linear space. This new evolution ODE, named determining form, induces an infinite-dimensional dynamical system in the space X which is noteworthy for two reasons. One is that F is globally Lipschitz from X into itself. The other is that the long-term dynamics of the determining form contains that of the NSE; the traveling wave solutions of the determining form, i.e., those of the form υ(t, s) = υ0(t + s), correspond exactly to initial data v0 that are projections of solutions of the global attractor of the NSE onto the determining modes. The determining form is also shown to be dissipative; an estimate for the radius of an absorbing ball is derived in terms of the number of determining modes and the Grashof number (a dimensionless physical parameter).

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U2 - 10.1063/1.4766459

DO - 10.1063/1.4766459

M3 - Article

AN - SCOPUS:84870495869

VL - 53

JO - Journal of Mathematical Physics

JF - Journal of Mathematical Physics

SN - 0022-2488

IS - 11

M1 - 115623

ER -