A determining form for the two-dimensional Navier-Stokes equations: The Fourier modes case

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2012

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American Institute of Physics

Abstract

The determining modes for the two-dimensional incompressible Navier-Stokes equations (NSE) are shown to satisfy an ordinary differential equation (ODE) of the form $dv/dt = F(v)$, in the Banach space, $X$, of all bounded continuous functions of the variable $s\in\mathbb{R}$ 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 $v(t, s) = v_{0}(t + s)$, correspond exactly to initial data $v_{0}$ 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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Foias, C., Jolly, M. S., Kravchenko, R., & Titi, E. S. (2012). A determining form for the two-dimensional Navier-Stokes equations: The Fourier modes case. Journal of Mathematical Physics, 53(11), 115623. http://dx.doi.org/10.1063/1.4766459

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© 2012 American Institute of Physics

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Article