Entire solutions of hydrodynamical equations with exponential dissipation
Bardos, C., Frisch, U., Pauls, W., Ray, S.S. and Titi, E.S.

Sunday 28 November 2010 by Ponty Yannick

 Abstract: We consider amodification of the three-dimensional Navier–Stokes equations
and other hydrodynamical evolution equations with space-periodic initial conditions in
which the usual Laplacian of the dissipation operator is replaced by an operator whose
Fourier symbol grows exponentially as e|k|/kd at high wavenumbers |k|. Using estimates
in suitable classes of analytic functions, we show that the solutions with initially finite
energy become immediately entire in the space variables and that the Fourier coefficients
decay faster than exp(−C(k/kd) ln(|k|/kd)) for any C < 1/(2 ln 2). The same result holds for the
one-dimensional Burgers equation with exponential dissipation but can be improved:
heuristic arguments and very precise simulations, analyzed by the method of asymptotic
extrapolation of van der Hoeven, indicate that the leading-order asymptotics is precisely
of the above form with C = C* = 1/ ln 2. The same behavior with a universal constant
C* is conjectured for the Navier–Stokes equations with exponential dissipation in any
space dimension. This universality prevents the strong growth of intermittency in the
far dissipation range which is obtained for ordinary Navier–Stokes turbulence. Possible
applications to improved spectral simulations are briefly discussed.

 

 Bardos, C., Frisch, U., Pauls, W., Ray, S.S. and Titi, E.S., "Entire solutions of hydrodynamical equations with exponential dissipation", Communications in Mathematical Physics, 293, Issue 2, pp. 519-543 (2010) (doi:10.1007/s00220-009-0916-z)