Entire Solutions of Hydrodynamical Equations with Exponential Dissipation |
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Authors: | Claude Bardos Uriel Frisch Walter Pauls Samriddhi Sankar Ray Edriss S Titi |
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Institution: | (1) Mathematical Institute, University of Oxford, 24-29 St Giles’, Oxford, OX1 3LB, UK |
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Abstract: | We consider a modification 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{{{\rm e}^{|k|/k_{\rm d}}}} 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 e-C(k/kd) ln(|k|/kd){{{\rm e}^{-C(k/k_{\rm d})\,{\rm ln}(|k|/k_{\rm d})}}} 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. |
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