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Entire Solutions of Hydrodynamical Equations with Exponential Dissipation

Bardos, Claude and Frisch, Uriel and Pauls, Walter and Ray, Samriddhi Sankar and Titi, Edriss S (2010) Entire Solutions of Hydrodynamical Equations with Exponential Dissipation. In: Communications in Mathematical Physics, 293 (2). pp. 519-543.

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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(vertical bar k vertical bar/kd) at high wavenumbers vertical bar k vertical bar. 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(vertical bar k vertical bar/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.

Item Type: Journal Article
Additional Information: Copyright for this article belongs to Springer.
Department/Centre: Division of Physical & Mathematical Sciences > Physics
Date Deposited: 04 Dec 2009 05:23
Last Modified: 19 Sep 2010 05:53
URI: http://eprints.iisc.ernet.in/id/eprint/25066

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