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 \({{{\rm e}^{|k|/k_{\rm d}}}}\) 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 \({{{\rm e}^{-C(k/k_{\rm d})\,{\rm ln}(|k|/k_{\rm d})}}}\) 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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Majda, A.J., Bertozzi, A.L.: Vorticity and Incompressible Flow. Cambridge Texts in Applied Mathematics. Cambridge: Cambridge University Press, 2001
Frisch U., Matsumoto T., Bec J.: Singularities of Euler flow? Not out of the blue!. J. Stat. Phys. 113, 761–781 (2003)
Bardos, C., Titi, E.S.: Euler equations of incompressible ideal fluids. Usp. Mat. Nauk 62, 5–46 (2007). English version Russ. Math. Surv. 62, 409–451 (2007)
Constantin P.: On the Euler equations of incompressible fluids. Bull. Amer. Math. Soc. 44, 603–621 (2007)
Eyink, G., Frisch, U., Moreau, R., Sobolevskiĭ, A.: Proceedings of “Euler Equations: 250 Years On”, (Aussois, June 18–23, 2007), Physica D 237(14–17) (2008)
Lions J.L.: Quelques Méthodes de Résolution des Problèmes aux Limites non Linéaires. Gauthier- Villars, Paris (1969)
Constantin, P., Foias, C.: Navier–Stokes Equations. Chicago Lectures in Mathematics. Chicago: University of Chicago Press, 1988
Fefferman, C.: Existence & smoothness of the Navier–Stokes equation. Millenium problems of the Clay Mathematics Institute (2000). Available at www.claymath.org/millennium/Navier-Stokes_Equations/Official_Problem_Description.pdf
Temam, R.: Navier–Stokes Equations. Theory and numerical analysis. Revised edition. With an appendix by F. Thomasset. Published by AMS Bookstore. Providence, RI: Amer. Math. Soc., 2001
Sohr H.: The Navier–Stokes Equations. Birkhäuser, Basel (2001)
Brachet M.-E., Meiron D.I., Orszag S.A., Nickel B.G., Morf R.H., Frisch U.: Small-scale structure of the Taylor-Green vortex, J. Fluid Mech. 130, 411–452 (1983)
von Neumann, J.: Recent theories of turbulence (1949), In: Collected works (1949–1963) 6, ed. A.H. Taub, New York: Pergamon Press, 1963, pp. 37–472
Li D., Sinai Ya.G.: Blow-ups of complex solutions of the 3D Navier–Stokes system and renormalization group method. J. Eur. Math. Soc. 10, 267–313 (2008)
Oliver M., Titi E.S.: On the domain of analyticity for solutions of second order analytic nonlinear differential equations. J. Diff. Eq. 174, 55–74 (2001)
Pauls W., Matsumoto T.: Lagrangian singularities of steady two-dimensional flow. Geophys. Astrophys. Fluid. Dyn. 99, 61–75 (2005)
Senouf D., Caflisch R., Ercolani N.: Pole dynamics and oscillation for the complex Burgers equation in the small-dispersion limit. Nonlinearity 9, 1671–1702 (1996)
Poláčik, P., Šverák, V.: Zeros of complex caloric functions and singularities of complex viscous Burgers equation. Preprint. 2008, http://arXiv.org/abs/math/0612506v1 [math.AP], 2006
Ladyzhenskaya O.A.: The Mathematical Theory of Viscous Incompressible Flow (1st ed). Gordon and Breach, New York (1963)
Holloway G.: Representing topographic stress for large-scale ocean models. J. Phys. Oceanogr. 22, 1033–1046 (1992)
Foias C., Temam R.: Gevrey class regularity for the solutions of the Navier–Stokes equations. J. Funct. Anal. 87, 359–369 (1989)
Ferrari A., Titi E.S.: Gevrey regularity for nonlinear analytic parabolic equations. Commun. Part. Diff. Eq. 23, 1–16 (1998)
Agmon, S.: Lectures on Elliptic Boundary Value Problems. Mathematical Studies, Princeton, NJ: Van Nostrand, 1965
Cao C., Rammaha M., Titi E.S.: The Navier–Stokes equations on the rotating 2–D sphere: Gevrey regularity and asymptotic degrees of freedom. Zeits. Ange. Math. Phys. (ZAMP) 50, 341–360 (1999)
Doelman A., Titi E.S.: Regularity of solutions and the convergence of the Galerkin method in the Ginzburg–Landau equation. Num. Funct. Anal. Optim. 14, 299–321 (1993)
Masuda K.: On the analyticity and the unique continuation theorem for solutions of the Navier–Stokes equation. Proc. Japan Acad. 43, 827–832 (1967)
Doering C.R., Titi E.S.: Exponential decay rate of the power spectrum for solutions of the Navier–Stokes equations. Phys. Fluids 7, 1384–1390 (1995)
Gottlieb D., Orszag S.A.: Numerical Analysis of Spectral Methods. SIAM, Philadelphia (1977)
Frisch U., She Z.S., Thual O.: Viscoelastic behaviour of cellular solutions to the Kuramoto-Sivashinsky model. J. Fluid Mech. 168, 221–240 (1986)
Cox C.M., Matthews P.C.: Exponential time differencing for stiff systems. J. Comput. Phys. 76, 430–455 (2002)
Hoeven J.: On asymptotic extrapolation. J. Symb. Comput. 44, 1000–1016 (2009)
Ecalle, J.: Introduction aux Fonctions Analysables et Preuve Constructive de la Conjecture de Dulac. Actualités mathématiques. Paris: Hermann, 1992
van der Hoeven, J.: Transseries and Real Differential Algebra. Lecture Notes in Math. 1888, Berlin: Springer, 2006
Pauls W., Frisch U.: A Borel transform method for locating singularities of Taylor and Fourier series. J. Stat. Phys. 127, 1095–1119 (2007)
Sinai Ya.G.: Diagrammatic approach to the 3D Navier-Stokes system. Russ. Math. Surv. 60, 849–873 (2005)
Caflisch R.E.: Singularity formation for complex solutions of the 3D incompressible Euler equations. Physica D 67, 1–18 (1993)
Kraichnan R.H.: Intermittency in the very small scales of turbulence. Phys. Fluids 10, 2080–2082 (1967)
Frisch U., Morf R.: Intermittency in nonlinear dynamics and singularities at complex times. Phys. Rev. A 10, 2673–2705 (1981)
Frisch U., Kurien S., Pandit R., Pauls W., Ray S.S., Wirth A., Zhu J.-Z.: Hyperviscosity, Galerkin truncation and bottlenecks in turbulence. Phys. Rev. Lett. 101, 144501 (2008)
Wirth, A. Private communication, 1996
Zhu, J.-Z. Private communication, 2008
Orszag, S.A. Private communication, 1979
Acknowledgments
We thank J.-Z. Zhu and A. Wirth for important input and M. Blank, K. Khanin, B. Khesin and V. Zheligovsky for many remarks. CB acknowledges the warm hospitality of the Weizmann Institute and SSR that of the Observatoire de la Côte d’Azur, places where parts of this work were carried out. The work of EST was supported in part by the NSF grant No. DMS-0708832 and the ISF grant No. 120/06. SSR thanks R. Pandit, D. Mitra and P. Perlekar for useful discussions and acknowledges DST and UGC (India) for support and SERC (IISc) for computational resources. UF, WP and SSR were partially supported by ANR “OTARIE” BLAN07-2_183172 and used the Mésocentre de calcul of the Observatoire de la Côte d’Azur for computations.
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Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
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Bardos, C., Frisch, U., Pauls, W. et al. Entire Solutions of Hydrodynamical Equations with Exponential Dissipation. Commun. Math. Phys. 293, 519–543 (2010). https://doi.org/10.1007/s00220-009-0916-z
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DOI: https://doi.org/10.1007/s00220-009-0916-z