Turbulence for the generalised Burgers equation

A. A. Boritchev

Trudy Matematicheskogo Instituta imeni V.A. Steklova, Tome 69 (2014) no. 6, pp. 957-994

Résumé

This survey reviews rigorous results obtained by A. Biryuk and the author on turbulence for the generalised space-periodic Burgers equation $$ u_t+f'(u)u_x=\nu u_{xx}+\eta,\qquad x \in S^1=\mathbb{R}/\mathbb{Z}, $$ where $f$ is smooth and strongly convex, and the constant $0\nu\ll 1$ corresponds to the viscosity coefficient. Both the unforced case ($\eta=0$) and the case when $\eta$ is a random force which is smooth with respect to $x$ and irregular (kick or white noise) with respect to $t$ are considered. In both cases sharp bounds of the form $C\nu^{-\delta}$, $\delta\geqslant 0$, are obtained for the Sobolev norms of $u$ averaged over time and over the ensemble, with the same value of $\delta$ for upper and lower bounds. These results yield sharp bounds for small-scale quantities characterising turbulence, confirming the physical predictions. Bibliography: 56 titles.

Détail

Keywords: Burgers equation, stochastic partial differential equations, intermittency, stationary measure.
Mots-clés : turbulence

Bibliographie

[1] R. A. Adams, Sobolev spaces, Pure Appl. Math., 65, Academic Press, New York–London, 1975, xviii+268 pp. | MR | Zbl

[2] E. Aurell, U. Frisch, J. Lutsko, M. Vergassola, “On the multifractal properties of the energy dissipation derived from turbulence data”, J. Fluid Mech., 238 (1992), 467–486 | DOI | MR | Zbl

[3] Y. Bakhtin, “Ergodic theory of the Burgers equation with random force”, St. Petersburg summer school in probability and statistical physics (St. Petersburg, 2012), eds. V. Sidoravicius, S. Smirnov (to appear)

[4] G. K. Batchelor, The theory of homogeneous turbulence, Cambridge Monogr. Mech. Appl. Math., Cambridge Univ. Press, Cambridge, 1953, x+197 pp. | MR | Zbl

[5] H. Bateman, “Some recent researches on the motion of fluids”, Mon. Wea. Rev., 1915, no. 43, 163–170 | 2.0.CO;2 class='badge bg-secondary rounded-pill ref-badge extid-badge'>DOI

[6] J. Bec, K. Khanin, “Burgers turbulence”, Phys. Rep., 447:1-2 (2007), 1–66 | DOI | MR

[7] A. È. Biryuk, “Spectral properties of solutions of the Burgers equation with small dissipation”, Funct. Anal. Appl., 35:1 (2001), 1–12 | DOI | DOI | MR | Zbl

[8] A. Biryuk, Note on the transformation that reduces the Burgers equation to the heat equation, Mathematical Physics Preprint Archive, MP-ARC-2003-370, 2003, 4 pp.

[9] A. Boritchev, “Decaying turbulence in the generalised Burgers equation”, Arch. Ration. Mech. Anal., 214:1 (2014), 331–357 ; (2012 (v4 – 2014)), 28 pp., arXiv: 1208.5241 | DOI | MR | Zbl

[10] A. Boritchev, Generalised Burgers equation with random force and small viscosity, PhD thesis, École Polytechnique, 2012, 156 pp.

[11] A. Boritchev, “Estimates for solutions of a low-viscosity kick-forced generalised Burgers equation”, Proc. Roy. Soc. Edinburgh Sect. A, 143:2 (2013), 253–268 | DOI | MR | Zbl

[12] A. Boritchev, “Sharp estimates for turbulence in white-forced generalised Burgers equation”, Geom. Funct. Anal., 23:6 (2013), 1730–1771 | DOI | MR | Zbl

[13] J. M. Burgers, “A mathematical model illustrating the theory of turbulence”, Adv. Appl. Mech., 1, Academic Press, Inc., New York, NY, 1948, 171–199 | DOI | MR

[14] J. M. Burgers, The nonlinear diffusion equation. Asymptotic solutions and statistical problems, Reidel, Dordrecht–Boston, 1974, x+173 pp. | Zbl

[15] A. J. Chorin, Lectures on turbulence theory, Math. Lecture Ser., 5, Publish or Perish, Inc., Boston, MA, 1975, iii+159 pp. | MR | Zbl

[16] J. D. Cole, “On a quasilinear parabolic equation occurring in aerodynamics”, Quart. Appl. Math., 9 (1951), 225–236 | MR | Zbl

[17] G. Da Prato, J. Zabczyk, Stochastic equations in infinite dimensions, Encyclopedia Math. Appl., 44, Cambridge Univ. Press, Cambridge, 1992, xviii+454 pp. | DOI | MR | Zbl

[18] G. Da Prato, J. Zabczyk, Ergodicity for infinite dimensional systems, London Math. Soc. Lecture Note Ser., 229, Cambridge Univ. Press, Cambridge, 1996, xii+339 pp. | DOI | MR | Zbl

[19] C. M. Dafermos, Hyperbolic conservation laws in continuum physics, Grundlehren Math. Wiss., 325, 3rd ed., Springer-Verlag, Berlin, 2010, xxxvi+708 pp. | DOI | MR | Zbl

[20] C. R. Doering, J. D. Gibbon, Applied analysis of the Navier–Stokes equations, Cambridge Texts Appl. Math., Cambridge Univ. Press, Cambridge, 1995, xiv+217 pp. | DOI | MR | Zbl

[21] W. E, K. Khanin, A. Mazel, Ya. Sinai, “Invariant measures for Burgers equation with stochastic forcing”, Ann. of Math. (2), 151:3 (2000), 877–960 | DOI | MR | Zbl

[22] L. C. Evans, Partial differential equations, Grad. Stud. Math., 19, 2nd ed., Amer. Math. Soc., Providence, RI, 2010, xxii+749 pp. | DOI | MR | Zbl

[23] V. A. Florin, “Nekotorye prosteishie nelineinye zadachi konsolidatsii vodonasyschennoi zemlyanoi sredy”, Izv. AN SSSR. OTN, 1948, no. 9, 1389–1402 | MR

[24] A. R. Forsyth, Theory of differential equations, v. 5, 6, Partial differential equations, Cambridge Univ. Press, London, 1906, xx+478 pp., xiii+596 pp. | MR | Zbl

[25] J.-D. Fournier, U. Frisch, “L'équation de Burgers déterministe et statistique”, J. Méc. Théor. Appl., 2:5 (1983), 699–750 | MR | Zbl

[26] U. Frisch, Turbulence. The legacy of A. N. Kolmogorov, Cambridge Univ. Press, Cambridge, 1995, xiv+296 pp. | MR | Zbl | Zbl

[27] U. Frisch, J. Bec, “Burgulence”, Turbulence: nouveaux aspects/New trends in turbulence (Les Houches, 2000), eds. M. Lesieur, A. Yaglom, F. David, EDP Sci., Les Ulis, 2001, 341–383 | DOI | MR | Zbl

[28] D. Gomes, R. Iturriaga, K. Khanin, P. Padilla, “Viscosity limit of stationary distributions for the random forced Burgers equation”, Mosc. Math. J., 5:3 (2005), 613–631 | MR | Zbl

[29] T. Gotoh, R. H. Kraichnan, “Steady-state Burgers turbulence with large-scale forcing”, Phys. Fluids, 10:11 (1998), 2859–2866 | DOI | MR | Zbl

[30] E. Hopf, “The partial differential equation $u_t+uu_x=\mu u_{xx}$”, Comm. Pure Appl. Math., 3:3 (1950), 201–230 | DOI | MR | Zbl

[31] R. Iturriaga, K. Khanin, “Burgers turbulence and random Lagrangian systems”, Comm. Math. Phys., 232:3 (2003), 377–428 | DOI | MR | Zbl

[32] S. Kida, “Asymptotic properties of Burgers turbulence”, J. Fluid Mech., 93:2 (1979), 337–377 | DOI | MR | Zbl

[33] A. N. Kolmogorov, “Dissipation of energy in the locally isotropic turbulence”, Proc. Roy. Soc. London Ser. A, 434:1890 (1991), 15–17 | DOI | MR | MR | Zbl | Zbl

[34] A. N. Kolmogorov, “On the degeneration of isotropic turbulence in an incompressible viscous fluid”, Selected works of A. N. Kolmogorov, v. I, Math. Appl. (Soviet Ser.), 25, Mathematics and mechanics, Kluwer Academic Publishers Group, Dordrecht, 1991, 319–323 | DOI | MR | MR | Zbl | Zbl

[35] A. N. Kolmogorov, “The local structure of turbulence in incompressible viscous fluid for very large Reynolds number”, Proc. Roy. Soc. London Ser. A, 434:1890 (1991), 9–13 | DOI | MR | MR | Zbl | Zbl

[36] A. N. Kolmogorov, “A refinement of previous hypotheses concerning the local structure of turbulence in a viscous incompressible fluid at high Reynolds number”, J. Fluid Mech., 13:1 (1962), 82–85 | DOI | MR | Zbl

[37] R. H. Kraichnan, “Lagrangian-history statistical theory for Burgers' equation”, Phys. Fluids, 11:2 (1968), 265–277 | DOI | Zbl

[38] H.-O. Kreiss, “Fourier expansions of the solutions of the Navier–Stokes equations and their exponential decay rate”, Analyse mathématique et applications, Gauthier-Villars, Montrouge, 1988, 245–262 | MR | Zbl

[39] H.-O. Kreiss, J. Lorenz, Initial-boundary value problems and the Navier–Stokes equations, Pure Appl. Math., 136, Academic Press, Inc., Boston, MA, 1989, xii+402 pp. | MR | Zbl

[40] S. N. Kruzhkov, “The Cauchy problem in the large for nonlinear equations and for certain quasilinear systems of the first-order with several variables”, Soviet Math. Dokl., 5 (1964), 493–496 | MR | Zbl

[41] S. B. Kuksin, “On turbulence in nonlinear Schrödinger equations”, Geom. Funct. Anal., 7:4 (1997), 783–822 | DOI | MR | Zbl

[42] S. B. Kuksin, “Spectral properties of solutions for nonlinear PDEs in the turbulent regime”, Geom. Funct. Anal., 9:1 (1999), 141–184 | DOI | MR | Zbl

[43] S. Kuksin, A. Shirikyan, Mathematics of two-dimensional turbulence, Cambridge Tracts in Math., 194, Cambridge Univ. Press, Cambridge, 2012, xvi+320 pp. | DOI | Zbl

[44] H.-H. Kuo, Gaussian measures in Banach spaces, Lecture Notes in Math., 463, Springer-Verlag, Berlin–New York, 1975, vi+224 pp. | DOI | MR | Zbl

[45] P. D. Lax, Hyperbolic partial differential equations, Courant Lect. Notes Math., 14, New York Univ., Courant Inst. Math. Sci., New York, NY; Amer. Math. Soc., Providence, RI, 2006, viii+217 pp. | MR | Zbl

[46] A. M. Obukhov, “O raspredelenii energii v spektre turbulentnogo potoka”, Dokl. AN SSSR, 32:1 (1941), 22–24 | MR | Zbl

[47] A. M. Obukhov, “O raspredelenii energii v spektre turbulentnogo potoka”, Izv. AN SSSR. Ser. geogr. i geofiz., 5:4 (1941), 453–466 | MR

[48] G. Parisi, U. Frisch, “Fully developed turbulence and intermittency”, Turbulence and predictability in geophysical fluid dynamics and climate dynamics, Proceedings of the International School of Physics “Enrico Fermi”, eds. M. Ghil, R. Benzi, G. Parisi, North-Holland, Amsterdam, 1985, 71–88

[49] D. Serre, Systems of conservation laws, v. 1, Hyperbolicity, entropies, shock waves, Cambridge Univ. Press, Cambridge, 1999, xxii+263 pp. | DOI | MR | Zbl

[50] Z.-S. She, S. A. Orszag, “Physical model of intermittency in turbulence: inertial-range non-Gaussian statistics”, Phys. Rev. Lett., 66:13 (1991), 1701–1704 | DOI

[51] Ya. G. Sinai, “Two results concerning asymptotic behavior of solutions of the Burgers equation with force”, J. Statist. Phys., 64:1-2 (1991), 1–12 | DOI | MR | Zbl

[52] Ya. G. Sinai, “Burgers system driven by a periodic stochastic flow”, Itô's stochastic calculus and probability theory, Springer, Tokyo, 1996, 347–353 | DOI | MR | Zbl

[53] E. Tadmor, “Total variation and error estimates for spectral viscosity approximations”, Math. Comp., 60:201 (1993), 245–256 | DOI | MR | Zbl

[54] M. E. Taylor, Partial differential equations, v. I, Appl. Math. Sci., 115, Basic theory, Springer-Verlag, New York, 1996, xxiv+563 pp. | MR | Zbl

[55] A. Tsinober, An informal conceptual introduction to turbulence, Fluid Mech. Appl., 92, Springer, Dordrecht, 2009, xx+464 pp. | DOI | MR | Zbl

[56] J. von Neumann, Collected works (1949–63), v. 6, Pergamon Press Book, Theory of games, astrophysics, hydrodynamics and meteorology, The Macmillan Co., New York, 1963, x+538 pp. | MR | Zbl