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Anomalous Dissipation in Passive Scalar Transport

Archive for Rational Mechanics and Analysis volume 243, pages 1151–1180 (2022)Cite this article

Abstract

We study anomalous dissipation in hydrodynamic turbulence in the context of passive scalars. Our main result produces an incompressible \(C^\infty ([0,T)\times {\mathbb {T}}^d)\cap L^1([0,T]; C^{1-}({\mathbb {T}}^d))\) velocity field which explicitly exhibits anomalous dissipation. As a consequence, this example also shows the non-uniqueness of solutions to the transport equation with an incompressible \(L^1([0,T]; C^{1-}({\mathbb {T}}^d))\) drift, which is smooth except at one point in time. We also give a sufficient condition for anomalous dissipation based on solutions to the inviscid equation becoming singular in a controlled way. Finally, we discuss connections to the Obukhov-Corrsin monofractal theory of scalar turbulence along with other potential applications.

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Fig. 1

Notes

  1. We remark that it is also unknown whether this double exponential lower bound above is attained for any flow. In discrete time [38] produce an example where is in fact attained. In continuous time, however, there are no examples exhibiting the double exponential decay. Moreover, Miles and Doering [49] provide numerical evidence and a heuristic argument that the Batchelor scale limits the effectiveness of mixing, suggesting that the \(L^2\) energy can only decay exponentially.

References

  1. Alberti, G., Bianchini, S., Crippa, G.: Structure of level sets and Sard-type properties of Lipschitz maps. Annali della Scuola Normale Superiore di Pisa-Classe di Scienze 12(4), 863–902, 2013

    MathSciNet  MATH  Google Scholar 

  2. Alberti, G., Bianchini, S., Crippa, G.: A uniqueness result for the continuity equation in two dimensions. J. Eur. Math. Soc. (JEMS) 16(2), 201–234, 2014. https://doi.org/10.4171/JEMS/431.

    Article  MathSciNet  MATH  Google Scholar 

  3. Alberti, G., Crippa, G., Mazzucato, A.L.: Exponential self-similar mixing by incompressible flows. J. Am. Math. Soc. 32(2), 445–490, 2019. https://doi.org/10.1090/jams/913.

    Article  MathSciNet  MATH  Google Scholar 

  4. Alberti, G., Crippa, G., Mazzucato, A.L.: Loss of regularity for the continuity equation with non-Lipschitz velocity field. Ann. PDE 5(1), 19, 2019. https://doi.org/10.1007/s40818-019-0066-3.

    Article  MathSciNet  MATH  Google Scholar 

  5. Aizenman, M.: On vector fields as generators of flows: a counterexample to Nelson’s conjecture. Ann. Math. (2) 107(2), 287–296, 1978. https://doi.org/10.2307/1971145.

    Article  MathSciNet  MATH  Google Scholar 

  6. Arnold, V.I., Khesin, B.A.: Topological Methods in Hydrodynamics, vol. 125. Applied Mathematical Sciences. Springer-Verlag, New York (1998)

    Book  Google Scholar 

  7. Ambrosio, L.: Transport equation and Cauchy problem for \(BV\) vector fields. Invent. Math. 158(2), 227–260, 2004. https://doi.org/10.1007/s00222-004-0367-2.

    Article  ADS  MathSciNet  MATH  Google Scholar 

  8. Bedrossian, J., Blumenthal, A., Punshon-Smith, S.: Almost-sure exponential mixing of passive scalars by the stochastic Navier-Stokes equations (2019). arXiv:1905.03869

  9. Bedrossian, J., Blumenthal, A., Punshon-Smith, S.: The Batchelor spectrum of passive scalar turbulence in stochastic fluid mechanics (2019). arXiv:1911.11014

  10. Buckmaster, T., De Lellis, C., Székelyhidi, L., Jr., Vicol, V.: Onsager’s conjecture for admissible weak solutions. Commun. Pure Appl. Math. 72(2), 229–274, 2019. https://doi.org/10.1002/cpa.21781.

    Article  MathSciNet  MATH  Google Scholar 

  11. Bernard, D., Gawȩdzki, K., Kupiainen, A.: Slow modes in passive advection. J. Stat. Phys. 90(3–4), 519–569, 1998. https://doi.org/10.1023/A:1023212600779.

    Article  ADS  MathSciNet  MATH  Google Scholar 

  12. Bressan, A.: A lemma and a conjecture on the cost of rearrangements. Rend. Sem. Mat. Univ. Padova 110, 97–102, 2003

    MathSciNet  MATH  Google Scholar 

  13. Bressan, A.: Prize offered for the solution of a problem on mixing flows. (2006)

  14. Brué, Elia, Nguyen, Quoc-Hung: Advection diffusion equation with Sobolev vector field (2020). arXiv:2003.08198

  15. Crippa, G., De Lellis, C.: Regularity and compactness for the DiPerna-Lions flow. In: Hyperbolic problems: theory, numerics, applications, pp. 423–430. Springer, Berlin (2008). https://doi.org/10.1007/978-3-540-75712-2_39

  16. Crippa, G., De Lellis, C.: Estimates and regularity results for the DiPerna-Lions flow. J. Reine Angew. Math 616, 15–46, 2008

    MathSciNet  MATH  Google Scholar 

  17. Constantin, P., W, E., Titi, E.S.: Onsager’s conjecture on the energy conservation for solutions of Euler’s equation. Commun. Math. Phys. 165(1):207–209 (1994). http://projecteuclid.org/euclid.cmp/1104271041

  18. Crippa, G., Gusev, N., Spirito, S., Wiedemann, E.: Non-uniqueness and prescribed energy for the continuity equation. Commun. Math. Sci. 13(7), 1937–1947, 2015. https://doi.org/10.4310/CMS.2015.v13.n7.a12.

    Article  MathSciNet  MATH  Google Scholar 

  19. Constantin, P., Kiselev, A., Ryzhik, L., Zlatoš, A.: Diffusion and mixing in fluid flow. Ann. Math. (2) 168(2), 643–674, 2008. https://doi.org/10.4007/annals.2008.168.643.

    Article  MathSciNet  MATH  Google Scholar 

  20. Colombini, F., Luo, T., Rauch, J.: Uniqueness and nonuniqueness for nonsmooth divergence free transport. In Seminaire: Équations aux Dérivées Partielles, 2002–2003, Sémin. Équ. Dériv. Partielles, pages Exp. No. XXII, 21. École Polytech., Palaiseau (2003)

  21. Corrsin, S.: On the spectrum of isotropic temperature fluctuations in an isotropic turbulence. J. Appl. Phys. 22, 469–473, 1951

    Article  ADS  MathSciNet  Google Scholar 

  22. Constantin, P., Procaccia, I.: Scaling in fluid turbulence: a geometric theory. Phys. Rev. E (3) 47(5), 3307–3315, 1993. https://doi.org/10.1103/PhysRevE.47.3307.

    Article  ADS  MathSciNet  Google Scholar 

  23. Constantin, P., Procaccia, I.: The geometry of turbulent advection: sharp estimates for the dimensions of level sets. Nonlinearity 7(3), 1045–1054, 1994

    Article  ADS  MathSciNet  Google Scholar 

  24. Zelati, M. Coti, Delgadino, M.G., Elgindi, T.M.: On the relation between enhanced dissipation time-scales and mixing rates (2018). arXiv:1806.03258

  25. Drivas, T.D., Eyink, G.L.: A Lagrangian fluctuation-dissipation relation for scalar turbulence. Part I. Flows with no bounding walls. J. Fluid Mech. 829, 153–189, 2017. https://doi.org/10.1017/jfm.2017.567.

    Article  ADS  MathSciNet  MATH  Google Scholar 

  26. Drivas, T.D., Eyink, G.L.: An Onsager singularity theorem for Leray solutions of incompressible Navier-Stokes. Nonlinearity 32(11), 4465–4482, 2019. https://doi.org/10.1088/1361-6544/ab2f42.

    Article  ADS  MathSciNet  MATH  Google Scholar 

  27. Depauw, N.: Non unicité des solutions bornées pour un champ de vecteurs BV en dehors d’un hyperplan. C. R. Math. Acad. Sci. Paris 337(4), 249–252, 2003. https://doi.org/10.1016/S1631-073X(03)00330-3.

    Article  MathSciNet  MATH  Google Scholar 

  28. DiPerna, R.J., Lions, P.-L.: Ordinary differential equations, transport theory and Sobolev spaces. Invent. Math. 98(3), 511–547, 1989. https://doi.org/10.1007/BF01393835.

    Article  ADS  MathSciNet  MATH  Google Scholar 

  29. De Lellis, C., Székelyhidi, L., Jr.: The Euler equations as a differential inclusion. Ann. Math. (2) 170(3), 1417–1436, 2009. https://doi.org/10.4007/annals.2009.170.1417.

    Article  MathSciNet  MATH  Google Scholar 

  30. De Lellis, C., Székelyhidi, L., Jr.: On admissibility criteria for weak solutions of the Euler equations. Arch. Ration. Mech. Anal. 195(1), 225–260, 2010. https://doi.org/10.1007/s00205-008-0201-x.

    Article  MathSciNet  MATH  Google Scholar 

  31. De Lellis, C., Székelyhidi, L., Jr.: The \(h\)-principle and the equations of fluid dynamics. Bull. Am. Math. Soc. (N.S.) 49(3), 347–375, 2012. https://doi.org/10.1090/S0273-0979-2012-01376-9.

    Article  MathSciNet  MATH  Google Scholar 

  32. Donzis, D.A., Sreenivasan, K.R., Yeung, P.K.: Scalar dissipation rate and dissipative anomaly in isotropic turbulence. J. Fluid Mech. 532, 199–216, 2005. https://doi.org/10.1017/S0022112005004039.

    Article  ADS  MathSciNet  MATH  Google Scholar 

  33. Eyink, G.L., Drivas, T.D.: Spontaneous stochasticity and anomalous dissipation for Burgers equation. J. Stat. Phys. 158(2), 386–432, 2015. https://doi.org/10.1007/s10955-014-1135-3.

    Article  ADS  MathSciNet  MATH  Google Scholar 

  34. Eyink, G.L.: Energy dissipation without viscosity in ideal hydrodynamics. I. Fourier analysis and local energy transfer. Phys. D 78(3–4), 222–240, 1994. https://doi.org/10.1016/0167-2789(94)90117-1.

    Article  MathSciNet  MATH  Google Scholar 

  35. Eyink, G.L.: Intermittency and anomalous scaling of passive scalars in any space dimension. Phys. Rev. E 54, 1497–1503, 1996. https://doi.org/10.1103/PhysRevE.54.1497.

    Article  ADS  Google Scholar 

  36. Elgindi, T.M., Zlatoš, A.: Universal mixers in all dimensions (2018). arXiv:1809.09614

  37. Falkovich, G., Gawȩdzki, K., Vergassola, M.: Particles and fields in fluid turbulence. Rev. Mod. Phys. 73(4), 913–975, 2001. https://doi.org/10.1103/revmodphys.73.913.

    Article  ADS  MathSciNet  MATH  Google Scholar 

  38. Feng, Y., Iyer, G.: Dissipation enhancement by mixing. Nonlinearity 32(5), 1810–1851, 2019. https://doi.org/10.1088/1361-6544/ab0e56.

    Article  ADS  MathSciNet  MATH  Google Scholar 

  39. Gawȩdzki, K.: Soluble models of turbulent transport. In: Non-equilibrium statistical mechanics and turbulence. London Mathematical Society Lecture Note Series, vol. 355, pp. 44–107. Cambridge University Press, Cambridge (2008)

  40. Iyer, G., Kiselev, A., Xu, X.: Lower bounds on the mix norm of passive scalars advected by incompressible enstrophy-constrained flows. Nonlinearity 27(5), 973–985, 2014. https://doi.org/10.1088/0951-7715/27/5/973.

    Article  ADS  MathSciNet  MATH  Google Scholar 

  41. Isett, P.: A proof of Onsager’s conjecture. Ann. Math. (2) 188(3), 871–963, 2018. https://doi.org/10.4007/annals.2018.188.3.4.

    Article  MathSciNet  MATH  Google Scholar 

  42. Iyer, G., Xu, X., Zlatoš, A.: Convection-induced singularity suppression in the Keller-Segel and other non-linear PDEs (2019). arXiv:1908.01941

  43. Jeong, I.-J., Yoneda, T.: Enstrophy dissipation and vortex thinning for the incompressible 2D Navier-Stokes equations. Nonlinearity 34(4), 1837–1853, 2021

    Article  MathSciNet  Google Scholar 

  44. Kaneda, Y., Ishihara, T., Yokokawa, M., Itakura, K., Uno, A.: Energy dissipation rate and energy spectrum in high resolution direct numerical simulations of turbulence in a periodic box. Phys. Fluids 15(2), L21–L24, 2003. https://doi.org/10.1063/1.1539855.

    Article  ADS  MATH  Google Scholar 

  45. Kolmogorov, A.N.: Energy dissipation in locally isotropic turbulence. Dokl. Akad. Nauk. SSSR 32, 19–21, 1941

    ADS  MATH  Google Scholar 

  46. Le Jan, Y., Raimond, O.: Integration of Brownian vector fields. Ann. Probab. 30(2), 826–873, 2002. https://doi.org/10.1214/aop/1023481009.

    Article  MathSciNet  MATH  Google Scholar 

  47. Le Jan, Y., Raimond, O.: Flows, coalescence and noise. Ann. Probab. 32(2), 1247–1315, 2004. https://doi.org/10.1214/009117904000000207.

    Article  MathSciNet  MATH  Google Scholar 

  48. Lunasin, E., Lin, Z., Novikov, A., Mazzucato, A., Doering, C.R.: Optimal mixing and optimal stirring for fixed energy, fixed power, or fixed palenstrophy flows. J. Math. Phys. 53(11) (2012). https://doi.org/10.1063/1.4752098

  49. Miles, C.J., Doering, C.R.: Diffusion-limited mixing by incompressible flows. Nonlinearity 31(5), 2346, 2018. https://doi.org/10.1088/1361-6544/aab1c8.

    Article  ADS  MathSciNet  MATH  Google Scholar 

  50. Modena, S., Székelyhidi L., Jr.: Non-uniqueness for the transport equation with Sobolev vector fields. Ann. PDE 4(2):18–38 (2018). https://doi.org/10.1007/s40818-018-0056-x

  51. Obukhov, A.M.: Structure of temperature field in turbulent flow. Izv. Akad. Nauk. SSSR, Geogr. Geofiz, 13: 1949

  52. Onsager, L.: Statistical hydrodynamics. Nuovo Cimento (9) 6, 279–287, 1949

    Article  MathSciNet  Google Scholar 

  53. Pierrehumbert, R.: Tracer microstructure in the large-eddy dominated regime. Chaos Solitons Fractals 4(6), 1091–1110, 1994. https://doi.org/10.1016/0960-0779(94)90139-2. Special Issue: Chaos Applied to Fluid Mixing

    Article  ADS  Google Scholar 

  54. Pearson, B.R., Krogstad, P.A., van de Water, W.: Measurements of the turbulent energy dissipation rate. Phys. Fluids 14(3), 1288–1290, 2002. https://doi.org/10.1063/1.1445422.

    Article  ADS  Google Scholar 

  55. Poon, C.-C.: Unique continuation for parabolic equations. Commun. Partial Differ. Equ. 21(3–4), 521–539, 1996. https://doi.org/10.1080/03605309608821195.

    Article  MathSciNet  MATH  Google Scholar 

  56. Sreenivasan, K.R.: On the scaling of the turbulence energy dissipation rate. Phys. Fluids 27(5), 1048–1051, 1984. https://doi.org/10.1063/1.864731.

    Article  ADS  Google Scholar 

  57. Seis, C.: Maximal mixing by incompressible fluid flows. Nonlinearity 26(12), 3279, 2013

    Article  ADS  MathSciNet  Google Scholar 

  58. Seis, C.: Diffusion limited mixing rates in passive scalar advection (2020). arXiv:2003.08794

  59. Sreenivasan, K.R.: An update on the energy dissipation rate in isotropic turbulence. Phys. Fluids 10(2), 528–529, 1998. https://doi.org/10.1063/1.869575.

    Article  ADS  MathSciNet  MATH  Google Scholar 

  60. Sreenivasan, K.R.: Turbulent mixing: a perspective. Proc. Natl. Acad. Sci. 116(37), 18175–18183, 2019. https://doi.org/10.1073/pnas.1800463115.

    Article  ADS  MathSciNet  MATH  Google Scholar 

  61. Shraiman, B.I., Siggia, E.D.: Scalar turbulence. Nature 405(6787), 639, 2000. https://doi.org/10.1038/35015000.

    Article  ADS  Google Scholar 

  62. Thiffeault, J.-L.: Using multiscale norms to quantify mixing and transport. Nonlinearity 25(2), R1–R44, 2012. https://doi.org/10.1088/0951-7715/25/2/R1.

    Article  ADS  MathSciNet  MATH  Google Scholar 

  63. Wei, D.: Diffusion and mixing in fluid flow via the resolvent estimate (2018). arXiv:1811.11904

  64. Yao, Y., Zlatoš, A.: Mixing and un-mixing by incompressible flows. J. Eur. Math. Soc. (JEMS) 19(7), 1911–1948, 2017. https://doi.org/10.4171/JEMS/709.

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Stony Brook University, Stony Brook, NY, USA

    Theodore D. Drivas

  2. Department of Mathematics, Duke University, Durham, USA

    Tarek M. Elgindi

  3. Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, USA

    Gautam Iyer

  4. Department of Mathematical Sciences and RIM, Seoul National University, Seoul, South Korea

    In-Jee Jeong

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  1. Theodore D. Drivas
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  2. Tarek M. Elgindi
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  3. Gautam Iyer
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  4. In-Jee Jeong
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Correspondence to In-Jee Jeong.

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Communicated by P. Constantin.

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This work has been partially supported by the National Science Foundation under grants DMS-1703997 to TD, DMS-1817134 to TE, DMS-1814147 to GI, as well as by the Center for Nonlinear Analysis. IJ has been supported by the Samsung Science and Technology Foundation under Project Number SSTF-BA2002-04. TD and TE would like to thank the Korean Institute for Advanced Study (KIAS) for its hospitality. TD would like to thank Navid Constantinou for useful discussions.

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Drivas, T.D., Elgindi, T.M., Iyer, G. et al. Anomalous Dissipation in Passive Scalar Transport. Arch Rational Mech Anal 243, 1151–1180 (2022). https://doi.org/10.1007/s00205-021-01736-2

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