Advection Diffusion Equations with Sobolev Velocity Field

Abstract

In this note we study advection-diffusion equations associated to incompressible \(W^{1,p}\) velocity fields with \(p>2\). We present new estimates on the energy dissipation rate and we discuss applications to the study of upper bounds on the enhanced dissipation rate, lower bounds on the \(L^2\) norm of the density, and quantitative vanishing viscosity estimates. The key tools employed in our argument are a propagation of regularity result, coming from the study of transport equations, and a new result connecting the energy dissipation rate to regularity estimates for transport equations. Eventually we provide examples which underline the sharpness of our estimates.

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Notes

  1. 1.

    Here and in the sequel we use the notation \(A \sim _c B\) to mean that \(C^{-1}A \le B \le C A\) where C depends only on c. Similar notation will be adopted for \(\lesssim _c\) and \( > rsim _c\).

References

  1. [ACM14]

    Alberti, G., Crippa, G., Mazzucato, A.-L.: Exponential self-similar mixing and loss of regularity for continuity equations. C. R. Math. Acad. Sci. Paris 352(11), 901–906 (2014)

    MathSciNet  Article  Google Scholar 

  2. [ACM16]

    Alberti, G., Crippa, G., Mazzucato, A.-L.: Exponential self-similar mixing by incompressible flows. J. Am. Math. Soc. 32(2), 445–490 (2019)

    MathSciNet  Article  Google Scholar 

  3. [ACM18]

    Alberti, G., Crippa, G., Mazzucato, A.-L.: Loss of regularity for the continuity equation with non-Lipschitz velocity field. Ann. PDE 5(1), 5:9 (2019)

    MathSciNet  Article  Google Scholar 

  4. [A04]

    Ambrosio, L.: Transport equation and Cauchy problem for \(BV\) vector fields. Invent. Mat. 158, 227–260 (2004)

    ADS  MathSciNet  Article  Google Scholar 

  5. [AC14]

    Ambrosio, L., Crippa, G.: Continuity equations and ODE flows with non-smooth velocity. Proc. R. Soc. Edinb. Sect. A 144, 1191–1244 (2014)

    MathSciNet  Article  Google Scholar 

  6. [BBJ19]

    Ben Belgacem, F., Jabin, P.-E.: Convergence of numerical approximations to non-linear continuity equations with rough force fields. Arch. Ration. Mech. Anal. 234(2), 509–547 (2019)

    MathSciNet  Article  Google Scholar 

  7. [BCZ17]

    Bedrossian, J., Coti Zelati, M.: Enhanced dissipation, hypoellipticity, and anomalous small noise inviscid limits in shear flows. Arch. Ration. Mech. Anal. 224(3), 1161–1204 (2017)

    MathSciNet  Article  Google Scholar 

  8. [B03]

    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 

  9. [BCDL20]

    Brué, E., Colombo, M., De Lellis, C.: Positive solutions of transport equations and classical nonuniqueness of characteristic curves. Arch. Ration. Mech. Anal. (to appear), preprint arXiv:2003.00539

  10. [BN18b]

    Brué, E., Nguyen, Q.-H.: On the Sobolev space of functions with derivative of logarithmic order. Adv. Nonlinear Anal. 9(1), 836–849 (2020)

    MathSciNet  Article  Google Scholar 

  11. [BN18c]

    Brué, E., Nguyen, Q.-H.: Sharp regularity estimates for solutions of the continuity equation drifted by Sobolev vector fields. Anal. PDE (to appear)

  12. [BN19]

    Brué, E., Nguyen, Q.-H.: Sobolev estimates for solutions of the transport equation and ODE flows associated to non-Lipschitz drifts. Math. Ann. (2020). https://doi.org/10.1007/s00208-020-01988-5

  13. [CKRZ08]

    Constantin, P., Kiselev, A., Ryzhik, L., Zlatos, A.: Diffusion and mixing in fluid flow. Ann. Math. (2) 168(2), 643–674 (2008)

    MathSciNet  Article  Google Scholar 

  14. [CZDE18]

    Coti Zelati, M., Delgadino, M.-G., Elgindi, T.-M.: On the relation between enhanced dissipation time-scales and mixing rates. Commun. Pure Appl. Math. 73(6), 1205–1244 (2020)

    Article  Google Scholar 

  15. [CZDO19]

    Coti Zelati, M., Dolce, M.: Separation of time-scales in drift-diffusion equations on \({\mathbb{R}}^2\). J. Math. Pures Appl. (9) 142, 58–75 (2020)

    MathSciNet  Article  Google Scholar 

  16. [CZDR19]

    Coti Zelati, M., Drivas, T.-D.: A stochastic approach to enhanced diffusion. Ann. Scu. Norm. Sup. Pisa Cl. Sci. (to appear), preprint on arXiv:1911.09995

  17. [CDL08]

    Crippa, G., De Lellis, C.: Estimates and regularity results for the Di Perna–Lions flow. J. Reine Angew. Math. 616, 15–46 (2008)

    MathSciNet  MATH  Google Scholar 

  18. [DPL89]

    DiPerna, R.-J., Lions, P.-L.: Ordinary differential equations, transport theory and Sobolev spaces. Invent. Math. 98, 511–547 (1989)

    ADS  MathSciNet  Article  Google Scholar 

  19. [DEIJ2019]

    Drivas, T.-D., Elgindi, T.-M., Iyer, G., Jeong, I.-J.: Anomalous dissipation in passive scalar transport. Preprint on arXiv:1911.03271

  20. [EZ]

    Elgindi, T.-M., Zlatoš, A.: Universal mixers in all dimensions. Adv. Math. 356, 106807–33 (2019)

    MathSciNet  Article  Google Scholar 

  21. [FI19]

    Feng, Y., Iyer, G.: Dissipation enhancement by mixing. Nonlinearity 32(5), 1810–1851 (2019)

    ADS  MathSciNet  Article  Google Scholar 

  22. [IKX14]

    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)

    ADS  MathSciNet  Article  Google Scholar 

  23. [JY20]

    Jeong, I.-J., Yoneda, T.: Vortex stretching and a modified zeroth law for the incompressible 3D Navier–Stokes equations. Math. Ann. (2020). https://doi.org/10.1007/s00208-020-02019-z

  24. [J16]

    Jabin, P.-E.: Critical non-Sobolev regularity for continuity equations with rough velocity fields. J. Differ. Equ. 260(5), 4739–4757 (2016)

    ADS  MathSciNet  Article  Google Scholar 

  25. [K97]

    Kunita, H.: Stochastic Flows and Stochastic Differential Equations, Cambridge Studies in Advanced Mathematics. Cambridge Studies in Advanced Mathematics, vol. 24. Cambridge University Press, Cambridge (1997). (Reprint of the 1990 original)

    Google Scholar 

  26. [LF16]

    Léger, F.: A new approach to bounds on mixing. Math. Models Methods Appl. Sci. 28(5), 829–849 (2018)

    MathSciNet  Article  Google Scholar 

  27. [MD18]

    Miles, C.-J., Doering, C.-R.: Diffusion-limited mixing by incompressible flows. Nonlinearity 31(5), 2346–2350 (2018)

    ADS  MathSciNet  Article  Google Scholar 

  28. [MS19]

    Modena, S., Sattig, G.: Convex integration solutions to the transport equation with full dimensional concentration. Annales de l’Institut Henri Poincaré C, Analyse non linéaire, 2020 (to appear), preprint on arXiv:1902.08521

  29. [MSz18]

    Modena, S., Székelyhidi Jr., L.: Non-uniqueness for the transport equation with Sobolev vector fields. Ann. PDE 4(2), 1–38 (2018)

    MathSciNet  Article  Google Scholar 

  30. [MSz19]

    Modena, S., Székelyhidi Jr., L.: Non-renormalized solutions to the continuity equation. Calc. Var. Partial Differ. Equ. 58(6), 1–30 (2019)

    MathSciNet  Article  Google Scholar 

  31. [QN18]

    Nguyen, Q.-H.: Quantitative estimates for regular Lagrangian flows with \(BV\) vector fields. Comm. Pure Appl. Math. (to appear), preprint on arXiv:1805.01182

  32. [Poon96]

    Poon, C.-C.: Unique continuation for parabolic equations. Commun. Partial Differ. Equ. 21(3–4), 521–539 (1996)

    MathSciNet  Article  Google Scholar 

  33. [S17]

    Seis, C.: A quantitative theory for the continuity equation. Ann. Inst. H. Poincaré Anal. Non Liné aire 34(7), 1837–1850 (2017)

    ADS  MathSciNet  Article  Google Scholar 

  34. [S18]

    Seis, C.: Optimal stability estimates for continuity equations. Proc. R. Soc. Edinb. Sect. A 148(6), 1279–1296 (2018)

    MathSciNet  Article  Google Scholar 

  35. [S20]

    Seis, C.: Diffusion limited mixing rates in passive scalar advection. Preprint on arXiv:2003.08794

  36. [YZ17]

    Yao, Y., Zlatoš, A.: Mixing and un-mixing by incompressible flows. J. Eur. Math. Soc. 19(7), 1911–1948 (2017)

    MathSciNet  Article  Google Scholar 

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Acknowledgements

Most of this work was developed while the first author was a PhD student at Scuola Normale Superiore, Pisa. The second author was supported by the ShanghaiTech University startup fund, and part of this work was done while he was visiting Scuola Normale Superiore. The authors wish to express their gratitude to this institution for the excellent working conditions and the stimulating atmosphere.

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Correspondence to Quoc-Hung Nguyen.

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Brué, E., Nguyen, QH. Advection Diffusion Equations with Sobolev Velocity Field. Commun. Math. Phys. (2021). https://doi.org/10.1007/s00220-021-03993-4

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