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Nonuniqueness of Weak Solutions for the Transport Equation at Critical Space Regularity

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Abstract

We consider the linear transport equations driven by an incompressible flow in dimensions \(d\ge 3\). For divergence-free vector fields \(u \in L^1_t W^{1,q}\), the celebrated DiPerna-Lions theory of the renormalized solutions established the uniqueness of the weak solution in the class \(L^\infty _t L^p\) when \(\frac{1}{p} + \frac{1}{q} \le 1\). For such vector fields, we show that in the regime \(\frac{1}{p} + \frac{1}{q} > 1\), weak solutions are not unique in the class \( L^1_t L^p\). One crucial ingredient in the proof is the use of both temporal intermittency and oscillation in the convex integration scheme.

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Notes

  1. Note that uniqueness result for positive \(\rho \) can go beyond the DiPerna-Lions range, see [7, Theorem 1.5]

  2. We identify [0, T] with an 1-dimensional torus.

  3. In fact, the singular set of u is dense, and as a result, there is no local regularity outside the singular set, cf. [8, 20].

References

  1. Ambrosio, L., Colombo, M., Figalli, A.: Existence and uniqueness of maximal regular flows for non-smooth vector fields. Arch. Ration. Mech. Anal. 218(2), 1043–1081 (2015)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  3. Aizenman, M.: On vector fields as generators of flows: a counterexample to Nelson’s conjecture. Ann. Math. (2) 107(2), 287–296 (1978)

    Article  MathSciNet  Google Scholar 

  4. Ambrosio, L.: Transport equation and Cauchy problem for \(BV\) vector fields. Invent. Math. 158(2), 227–260 (2004)

    Article  ADS  MathSciNet  Google Scholar 

  5. Ambrosio, L.: Well posedness of ODE’s and continuity equations with nonsmooth vector fields, and applications. Rev. Math. Complut. 30(3), 427–450 (2017)

    Article  MathSciNet  Google Scholar 

  6. Beekie, R., Buckmaster, T., Vicol, V.: Weak solutions of ideal MHD which do not conserve magnetic helicity. Ann. PDE 6(1), 40 (2020). Paper No. 1

    Article  MathSciNet  Google Scholar 

  7. Bruè, E., Colombo, M., De Lellis, C.: Positive solutions of transport equations and classical nonuniqueness of characteristic curves. arXiv:2003.00539, (2020)

  8. Buckmaster, T., Colombo, M., Vicol, V.: Wild solutions of the Navier–Stokes equations whose singular sets in time have Hausdorff dimension strictly less than 1. J. Eur. Math. Soc. (JEMS) (2020)

  9. Buckmaster, T., De Lellis, C., Isett, P., Székelyhidi Jr., L.: Anomalous dissipation for \(1/5\)-Hölder Euler flows. Ann. Math. (2) 182(1), 127–172 (2015)

    Article  MathSciNet  Google Scholar 

  10. Buckmaster, T., De Lellis, C., Székelyhidi Jr., L.: Dissipative Euler flows with Onsager-critical spatial regularity. Commun. Pure Appl. Math. 69(9), 1613–1670 (2016)

    Article  MathSciNet  Google Scholar 

  11. Buckmaster, T., De Lellis, C., Székelyhidi Jr. L., Vicol, V.: Onsager’s conjecture for admissible weak solutions. Commun. Pure Appl. Math. (2018)

  12. Bruè, E., Nguyen, Q.-H.: Sharp regularity estimates for solutions of the continuity equation drifted by sobolev vector fields. arXiv: 1806.03466 (2018)

  13. Buckmaster, T., Shkoller, S., Vicol, V.: Nonuniqueness of weak solutions to the SQG equation. Commun. Pure Appl. Math. 72(9), 1809–1874 (2019)

    Article  MathSciNet  Google Scholar 

  14. Buckmaster, T., Vicol, V.: Convex integration and phenomenologies in turbulence. EMS Surv. Math. Sci. 6(1), 173–263 (2019)

    Article  MathSciNet  Google Scholar 

  15. Buckmaster, T., Vicol, V.: Nonuniqueness of weak solutions to the Navier-Stokes equation. Ann. Math. (2) 189(1), 101–144 (2019)

    Article  MathSciNet  Google Scholar 

  16. Caravenna, L., Crippa, G.: Uniqueness and Lagrangianity for solutions with lack of integrability of the continuity equation. C. R. Math. Acad. Sci. Paris 354(12), 1168–1173 (2016)

    Article  MathSciNet  Google Scholar 

  17. Caravenna, L., Crippa, G.: A directional Lipschitz extension lemma, with applications to uniqueness and Lagrangianity for the continuity equation. arXiv: 1812.06817 (2018)

  18. 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 

  19. 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)

    Article  MathSciNet  Google Scholar 

  20. Caffarelli, L., Kohn, R., Nirenberg, L.: Partial regularity of suitable weak solutions of the Navier–Stokes equations. Commun. Pure Appl. Math. 35(6), 771–831 (1982)

    Article  ADS  MathSciNet  Google Scholar 

  21. Constantin, P., Kukavica, I., Vicol, V.: Contrast between Lagrangian and Eulerian analytic regularity properties of Euler equations. Ann. Inst. H. Poincaré Anal. Non Linéaire 33(6), 1569–1588 (2016)

    Article  ADS  MathSciNet  Google Scholar 

  22. Colombini, F., Lerner, N.: Uniqueness of continuous solutions for BV vector fields. Duke Math. J. 111(2), 357–384 (2002)

    Article  MathSciNet  Google Scholar 

  23. Cheskidov, A., Luo, X.: Stationary and discontinuous weak solutions of the Navier–Stokes equations. arXiv:1901.07485 ( 2019)

  24. 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)

  25. Cheskidov, A., Shvydkoy, R.: Euler equations and turbulence: analytical approach to intermittency. SIAM J. Math. Anal. 46(1), 353–374 (2014)

    Article  MathSciNet  Google Scholar 

  26. Dai, M.: Non-uniqueness of Leray-Hopf weak solutions of the 3D Hall-MHD system. arXiv:1812.11311 (2018)

  27. Drivas, T. D., Elgindi, T. M., Iyer, G., Jeong, I.-J.: Anomalous dissipation in passive scalar transport. arXiv:1911.03271 (2019)

  28. 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)

    Article  MathSciNet  Google Scholar 

  29. DiPerna, R.J., Lions, P.-L.: Ordinary differential equations, transport theory and Sobolev spaces. Invent. Math. 98(3), 511–547 (1989)

    Article  ADS  MathSciNet  Google Scholar 

  30. De Lellis, C.: ODEs with Sobolev coefficients: the Eulerian and the Lagrangian approach. Discrete Contin. Dyn. Syst. Ser. S 1(3), 405–426 (2008)

    MathSciNet  MATH  Google Scholar 

  31. De Lellis, C., Székelyhidi Jr., L.: The Euler equations as a differential inclusion. Ann. Math. (2) 170(3), 1417–1436 (2009)

    Article  MathSciNet  Google Scholar 

  32. De Lellis, C., Székelyhidi Jr., L.: Dissipative continuous Euler flows. Invent. Math. 193(2), 377–407 (2013)

    Article  ADS  MathSciNet  Google Scholar 

  33. De Lellis, C., Székelyhidi Jr., L.: Dissipative Euler flows and Onsager’s conjecture. J. Eur. Math. Soc. (JEMS) 16(7), 1467–1505 (2014)

    Article  MathSciNet  Google Scholar 

  34. DiPerna, R.J., Majda, A.J.: Concentrations in regularizations for \(2\)-D incompressible flow. Commun. Pure Appl. Math. 40(3), 301–345 (1987)

    Article  MathSciNet  Google Scholar 

  35. DiPerna, R.J., Majda, A.J.: Oscillations and concentrations in weak solutions of the incompressible fluid equations. Commun. Math. Phys. 108(4), 667–689 (1987)

    Article  ADS  MathSciNet  Google Scholar 

  36. Daneri, S., Székelyhidi Jr., L.: Non-uniqueness and h-principle for Hölder-continuous weak solutions of the Euler equations. Arch. Ration. Mech. Anal. 224(2), 471–514 (2017)

    Article  MathSciNet  Google Scholar 

  37. Frisch, U.: Turbulence. Cambridge University Press, Cambridge (1995). The legacy of A. N. Kolmogorov

    Book  Google Scholar 

  38. Isett, P.: A proof of Onsager’s conjecture. Ann. Math.(2) 188(3), 871–963 (2018)

    Article  MathSciNet  Google Scholar 

  39. Le Bris, C., Lions, P.-L.: Renormalized solutions of some transport equations with partially \(W^{1,1}\) velocities and applications. Ann. Math. Pura Appl. (4) 183(1), 97–130 (2004)

    Article  Google Scholar 

  40. Luo, X.: Stationary solutions and nonuniqueness of weak solutions for the Navier–Stokes equations in high dimensions. Arch. Ration. Mech. Anal. 233(2), 701–747 (2019)

    Article  MathSciNet  Google Scholar 

  41. Mandelbrot, B.: Intermittent turbulence and fractal dimension: kurtosis and the spectral exponent \(5/3+B\). In Turbulence and Navier-Stokes equations (Proc. Conf., Univ. Paris-Sud, Orsay, 1975), pages 121–145. Lecture Notes in Math., Vol. 565. Springer, Berlin, 1976

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  44. Modena, S., Sattig, G.: Convex integration solutions to the transport equation with full dimensional concentration. Annales de l’Institut Henri Poincaré C, Analyse non linèaire (2020)

  45. Novack, M.: Nonuniqueness of weak solutions to the 3 dimensional quasi-geostrophic equations. SIAM J. Math. Anal. 52(4), 3301–3349 (2020)

    Article  MathSciNet  Google Scholar 

  46. Robinson, J.C., Sadowski, W.: Almost-everywhere uniqueness of Lagrangian trajectories for suitable weak solutions of the three-dimensional Navier–Stokes equations. Nonlinearity 22(9), 2093–2099 (2009)

    Article  ADS  MathSciNet  Google Scholar 

  47. Robinson, J.C., Sadowski, W.: A criterion for uniqueness of Lagrangian trajectories for weak solutions of the 3D Navier–Stokes equations. Commun. Math. Phys. 290(1), 15–22 (2009)

    Article  ADS  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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Acknowledgements

AC was partially supported by the NSF grant DMS–1909849. The authors thank the anonymous referee for helpful comments.

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Authors and Affiliations

  1. Department of Mathematics, Statistics and Computer Science, University of Illinois at Chicago, Chicago, IL, 60607, USA

    Alexey Cheskidov

  2. Department of Mathematics, Duke University, Durham, NC, 27708, USA

    Xiaoyutao Luo

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Cheskidov, A., Luo, X. Nonuniqueness of Weak Solutions for the Transport Equation at Critical Space Regularity. Ann. PDE 7, 2 (2021). https://doi.org/10.1007/s40818-020-00091-x

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