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Non-uniqueness of Leray Solutions to the Hypodissipative Navier–Stokes Equations in Two Dimensions

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Abstract

We exhibit non-unique Leray solutions of the forced Navier–Stokes equations with hypodissipation in two dimensions. Unlike the solutions constructed in Albritton et al. (Ann Math 196(1):415–455, 2022), the solutions we construct live at a supercritical scaling, in which the hypodissipation formally becomes negligible as \(t \rightarrow 0^+\). In this scaling, it is possible to perturb the Euler non-uniqueness scenario of Vishik (Instability and non-uniqueness in the Cauchy problem for the Euler equations of an ideal incompressible fluid. Part I, 2018; Instability and non-uniqueness in the Cauchy problem for the Euler equations of an ideal incompressible fluid. Part II, 2018) to the hypodissipative setting at the nonlinear level. Our perturbation argument is quasilinear in spirit and circumvents the spectral theoretic approach to incorporating the dissipation in Albritton et al. (2022).

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Notes

  1. Indeed, the majority of the present work was actually written before we realized that the spectral problem in [ABC22b] was manageable. This work is independent from [ABC22b] except for the observation that Vishik’s vortex can be truncated.

  2. One must be more careful with the functional set-up in the critical case.

  3. A different approach is to work with the Lagrangian formulation, in which there is no derivative loss, see the construction of unstable manifolds in [LZ13, LZ14].

  4. Crucially, this is where we use assumptions on the semigroup \(e^{\tau \varvec{L}_\textrm{ss}}\) generated by the linearized operator. While the above Duhamel formula ‘loses derivatives’ in the sense that we are estimating in \(L^2\) according to ‘higher’ quantities, e.g., \(\Vert \nabla \Omega \Vert _{L^2}\), it is acceptable to lose derivatives at this level, though it will not be acceptable at the ‘top tier’ (\(\Vert \nabla \Omega \Vert _{L^4}\)). This is common in quasi-linear perturbation arguments and can already be seen in standard proofs of local-in-time existence for the Euler equations.

  5. When \(\beta \le 1\), one must argue existence and uniqueness in a quasilinear fashion, whereas when \(\beta \in (1,2)\) it is possible to develop the well-posedness theory in a semilinear fashion.

References

  1. Albritton, D., Brué, E., Colombo, M., De Lellis, C., Giri, V., Janisch, M., Kwon, H.: Instability and nonuniqueness for the \(2d\) Euler equations in vorticity form, after M. Vishik (2021)

  2. Albritton, D., Bruè, E., Colombo, M.: Gluing non-unique Navier–Stokes solutions (2022). arXiv preprint arXiv:2209.03530

  3. Albritton, D., Brué, E., Colombo, M.: Non-uniqueness of Leray solutions of the forced Navier–Stokes equations. Ann. Math. 196(1), 415–455 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bardos, C., Guo, Y., Strauss, W.: Stable and Unstable Ideal Plane Flows, vol. 23. Dedicated to the Memory of Jacques-Louis Lions, pp. 149–164 (2002)

  5. Bressan, A., Murray, R.: On self-similar solutions to the incompressible Euler equations. J. Differ. Equ. 269(6), 5142–5203 (2020)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  6. Bressan, A., Shen, W.: A posteriori error estimates for self-similar solutions to the Euler equations. Discrete Contin. Dyn. Syst. 41(1), 113–130 (2021)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  8. Córdoba, A., Córdoba, D.: A maximum principle applied to quasi-geostrophic equations. Commun. Math. Phys. 249(3), 511–528 (2004)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  9. Colombo, M., De Lellis, C., De Rosa, L.: Ill-posedness of Leray solutions for the hypodissipative Navier–Stokes equations. Commun. Math. Phys. 362(2), 659–688 (2018)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  10. Chickering, K.R., Moreno-Vasquez, R.C., Pandya, G.: Asymptotically self-similar shock formation for 1d fractal Burgers equation (2021). arXiv preprint arXiv:2105.15128

  11. Caffarelli, L., Silvestre, L.: An extension problem related to the fractional Laplacian. Commun. Partial Differ. Equ. 32(7–9), 1245–1260 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  12. Caffarelli, L., Vazquez, J.L.: Nonlinear porous medium flow with fractional potential pressure. Arch. Ration. Mech. Anal. 202(2), 537–565 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  13. De Rosa, L.: Infinitely many Leray–Hopf solutions for the fractional Navier–Stokes equations. Comm. Partial Differ. Equ. 44(4), 335–365 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  14. Guillod, J., Šverák, V.: Numerical investigations of non-uniqueness for the Navier–Stokes initial value problem in borderline spaces. ArXiv e-prints, April (2017)

  15. Jia, H., Šverák, V.: Local-in-space estimates near initial time for weak solutions of the Navier–Stokes equations and forward self-similar solutions. Invent. Math. 196(1), 233–265 (2014)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  16. Jia, H., Šverák, V.: Are the incompressible 3d Navier–Stokes equations locally ill-posed in the natural energy space? J. Funct. Anal. 268(12), 3734–3766 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  17. Leray, J.: Sur le mouvement d’un liquide visqueux emplissant l’espace. Acta Math. 63(1), 193–248 (1934)

    Article  MathSciNet  MATH  Google Scholar 

  18. Lin, Z., Zeng, C.: Unstable manifolds of Euler equations. Commun. Pure Appl. Math. 66(11), 1803–1836 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  19. Lin, Z., Zeng, C.: Corrigendum: Unstable manifolds of Euler equations [mr3103911]. Commun. Pure Appl. Math. 67(7), 1215–1217 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  20. Merle, F., Raphael, P., Rodnianski, I., Szeftel, J.: On the implosion of a three dimensional compressible fluid (2019). arXiv preprint arXiv:1912.11009

  21. Oh, S.-J., Pasqualotto, F.: Gradient blow-up for dispersive and dissipative perturbations of the Burgers equation (2021). arXiv preprint arXiv:2107.07172

  22. Tao, T.: Finite time blowup for an averaged three-dimensional Navier–Stokes equation. J. Am. Math. Soc. 29(3), 601–674 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  23. Vishik, M.: Instability and Non-uniqueness in the Cauchy Problem for the Euler Equations of an Ideal Incompressible Fluid. Part I (2018)

  24. Vishik, M.: Instability and Non-uniqueness in the Cauchy Problem for the Euler Equations of an Ideal Incompressible Fluid. Part II (2018)

  25. Yudovich, V.I.: Non-stationary flow of an ideal incompressible liquid. USSR Comput. Math. Math. Phys. 3(6), 1407–1456 (1963)

    Article  MATH  Google Scholar 

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Acknowledgements

DA was supported by NSF Postdoctoral Fellowship Grant No. 2002023 and Simons Foundation Grant No. 816048. MC was supported by the SNSF Grant 182565. We thank the anonymous referees for their valuable work.

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  1. Department of Mathematics, Princeton University, Princeton, NJ, 08544, USA

    Dallas Albritton

  2. EPFL SB, Station 8, 1015, Lausanne, Switzerland

    Maria Colombo

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  1. Dallas Albritton
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Correspondence to Dallas Albritton.

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Albritton, D., Colombo, M. Non-uniqueness of Leray Solutions to the Hypodissipative Navier–Stokes Equations in Two Dimensions. Commun. Math. Phys. 402, 429–446 (2023). https://doi.org/10.1007/s00220-023-04725-6

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