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Instability and nonuniqueness in mathematical fluid dynamics

Department of Decision Sciences, Bocconi University, Via Sarfatti 25, Milan 20136, Italy

https://doi.org/10.1142/S0219891624400034Cited by:0 (Source: Crossref)

This article is part of the issue:

Special Issue: Recent Advances in Nonlinear Hyperbolic PDEs and Fluid Dynamics
Special Issue Editors: Fabio Ancona (University of Padua, Italy), Stefano Bianchini (SISSA, Italy)

Abstract

In this note, we review and expand upon the material presented in the minicourse titled “Instability and Nonuniqueness in Fluid Dynamics,” which the author held during the “Meeting on Nonlinear Evolution PDEs, Fluid Dynamics, and Transport Equations” at the Ettore Majorana Centre in Erice in May 2023.

Communicated by F. Ancona and S. Bianchini

Keywords:

AMSC: 35Q30, 35Q35

References

1. D. Albritton , Lecture Notes on Instability and Non-uniqueness in the Navier–Stokes Equations, UMN Summer Workshop on Analysis of PDEs, University of Minnesota (2022). Google Scholar
2. D. Albritton, E. Bruè and M. Colombo , Nonuniqueness of Leray solutions of the forced Navier–Stokes equations, Ann. Math. 196(2) (2022) 415–455. Google Scholar
3. D. Albritton, E. Bruè and M. Colombo , Gluing non-unique Navier–Stokes solutions, Ann. PDE 9(2) (2023) Paper No. 17, 25 pp. Crossref, Web of Science, Google Scholar
4. D. Albritton, E. Bruè, M. Colombo, C. De Lellis, V. Giri, M. Janisch and H. Kwon , Instability and Non-uniqueness for the 2D Euler Equations, After M. Vishik, Annals of Mathematics Studies, Vol. 219 (Princeton University Press, Princeton, NJ, 2024). Crossref, Google Scholar
5. D. Albritton and M. Colombo , Non-uniqueness of Leray solutions to the hypodissipative Navier–Stokes equations in two dimensions, Comm. Math. Phys. 402(1) (2023) 429–446. Crossref, Web of Science, Google Scholar
6. L. Ambrosio , Transport equation and Cauchy problem for $B V$ vector fields, Invent. Math. 158 (2004) 227–260. Crossref, Web of Science, Google Scholar
7. A. Bensoussan and R. Temam , Equations stochastiques du type Navier–Stokes, J. Funct. Anal. 13 (1973) 195–222. Crossref, Google Scholar
8. A. Bertozzi and A. Majda , Vorticity and Incompressible Flows (Cambridge University Press, Cambridge, 2002). Google Scholar
9. Z. Bradshaw and T.-P. Tsai , Forward discretely self-similar solutions of the Navier–Stokes equations II, Ann. Henri Poincarè 18(3) (2017) 1095–1119. Crossref, Web of Science, Google Scholar
10. A. Bressan and R. Murray , On self-similar solutions to the incompressible Euler equations, J. Differential Equations 269(6) (2020) 5142–5203. Crossref, Web of Science, Google Scholar
11. A. Bressan and W. Shen , A posteriori error estimates for self-similar solutions to the Euler equations, Discrete Contin. Dyn. Syst. 41(1) (2021) 113–130. Crossref, Web of Science, Google Scholar
12. E. Bruè and M. Colombo , Nonuniqueness of solutions to the Euler equations with vorticity in a Lorentz space, Commun. Math. Phys. 403(2) (2023) 1171–1192. Crossref, Web of Science, Google Scholar
13. E. Bruè, M. Colombo and C. De Lellis , Positive solutions of transport equations and classical nonuniqueness of characteristic curves, Arch. Ration. Mech. Anal. 240(2) (2021) 1055–1090. Crossref, Web of Science, Google Scholar
14. E. Bruè, M. Colombo and A. Kumar, Sharp nonuniqueness in the transport equation with sobolev velocity field (2024), https://arxiv.org/abs/2405.01670. Google Scholar
15. E. Bruè, M. Colombo and A. Kumar, Flexibility of two-dimensional Euler flows with integrable vorticity (2024), https://arxiv.org/abs/2408.07934. Google Scholar
16. E. Bruè and C. De Lellis , Anomalous dissipation for the forced 3D Navier–Stokes equations, Comm. Math. Phys. 400(3) (2023) 1507–1533. Crossref, Web of Science, Google Scholar
17. E. Bruè, R. Jin, Y. Li and D. Zhang, Non-uniqueness in law of Leray solutions to 3D forced stochastic Navier–Stokes equations (2023), https://arxiv.org/pdf/2309.09753. Google Scholar
18. M. Buck and S. Modena , Non-uniqueness and energy dissipation for 2D Euler equations with vorticity in hardy spaces, J. Math. Fluid Mech. 26 (2024), Paper No. 26, 39. Crossref, Web of Science, Google Scholar
19. M. Buck and S. Modena , Non-uniqueness and energy dissipation for 2D Euler equations with vorticity in hardy spaces, J. Math. Fluid Mech. 26 (2024) 26. Crossref, Web of Science, Google Scholar
20. T. Buckmaster , Onsager’s conjecture almost everywhere in time, Comm. Math. Phys. 333(3) (2015) 1175–1198. Crossref, Web of Science, Google Scholar
21. T. Buckmaster, M. Colombo and V. Vicol , Wild solutions of the Navier–Stokes equations whose singular sets in time have Hausdorff dimension strictly less than 1, J. Eur. Math. Soc. 24(9) (2022) 3333–3378. Crossref, Web of Science, Google Scholar
22. T. Buckmaster, C. De Lellis, P. Isett and L. Székelyhidi Jr. , Anomalous dissipation for $1 / 5$ -Hölder Euler flows, Ann. Math. 182(1) (2015) 127–172. Crossref, Google Scholar
23. T. Buckmaster, C. De Lellis, L. Székelyhidi Jr. and V. Vicol , Onsager’s conjecture for admissible weak solutions, Comm. Pure Appl. Math. 72(2) (2019) 229–274. Crossref, Web of Science, Google Scholar
24. T. Buckmaster and V. Vicol , Nonuniqueness of weak solutions to the Navier–Stokes equation, Ann. Math. 189(1) (2019) 101–144. Crossref, Web of Science, Google Scholar
25. T. Buckmaster and V. Vicol , Convex integration constructions in hydrodynamics, Bull. Amer. Math. Soc. 58(1) (2021) 1–44. Crossref, Web of Science, Google Scholar
26. L. Caffarelli, R. Kohn and L. Nirenberg , Partial regularity of suitable weak solutions of the Navier–Stokes equations, Comm. Pure Appl. Math. 35(6) (1982) 771–831. Crossref, Web of Science, Google Scholar
27. A. Castro, D. Faraco, M. Mengual and M. Solera, A proof of Vishik’s nonuniqueness Theorem for the forced 2D Euler equation (2024), https://arxiv.org/abs/2404.15995. Google Scholar
28. A. Cheskidov and X. Luo , Sharp nonuniqueness for the Navier–Stokes equations, Invent. Math. 229(3) (2022) 987–1054. Crossref, Web of Science, Google Scholar
29. A. Cheskidov and X. Luo , $L^{2}$ -critical nonuniqueness for the 2D Navier–Stokes equations, Ann. PDE 9(2) (2023) 1–56. Crossref, Web of Science, Google Scholar
30. P. Constantin, Weinan E and E. S. Titi , Onsager’s conjecture on the energy conservation for solutions of Euler’s equation, Comm. Math. Phys. 165(1) (1994) 207–209. Crossref, Google Scholar
31. S. Daneri and L. Székelyhidi Jr. , Non-uniqueness and h-principle for Hölder-continuous weak solutions of the Euler equations, Arch. Ration. Mech. Anal. 224(2) (2017) 471–514. Crossref, Web of Science, Google Scholar
32. C. De Lellis and L. Székelyhidi Jr. , The Euler equations as a differential inclusion, Ann. Math. 170(3) (2009) 1417–1436. Crossref, Web of Science, Google Scholar
33. C. De Lellis and L. Székelyhidi Jr. , Dissipative continuous Euler flows, Invent. Math. 193(2) (2013) 377–407. Crossref, Web of Science, Google Scholar
34. C. De Lellis and L. Székelyhidi Jr. , Weak stability and closure in turbulence, Trans. Roy. Soc. A 380(2218) (2022) 1–16. Google Scholar
35. J.-M. Delort , Existence de nappes de tourbillon en dimension deux, J. Amer. Math. Soc. 4(3) (1991) 553–586. Crossref, Google Scholar
36. R.-J. DiPerna and P.-L. Lions , Ordinary differential equations, transport theory and Sobolev spaces, Invent. Math. 98 (1989) 511–547. Crossref, Web of Science, Google Scholar
37. R. J. DiPerna and A. J. Majda , Concentrations in regularizations for 2-D incompressible flow, Comm. Pure Appl. Math. 40(3) (1987) 301–345. Crossref, Web of Science, Google Scholar
38. P. G. Drazin and W. H. Reid , Hydrodynamic Stability, Cambridge Mathematical Library, 2nd edn. (Cambridge University Press, Cambridge, 2004). With a foreword by John Miles. Crossref, Google Scholar
39. T. M. Elgindi , Finite-time singularity formation for smooth Euler solutions, Ann. Math. 196(3) (2022) 1047–1074. Google Scholar
40. K.-J. Engel and R. Nagel , One-Parameter Semigroups for Linear Evolution Equations, Graduate Texts in Mathematics, Vol. 194 (Springer-Verlag, New York, 2000). With contributions by S. Brendle, M. Campiti, T. Hahn, G. Metafune, G. Nickel, D. Pallara, C. Perazzoli, A. Rhandi, S. Romanelli, and R. Schnaubelt. Google Scholar
41. C. Fefferman , Existence and Smoothness of the Navier–Stokes Equation (Clay Mathematics Institute, 2000). Google Scholar
42. D. Gilbarg and N.-S. Trudinger , Elliptic Partial Differential Equations of Second Order (Springer, 1977). Crossref, Google Scholar
43. V. Giri and R.-O. Radu, The $2 d$ Onsager conjecture: A Newton–Nash iteration (2024), https://arxiv.org/abs/2305.18105. Google Scholar
44. J. Guillod and V. Sverák, Numerical investigations of non-uniqueness for the Navier–Stokes initial value problem in borderline spaces (2017), https://arxiv.org/abs/arXiv:1704.00560. Google Scholar
45. N. Gunther , On the motion of fluid in a moving container, Izv. Akad. Nauk USSR, Ser. Fiz.-Mat. 20 (1927) 1323–1348. Google Scholar
46. M. Hofmanova, R. Zhu and X. Zhu, Non-uniqueness of Leray–Hopf solutions for stochastic forced Navier–Stokes equations (2023), https://arxiv.org/abs/2309.03668. Google Scholar
47. E. Hopf , Über die Anfangswertaufgabe für die hydrodynamischen Grundgleichungen, Math. Nachr. 4 (1951) 213–231. Crossref, Google Scholar
48. P. Isett , A proof of Onsager’s conjecture, Ann. Math. 188(3) (2018) 871–963. Crossref, Web of Science, Google Scholar
49. P. Isett , Nonuniqueness and existence of continuous, globally dissipative Euler flows, Arch. Ration. Mech. Anal. 244(3) (2022) 1223–1309. Crossref, Web of Science, Google Scholar
50. H. Jia and V. Sverák , Local-in-space estimates near initial time for weak solutions of the Navier–Stokes equations and forward self-similar solutions, Invent. Math. 196(1) (2014) 233–265. Crossref, Web of Science, Google Scholar
51. H. Jia and V. Sverák , Are the incompressible 3d Navier–Stokes equations locally ill-posed in the natural energy space?, J. Funct. Anal. 268(12) (2015) 3734–3766. Crossref, Google Scholar
52. T. Kato , Perturbation Theory for Linear Operators (Die Grundlehren der mathematischen Wissenschaften, Band 132) (Springer-Verlag New York, Inc., 1966). Crossref, Google Scholar
53. T. Kato , Strong $L^{p}$ -solutions of the Navier–Stokes equations in $ℝ^{m}$ , with applications to weak solutions, Math. Z. 187(4) (1984) 471–480. Crossref, Web of Science, Google Scholar
54. C. Khor, C. Miao and X. Su , Non-uniqueness of Leray–Hopf solutions to the forced fractional Navier–Stokes equations in three dimensions, up to the JL Lions exponent, Bull. London Math. Soc. 55(6) (2023) 2705–2717. Crossref, Google Scholar
55. H. Kielhöfer , Bifurcation Theory: An Introduction with Applications to Partial Differential Equations (Springer Science and Business Media, 2011). Google Scholar
56. A. Kolmogorov , Dissipation of energy in the locally isotropic turbulence, Dokl. Akad. Nauk SSSR 32 (1941) 16–18. Google Scholar
57. A. Kolmogorov , Local structure of turbulence in an incompressible fluid at very high Reynolds number, Dokl. Acad. Nauk SSSR 30(4) (1941) 299–303. Google Scholar
58. A. Kolmogorov , On degeneration of isotropic turbulence in an incompressible viscous liquid, Dokl. Akad. Nauk SSSR 31 (1941) 538–540. Google Scholar
59. O. A. Ladyzhenskaya , Unique global solvability of the three-dimensional Cauchy problem for the Navier–Stokes equations in the presence of axial symmetry, Zap. Naučn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 7 (1968) 155–177. Google Scholar
60. O. A. Ladyzhenskaya , An example of nonuniqueness in Hopf’s class of weak solutions of the Navier–Stokes equations, Izv. Akad. Nauk SSSR Ser. Mat. 33 (1969) 240–247. Google Scholar
61. O. Ladyzhenskaya , The Mathematical Theory of Viscous Incompressible Flows, 2nd edn. (Gordon and Breach, New York, 1969). Google Scholar
62. P. G. Lemarié-Rieusset , Recent Developments in the Navier–Stokes Problem, Chapman & Hall/CRC Research Notes in Mathematics, Vol. 431 (Chapman & Hall/CRC, Boca Raton, FL, 2002). Crossref, Google Scholar
63. J. Leray , Sur le mouvement d’un liquide visqueux emplissant l’espace, Acta Math. 63(1) (1934) 193–248. Crossref, Google Scholar
64. L. Lichtenstein , Über einige Existenzprobleme der Hydrodynamik homogener, unzusammendrückbarer, reibungsloser Flüssigkeiten und die Helmholtzschen Wirbelsätze, Math. Z. 23(1) (1925) 89–154. Crossref, Google Scholar
65. Z. Lin , Instability of some ideal plane flows, SIAM J. Math. Anal. 35(2) (2003) 318–356. Crossref, Web of Science, Google Scholar
66. F. Mengual, Non-uniqueness of admissible solutions for the 2D Euler equation with $L^{p}$ vortex data (2023), https://doi.org/10.48550/arXiv.2304.09578. Google Scholar
67. S. Modena and L. Székelyhidi , Non-uniqueness for the transport equation with Sobolev vector fields, Ann. PDE 4(2) (2018) 1–38. Crossref, Google Scholar
68. S. Muller and V. Sverák , Convex integration for Lipschitz mappings and counterexamples to regularity, Ann. Math. 157(3) (2003) 715–742. Crossref, Web of Science, Google Scholar
69. J. Nash , $C^{1}$ isometric embeddings, Ann. Math. 60 (1954) 383–396. Crossref, Web of Science, Google Scholar
70. M. Novack and V. Vicol , An intermittent onsager theorem, Invent. Math. 233 (2023) 223–323. Crossref, Web of Science, Google Scholar
71. L. Onsager , Statistical hydrodynamics, Nuovo Cimento (9) 6(Supplemento 2) (1949) 279–287 (Convegno Internazionale di Meccanica Statistica). Crossref, Google Scholar
72. L. Rayleigh , On the stability or instability of certain fluid motions, London Math. Soc. 9 (1880) 57–70. Google Scholar
73. V. Scheffer , An inviscid flow with compact support in space-time, J. Geom. Anal. 3(4) (1993) 343–401. Crossref, Web of Science, Google Scholar
74. A. Shnirelman , Weak solutions with decreasing energy of incompressible Euler equations, Comm. Math. Phys. 210(3) (2000) 541–603. Crossref, Web of Science, Google Scholar
75. T. Tao, Local Well-Posedness of the Navier–Stokes Equations, Terry Tao’s Blog, September 16, 2018. Google Scholar
76. W. Tollmien , Ein allgemeines Kriterium der Instabilitat laminarer Geschwindigkeitsverteilungen, Nachr. Ges. Wiss. Gottingen, Math. Phys. Klasse NF 1 (1935) 79–114. Google Scholar
77. T.-P. Tsai , Lectures on Navier–Stokes Equations, Graduate Studies in Mathematics, Vol. 192 (American Mathematical Society, Providence, RI, 2018). Crossref, Google Scholar
78. J. Vassilicos , Dissipation in turbulent flows, Annu. Rev. Fluid Mech. 47 (2015) 95–114. Crossref, Web of Science, Google Scholar
79. M. Vishik, Instability and non-uniqueness in the Cauchy problem for the Euler equations of an ideal incompressible fluid. Part I (2018), https://arxiv.org/abs/1805.09426. Google Scholar
80. M. Vishik, Instability and non-uniqueness in the Cauchy problem for the Euler equations of an ideal incompressible fluid. Part II (2018), https://arxiv.org/abs/1805.09440. Google Scholar
81. V. I. Yudovich , Non-stationary flows of an ideal incompressible fluid, Z. Vycisl. Mat i Mat. Fiz. 3 (1963) 1032–1066. Google Scholar