A new path to the non blow-up of incompressible flows

Agélas, Léo

doi:10.1016/j.anihpc.2019.04.003

Agélas, Léo

Annales de l'I.H.P. Analyse non linéaire, Tome 36 (2019) no. 6, pp. 1503-1537.

Le texte intégral des articles récents est réservé aux abonnés de la revue. Consultez l'article sur le site de la revue.

Résumé

One of the most challenging questions in fluid dynamics is whether the three-dimensional (3D) incompressible Navier-Stokes, 3D Euler and two-dimensional Quasi-Geostrophic (2D QG) equations can develop a finite-time singularity from smooth initial data. Recently, from a numerical point of view, Luo & Hou presented a class of potentially singular solutions to the Euler equations in a fluid with solid boundary [1, 2]. Furthermore, in two recent papers [3, 4], Tao indicates a significant barrier to establishing global regularity for the 3D Euler and Navier-Stokes equations, in that any method for achieving this, must use the finer geometric structure of these equations. In this paper, we show that the singularity discovered by Luo & Hou which lies right on the boundary is not relevant in the case of the whole domain $R^{3}$ . We reveal also that the translation and rotation invariance present in the Euler, Navier-Stokes and 2D QG equations are the key for the non blow-up in finite time of the solutions. The translation and rotation invariance of these equations combined with the anisotropic structure of regions of high vorticity allowed to establish a new geometric non blow-up criterion which yield us to the non blow-up of the solutions in all the Kerr's numerical experiments and to show that the potential mechanism of blow-up introduced in [5] cannot lead to the blow-up in finite time of solutions of Euler equations.

DOI : 10.1016/j.anihpc.2019.04.003

Classification : 35Q30, 35Q31, 76B60, 76B65, 76B03
Mots-clés : 3D Euler equations, 3D Navier-Stokes equations, 2D Quasi-Geostrophic equation, Finite time singularities, Geometric properties for non blow-up

@article{AIHPC_2019__36_6_1503_0,
     author = {Ag\'elas, L\'eo},
     title = {A new path to the non blow-up of incompressible flows},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {1503--1537},
     publisher = {Elsevier},
     volume = {36},
     number = {6},
     year = {2019},
     doi = {10.1016/j.anihpc.2019.04.003},
     mrnumber = {4002165},
     language = {en},
     url = {http://www.numdam.org/articles/10.1016/j.anihpc.2019.04.003/}
}

TY  - JOUR
AU  - Agélas, Léo
TI  - A new path to the non blow-up of incompressible flows
JO  - Annales de l'I.H.P. Analyse non linéaire
PY  - 2019
SP  - 1503
EP  - 1537
VL  - 36
IS  - 6
PB  - Elsevier
UR  - http://www.numdam.org/articles/10.1016/j.anihpc.2019.04.003/
DO  - 10.1016/j.anihpc.2019.04.003
LA  - en
ID  - AIHPC_2019__36_6_1503_0
ER  -

%0 Journal Article
%A Agélas, Léo
%T A new path to the non blow-up of incompressible flows
%J Annales de l'I.H.P. Analyse non linéaire
%D 2019
%P 1503-1537
%V 36
%N 6
%I Elsevier
%U http://www.numdam.org/articles/10.1016/j.anihpc.2019.04.003/
%R 10.1016/j.anihpc.2019.04.003
%G en
%F AIHPC_2019__36_6_1503_0

Agélas, Léo. A new path to the non blow-up of incompressible flows. Annales de l'I.H.P. Analyse non linéaire, Tome 36 (2019) no. 6, pp. 1503-1537. doi : 10.1016/j.anihpc.2019.04.003. http://www.numdam.org/articles/10.1016/j.anihpc.2019.04.003/

Bibliographie
Cité par

[1] Luo, G.; Hou, T.Y. Potentially singular solutions of the 3D axisymmetric Euler equations, Proc. Natl. Acad. Sci., Volume 111 (2014) no. 36, pp. 12968–12973

[2] Luo, G.; Hou, T.Y. Toward the finite time blowup of the 3D axisymmetric Euler equations: a numerical investigation, Multiscale Model. Simul., Volume 12 (2014) no. 4, pp. 1722–1776 | MR

[3] Tao, T. Finite time blow-up for an averaged three-dimensional Navier-Stokes equations, J. Am. Math. Soc., Volume 29 (2016), pp. 601–674 | MR

[4] Tao, T. Finite time blowup for Lagrangian modifications of the three-dimensional Euler equation, Ann. Partial Differ. Equ., Volume 2 (2016) no. 9 | MR

[5] Brenner, M.P.; Hormoz, S.; Pumir, A. Potential singularity mechanism for the Euler equations, Phys. Rev. Fluids, Volume 1 (2016) no. 084503

[6] Constantin, P. On the Euler equations of incompressible fluids, Bull. Am. Math. Soc., Volume 44 (2007), pp. 603–621 | DOI | MR | Zbl

[7] Bardos, C.; Titi, E.S. Euler equations for an ideal incompressible fluid, Usp. Mat. Nauk, Volume 62 (2007) no. 3, pp. 5–46 | MR | Zbl

[8] Leray, J. Sur le mouvement d'un liquide visqueux emplissant l'espace, Acta Math., Volume 63 (1934), pp. 193–248 | DOI | JFM | MR

[9] Hopf, E. Über die anfangwertaufgabe für die hydrohynamischen grundgleichungen, Math. Nachr., Volume 4 (1951), pp. 213–231 | MR | Zbl

[10] Ladyzhenskaya, O. The Mathematical Theory of Viscous Incompressible Flows, Gordon and Breach, 1969 | MR | Zbl

[11] Lions, J.L.; Prodi, G. Un théorème d'existence et d'unicité dans les équations de Navier-Stokes en dimension 2, C. R. Math. Acad. Sci. Paris, Volume 248 (1959), pp. 3519–3521 | MR | Zbl

[12] Lions, J.L. Quelques Méthodes de Résolution des Problèmes aux Limites non Linéaires, Dunod, Paris, 1969 | MR | Zbl

[13] Temam, R. Navier-Stokes Equations, North-Holland, Amsterdam, 1977 | MR | Zbl

[14] Furioli, G.; Lemarié-Rieusset, P.G.; Terraneo, E. Unicité dans $L^{3} (R^{3})$ et d'autres espaces fonctionnels limites pour Navier-Stokes, Rev. Mat. Iberoam., Volume 16 (2000) no. 3 | DOI | MR | Zbl

[15] Giga, Y. Weak and strong solutions of the Navier-Stokes initial value problem, RIMS Kokyuroku Univ., Volume 19 (1983), pp. 887–910 | MR | Zbl

[16] Monniaux, S. Unicité dans $l^{d}$ des solutions du système de Navier-Stokes: cas des domaines lipschitziens, Ann. Math. Blaise Pascal, Volume 10 (2000), pp. 107–116 | Numdam | MR | Zbl

[17] Lions, J.L. Sur la régularité et l'unicité des solutions turbulentes des équations de Navier-stokes, Rend. Semin. Mat. Univ. Padova, Volume 30 (1960), pp. 16–23 | Numdam | MR | Zbl

[18] Gallagher, I.; Planchon, F. On global infinite energy solutions to the Navier-Stokes equations in two dimensions, Arch. Ration. Mech. Anal., Volume 161 (2002), pp. 307–337 | DOI | MR | Zbl

[19] Serrin, J. On the interior regularity of weak solutions of the Navier-Stokes equations, Arch. Ration. Mech. Anal., Volume 9 (1962), pp. 187–191 | DOI | MR | Zbl

[20] Von Wahl, W., Proc. Symposia in Pure Mathematics, Volume vol. 45, Amer. Math. Soc., Providence, Rhode Island (1983), pp. 497–503 | MR | Zbl

[21] Giga, Y. Solutions for semilinear parabolic equations in $L^{q}$ and regularity of weak solutions of the Navier-Stokes system, J. Differ. Equ., Volume 62 (1986), pp. 186–212 | DOI | MR | Zbl

[22] Iskauriaza, L.; Serëgin, G.A.; Sverák, V. $L_{3, \infty}$ -solutions of Navier-Stokes equations and backward uniqueness, Usp. Mat. Nauk, Volume 58 (2003) no. 2, pp. 3–44 | MR | Zbl

[23] He, C. Regularity for solutions to the Navier-Stokes equations with one velocity component regular, Electron. J. Differ. Equ., Volume 29 (2002), pp. 1–13 | MR | Zbl

[24] Heywood, J.G. Epochs of regularity for weak solutions of the Navier-Stokes equations in unbounded domains, Tohoku Math. J., Volume 40 (1988), pp. 293–313 | DOI | MR | Zbl

[25] Kato, T. Strong $l^{q}$ solutions of the Navier-Stokes equations in $R^{m}$ , with application to weak solutions, Math. Z., Volume 187 (1984), pp. 471–480 | DOI | MR | Zbl

[26] Kato, T. Liapunov functions and monotonicity in the Navier-Stokes equations, Lect. Notes Math., Volume 1450 (1990), pp. 53–63 | DOI | MR | Zbl

[27] Beirão da Veiga, H. A new regularity class for the Navier-Stokes equations in $R^{n}$ , Chin. Ann. Math., Ser. B, Volume 16 (1995) no. 4, pp. 407–412 | MR | Zbl

[28] Chae, D.; Choe, H-J. Regularity of solutions to the Navier-Stokes equations, Electron. J. Differ. Equ., Volume 5 (1999), pp. 1–7 | MR | Zbl

[29] Zhou, Y. A new regularity criterion for the Navier-Stokes equations in terms of the gradient of one velocity component, Methods Appl. Anal., Volume 9 (2002) no. 4, pp. 563–578 | DOI | MR | Zbl

[30] Constantin, P.; Fefferman, C. Direction of vorticity and the problem of global regularity for the Navier-Stokes equations, Indiana Univ. Math. J., Volume 42 (1994), pp. 775–789 | MR | Zbl

[31] Yudovich, V.I. Non-stationary flows of an ideal incompressible fluid, Ž. Vyčisl. Mat. Mat. Fiz., Volume 6 (1963) no. 3, pp. 1032–1066 | MR | Zbl

[32] Yudovich, V.I. Uniqueness theorem for the basic nonstationary problem in the dynamics of an ideal incompressible fluid, Math. Res. Lett., Volume 2 (1995) no. 1, pp. 27–38 | DOI | MR | Zbl

[33] Vishik, M. Incompressible flows of an ideal fluid with vorticity in borderline spaces of Besov type, Ann. Sci. Éc. Norm. Supér., Volume 32 (1999) no. 6, pp. 769–812 | Numdam | MR | Zbl

[34] DiPerna, R.J.; Majda, A.J.; Diperna, R.J.; Majda, A.J. Commun. Math. Phys., 108 (1987) no. 4, pp. 667–689 | DOI | MR | Zbl

[35] Camillo De Lellis, C.; Székelyhidi, L. Jr. The $h$ -principle and the equations of fluid dynamics, Bull. Am. Math. Soc., Volume 49 (2012) no. 3, pp. 347–375 | MR | Zbl

[36] Villani, C. Paradoxe de Scheffer-Shnirelman revu sous l'angle de l'intégration convexe, Séminaire Bourbaki 61e Année, vol. 1001, 2008 (D'après C. de Lellis et L. Székelyhidi) | Numdam | MR | Zbl

[37] Beale, J.T.; Kato, T.; Majda, A. Remarks on the breakdown of smooth solutions for the 3-D Euler equations, Commun. Math. Phys., Volume 94 (1984) no. 1, pp. 61–66 | DOI | MR | Zbl

[38] Ponce, G. Remarks on a paper by J. T. Beale, T. Kato, and A. Majda, Commun. Math. Phys., Volume 98 (1985), pp. 349–353 | DOI | MR | Zbl

[39] Ferrari, A.B. On the blow-up of solutions of the 3-D Euler equations in a bounded domain, Commun. Math. Phys., Volume 155 (1993), pp. 277–294 | DOI | MR | Zbl

[40] Shirota, T.; Yanagisawa, T. A continuation principle for the 3-D Euler equations for incompressible fluids in a bounded domain, Proc. Jpn. Acad., Ser. A, Math. Sci., Volume 69 (1993), pp. 77–82 | DOI | MR | Zbl

[41] Constantin, P.; Fefferman, C.; Majda, A.J. Geometric constraints on potentially singular solutions for the 3-D Euler equation, Commun. Partial Differ. Equ., Volume 21 (1996), pp. 559–571 | MR | Zbl

[42] Deng, J.; Hou, T.Y.; Yu, X. Geometric properties and non-blow-up of 3-D incompressible Euler flow, Commun. Partial Differ. Equ., Volume 30 (2005) no. 1, pp. 225–243 | MR | Zbl

[43] Gibbon, J.D.; Titi, E.S. The 3D incompressible Euler equations with a passive scalar: a road to blow-up?, J. Nonlinear Sci., Volume 23 (2013), pp. 993–1000 | DOI | MR | Zbl

[44] Held, I.M.; Pierrehumbert, R.T.; Garner, S.T.; Swanson, K.L. Surface quasi-geostrophic dynamics, J. Fluid Mech., Volume 282 (1995), pp. 1–20 | MR | Zbl

[45] Pedlosky, J. Geophysical Fluid Dynamics, Springer, New York, 1987 | DOI | Zbl

[46] Constantin, P.; Majda, A.; Tabak, E. Formation of strong fronts in the 2D quasi-geostrophic thermal active scalar, Nonlinearity, Volume 7 (1994), pp. 1495–1533 | DOI | MR | Zbl

[47] Córdoba, D. Nonexistence of simple hyperbolic blow-up for the quasi-geostrophic equation, Ann. Math., Volume 148 (1998), pp. 1135–1152 | DOI | MR | Zbl

[48] Córdoba, D.; Fefferman, C. Growth of solutions for QG and 2D Euler equations, J. Am. Math. Soc., Volume 15 (2002), pp. 665–670 | DOI | MR | Zbl

[49] Chae, D. The quasi-geostrophic equation in the Triebel-Lizorkin spaces, Nonlinearity, Volume 16 (2003), pp. 479–495 | DOI | MR | Zbl

[50] Ohkitani, K.; Yamada, M. Inviscid and inviscid-limit behavior of a surface quasi-geostrophic flow, Phys. Fluids, Volume 9 (1997) no. 4, pp. 876–882 | DOI | MR | Zbl

[51] Constantin, P.; Lai, M-C.; Sharma, R.; Tseng, Y-H.; Wu, J. New numerical results for the surface quasi-geostrophic equation, J. Sci. Comput., Volume 50 (2012) no. 1, pp. 1–28 | DOI | MR | Zbl

[52] Chae, D.; Constantin, P.; Wu, J. Deformation and symmetry in the inviscid SQG and the 3D Euler equations, J. Nonlinear Sci., Volume 22 (2012) no. 5, pp. 665–688 | DOI | MR | Zbl

[53] Montgomery-Smith, S. Finite time blow up for a Navier-Stokes like equation, Proc. Am. Math. Soc., Volume 129 (2001) no. 10, pp. 3025–3029 | DOI | MR | Zbl

[54] Gallagher, I.; Paicu, M. Remarks on the blow-up of solutions to a toy model for the Navier-Stokes equations, Proc. Am. Math. Soc., Volume 137 (2009) no. 6, pp. 2075–2083 | DOI | MR | Zbl

[55] Li, D.; Sinai, Ya. Blow ups of complex solutions of the 3D-Navier-Stokes system and renormalization group method, J. Eur. Math. Soc., Volume 10 (2008) no. 2, pp. 267–313 | MR | Zbl

[56] Plecháç, P.; Sverák, V. Singular and regular solutions of a nonlinear parabolic system, Nonlinearity, Volume 16 (2003) no. 6, pp. 2083–2097 | MR | Zbl

[57] Katz, N.H.; Pavlovic, N. Finite time blow-up for a dyadic model of the Euler equations, Trans. Am. Math. Soc., Volume 357 (2005) no. 2, pp. 695–708 | MR | Zbl

[58] Kambe, T. Gauge principle and variational formulation for flows of an ideal fluid, Acta Mech. Sin., Volume 19 (2003) no. 5, pp. 437–452 | MR

[59] Kambe, T. Gauge principle for flows of an ideal fluid, Fluid Dyn. Res., Volume 32 (2003), pp. 193–199 | DOI | MR | Zbl

[60] Constantin, P. Geometric statistic in turbulence, SIAM Rev., Volume 36 (1994) no. 1, pp. 73–98 | DOI | MR | Zbl

[61] Beirão da Veiga, H.; Berselli, L.C. Navier-Stokes equations: Green's matrices, vorticity direction, and regularity up to the boundary, J. Differ. Equ., Volume 246 (2009), pp. 597–628 | DOI | MR | Zbl

[62] Berselli, L.C. Some geometric constraints and the problem of global regularity for the Navier-Stokes equations, Nonlinearity, Volume 22 (2009) no. 10, pp. 2561–2581 | DOI | MR | Zbl

[63] Hou, T.Y.; Li, C. Dynamic stability of the 3D axi-symmetric Navier-Stokes equations with swirl, Commun. Pure Appl. Math., Volume 61 (2008), pp. 661–697 | MR | Zbl

[64] Deng, J.; Hou, T.Y.; Yu, X. Improved geometric conditions for non-blow-up of 3D incompressible Euler equation, Commun. Partial Differ. Equ., Volume 31 (2006), pp. 293–306 | DOI | MR | Zbl

[65] Deng, J.; Hou, T.Y.; Li, R.; Yu, X. Level set dynamics and the non-blow-up of the 2D quasi-geostrophic equation, Methods Appl. Anal., Volume 13 (2006) no. 2, pp. 157–180 | DOI | MR | Zbl

[66] Kerr, R.M.; Bustamante, M.D.; Robinson, J.C.; Rodrigo, J.L.; Sadowski, W. Exploring symmetry plane conditions in numerical Euler solutions, Mathematical Aspects of Fluid Mechanics, Cambridge University Press, 2012 | MR | Zbl

[67] Grafke, T. Finite-Time Euler Singularities: A Lagrangian Perspective, 2012 (PhD thesis in der Fakultät für Physik und Astronomie der Ruhr-Universität Bochum)

[68] Pshenichny, B.N. Necessary Conditions for an Extremum, Marcel Dekker, Inc., New York, NY, 1971 (Translated from Russian 1969) | MR | Zbl

[69] Kerr, R.M. Velocity and scaling of collapsing Euler vortices, Phys. Fluids, Volume 14 (2005) no. 075103 | MR | Zbl

[70] Kerr, R.M.; Tsinober; Gyr The outer regions in singular Euler, Fundamental Problematic Issues in Turbulence, Birkhäuser, Boston, 1998 | MR | Zbl

[71] Kerr, R.M. Euler singularities and turbulence, 19th ICTAM Kyoto '96, Elsevier Science, 1997

[72] Kuznetsov, E.A.; Ruban, V.P. Collapse of vortex lines in hydrodynamics, J. Exp. Theor. Phys., Volume 91 (2000) no. 4, pp. 775–785 | DOI

[73] Kuznetsov, E.A.; Podvigina, O.M.; Zheligovsky, V.A.; Bajer, K.; Moffatt, H.K. Numerical evidence of breaking of vortex lines in an ideal fluid, Tubes, Sheets and Singularities in Fluid Dynamics, Kluwer Academic Publishers, NATO ARW, 2001, pp. 305–316 | MR | Zbl

[74] Agafontsev, D.S.; Kuznetsov, E.A.; Mailybaev, A.A. Development of high vorticity structures in incompressible 3D Euler equations, Phys. Fluids, Volume 27 (2015) no. 085102

[75] Agafontsev, D.S.; Kuznetsov, E.A.; Mailybaev, A.A. Asymptotic solution for high-vorticity regions in incompressible three-dimensional Euler equations, J. Fluid Mech., Volume 813 (2017) | DOI | MR

[76] Grafke, T.; Grauer, R. Finite-time Euler singularities: a Lagrangian perspective, Appl. Math. Lett., Volume 26 (2013), pp. 500–505 | DOI | MR | Zbl

[77] Stein, E. Singular Integrals and Differentiability Properties of Functions, Princeton University Press, Princeton, NJ, 1970 | MR | Zbl

[78] Kato, T.; Ponce, G. Commutator estimates and the Euler and Navier-Stokes equations, Commun. Pure Appl. Math., Volume 41 (1988) no. 7, pp. 891–907 | DOI | MR | Zbl

[79] Bourguignon, J.P.; Brezis, H. Remarks on the Euler equation, J. Funct. Anal., Volume 15 (1974), pp. 341–363 | DOI | MR | Zbl

[80] Kato, T.; Ponce, G. Well-posedness of the Euler and Navier-Stokes equations in the Lebesgue spaces $Lsp (R^{2})$ , Rev. Mat. Iberoam., Volume 2 (1986), pp. 73–88 | DOI | MR | Zbl

[81] Kozono, H.; Taniuchi, Y. Bilinear estimates and critical Sobolev inequality in BMO, with applications to the Navier-Stokes and the Euler equations, RIMS Kokyuroku, Volume 1146 (2000), pp. 39–52 | MR | Zbl

[82] Agélas, L. Global regularity for logarithmically critical 2D MHD equations with zero viscosity, Monatshefte Math., Volume 181 (2016) no. 2, pp. 245–266 | DOI | MR

[83] Cordoba, D.; Fefferman, C. On the collapse of tubes carried by 3D incompressible flows, Comment. Phys.-Math., Volume 222 (2001), pp. 293–298 | MR | Zbl

[84] Kerr, R.M. Evidence for a singularity of the three-dimensional incompressible Euler equations, Phys. Fluids A, Volume 5 (1993), pp. 1725–1746 | MR | Zbl

[85] Kerr, R.M.; Pouquet, A.; Sulem, P.L. The role of singularities in Euler, Small-Scale Structure in Hydro and Magnetohydrodynamic Turbulence, Lecture Notes, Springer-Verlag, 1995

[86] Pelz, R.B. Locally self-similar, finite-time collapse in a high-symmetry vortex filament model, Phys. Rev. E, Volume 55 (1997) no. 2, pp. 1617–1620

[87] Pelz, R.B. Symmetry and the hydrodynamic blow-up problem, J. Fluid Mech., Volume 444 (2001), pp. 299–320 | MR | Zbl

[88] Borisenko, O.F.; Minchenko, L.I. Directional derivatives of the maximum function, Cybern. Syst. Anal., Volume 28 (1992) no. 2, pp. 309–312 | DOI | MR | Zbl

[89] Chorin, A.J.; Marsden, J.E. A Mathematical Introduction to Fluid Mechanics, Springer-Verlag, New York, 1993 | MR | Zbl

[90] Melander, M.V.; Hussain, F. Cross-linking of two antiparallel vortex tubes, Phys. Fluids, Volume 1 (1989), pp. 633–636 | DOI

[91] Kerr, R.M.; Hussain, F. Simulation of vortex reconnection, Physica D, Volume 37 (1989), pp. 474–484

[92] Pumir, A.; Siggia, E.D. Collapsing solutions to the 3-D Euler equations, Phys. Fluids A, Volume 2 (1990), pp. 220–241 | DOI | MR | Zbl

[93] Grauer, R.; Marliani, C.; Germaschewski, K. Adaptive mesh refinement for singular solutions of the incompressible Euler equations, Phys. Rev. Fluids, Volume 80 (1998), pp. 4177–4180

[94] Hou, T.Y.; Li, R. Dynamic depletion of vortex stretching and non-blow-up of the 3-D incompressible Euler equations, J. Nonlinear Sci., Volume 16 (2006), pp. 639–664 | MR | Zbl

[95] R.M. Kerr, Computational Euler history, 2006.

[96] Hormoz, S.; Brenner, M.P. Absence of singular stretching of interacting vortex filaments, J. Fluid Mech., Volume 707 (2012), pp. 191–204 | DOI | MR | Zbl

[97] Majda, A.; Bertozzi, A. Vorticity and Incompressible Flow, Cambridge Univ. Press, 2002 | MR | Zbl

[98] Teschl, G. Ordinary Differential Equations and Dynamical Systems, Graduate Studies in Mathematics, vol. 140, Amer. Math. Soc., 2012 | DOI | MR | Zbl

[99] Schmidt, S.; Schulz, V. Shape derivatives for general objective functions and the incompressible Navier-Stokes equations, Control Cybern., Volume 39 (2010) no. 3 | MR | Zbl

[100] Delfour, M.C.; Zolésio, J.-P. Shapes and Geometries: Metrics, Analysis, Differential Calculus, and Optimization, Siam, 2011 | MR | Zbl

[101] Wu, J.C. Elements of Vorticity Aerodynamics, Springer Tracts in Mechanical Engineering, vol. 140, Springer-Verlag, Berlin, Heidelberg, 2018 | DOI

Cité par Sources :