On self-similar finite-time blowups of the De Gregorio model on the real line. (English) Zbl 1529.35388
Summary: We show that the De Gregorio model on the real line admits infinitely many compactly supported, self-similar solutions that are distinct under rescaling and will blow up in finite time. These self-similar solutions fall into two classes: the basic class and the general class. The basic class consists of countably infinite solutions that are eigenfunctions of a self-adjoint compact operator. In particular, the leading eigenfunction coincides with the finite-time singularity solution of the De Gregorio model recently obtained by numerical approaches. The general class consists of more complicated solutions that can be obtained by solving nonlinear eigenvalue problems associated with the same compact operator.
MSC:
| 35Q35 | PDEs in connection with fluid mechanics |
| 35Q31 | Euler equations |
| 76B03 | Existence, uniqueness, and regularity theory for incompressible inviscid fluids |
| 35C06 | Self-similar solutions to PDEs |
| 35B44 | Blow-up in context of PDEs |
| 35B65 | Smoothness and regularity of solutions to PDEs |
| 35P30 | Nonlinear eigenvalue problems and nonlinear spectral theory for PDEs |
References:
| [1] | Carlen, E., Trace inequalities and quantum entropy: an introductory course, Entropy Quantum, 529, 73-140 (2010) · Zbl 1218.81023 · doi:10.1090/conm/529/10428 |
| [2] | Castro, A.; Córdoba, D., Infinite energy solutions of the surface quasi-geostrophic equation, Adv. Math., 225, 4, 1820-1829 (2010) · Zbl 1205.35219 · doi:10.1016/j.aim.2010.04.018 |
| [3] | Córdoba, A.; Córdoba, D.; Fontelos, MA, Formation of singularities for a transport equation with nonlocal velocity, Ann. Math., 162, 1377-1389 (2005) · Zbl 1101.35052 · doi:10.4007/annals.2005.162.1377 |
| [4] | Chen, J.: On the regularity of the De Gregorio model for the 3D Euler equations. J. Eur. Math. Soc (to appear). Preprint arXiv:2107.04777 |
| [5] | Chen, J., Singularity formation and global well-posedness for the generalized Constantin-Lax-Majda equation with dissipation, Nonlinearity, 33, 5, 2502 (2020) · Zbl 1481.35320 · doi:10.1088/1361-6544/ab74b0 |
| [6] | Chen, J., On the slightly perturbed De Gregorio model on \({S}^1\), Arch. Ration. Mech. Anal., 241, 3, 1843-1869 (2021) · Zbl 1475.35265 · doi:10.1007/s00205-021-01685-w |
| [7] | Chen, J., Hou, T.Y., Huang, D.: Matlab codes for computer-aided proofs in the paper “on the finite time blowup of the De Gregorio model for the 3D Euler equations”. https://www.dropbox.com/sh/jw3hkkgx9q6m9fj/AAC46ea4QorTCNAzVTkMQQnOa?dl=0 |
| [8] | Chen, J.; Hou, TY; Huang, D., On the finite time blowup of the De Gregorio model for the 3D Euler equations, Commun. Pure Appl. Math., 74, 6, 1282-1350 (2021) · Zbl 1469.35053 · doi:10.1002/cpa.21991 |
| [9] | Constantin, P.; Lax, PD; Majda, A., A simple one-dimensional model for the three-dimensional vorticity equation, Commun. Pure Appl. Math., 38, 6, 715-724 (1985) · Zbl 0615.76029 · doi:10.1002/cpa.3160380605 |
| [10] | De Gregorio, S., On a one-dimensional model for the three-dimensional vorticity equation, J. Stat. Phys., 59, 5, 1251-1263 (1990) · Zbl 0712.76027 · doi:10.1007/BF01334750 |
| [11] | De Gregorio, S., A partial differential equation arising in a 1D model for the 3D vorticity equation, Math. Methods Appl. Sci., 19, 15, 1233-1255 (1996) · Zbl 0860.35101 · doi:10.1002/(SICI)1099-1476(199610)19:15<1233::AID-MMA828>3.0.CO;2-W |
| [12] | Dong, H., Well-posedness for a transport equation with nonlocal velocity, J. Funct. Anal., 255, 11, 3070-3097 (2008) · Zbl 1170.35004 · doi:10.1016/j.jfa.2008.08.005 |
| [13] | Elgindi, TM; Jeong, I-J, On the effects of advection and vortex stretching, Arch. Ration. Mech. Anal., 235, 3, 1763-1817 (2020) · Zbl 1434.35091 · doi:10.1007/s00205-019-01455-9 |
| [14] | Hou, TY; Li, R., Dynamic depletion of vortex stretching and non-blowup of the 3-D incompressible Euler equations, J. Nonlinear Sci., 16, 6, 639-664 (2006) · Zbl 1370.76015 · doi:10.1007/s00332-006-0800-3 |
| [15] | Hou, TY; Li, C., Dynamic stability of the three-dimensional axisymmetric Navier-Stokes equations with swirl, Commun. Pure Appl. Math. A J. Issued Courant Inst. Math. Sci., 61, 5, 661-697 (2008) · Zbl 1138.35077 · doi:10.1002/cpa.20212 |
| [16] | Jia, H.; Stewart, S.; Sverak, V., On the De Gregorio modification of the Constantin-Lax-Majda model, Arch. Ration. Mech. Anal., 231, 2, 1269-1304 (2019) · Zbl 1408.35152 · doi:10.1007/s00205-018-1298-1 |
| [17] | Kiselev, A., Regularity and blow up for active scalars, Math. Model. Nat. Phenomena, 5, 4, 225-255 (2010) · Zbl 1194.35490 · doi:10.1051/mmnp/20105410 |
| [18] | Lei, Z.; Hou, TY, On the stabilizing effect of convection in three-dimensional incompressible flows, Commun. Pure Appl. Math. A J. Issued by the Courant Inst. Math. Sci., 62, 4, 501-564 (2009) · Zbl 1171.35095 |
| [19] | Lei, Z.; Liu, J.; Ren, X., On the Constantin-Lax-Majda model with convection, Commun. Math. Phys., 375, 1, 765-783 (2020) · Zbl 1439.35402 · doi:10.1007/s00220-019-03584-4 |
| [20] | Landman, MJ; Papanicolaou, GC; Sulem, C.; Sulem, P-L, Rate of blowup for solutions of the nonlinear Schrödinger equation at critical dimension, Phys. Rev. A, 38, 8, 3837 (1988) · doi:10.1103/PhysRevA.38.3837 |
| [21] | Li, D.; Rodrigo, J., Blow-up of solutions for a 1D transport equation with nonlocal velocity and supercritical dissipation, Adv. Math., 217, 6, 2563-2568 (2008) · Zbl 1138.35381 · doi:10.1016/j.aim.2007.11.002 |
| [22] | Lushnikov, PM; Silantyev, DA; Siegel, M., Collapse versus blow-up and global existence in the generalized Constantin-Lax-Majda equation, J. Nonlinear Sci., 31, 5, 1-56 (2021) · Zbl 1479.35684 · doi:10.1007/s00332-021-09737-x |
| [23] | Martınez, A.C.: Nonlinear and nonlocal models in fluid mechanics (2010) |
| [24] | McLaughlin, DW; Papanicolaou, GC; Sulem, C.; Sulem, P-L, Focusing singularity of the cubic Schrödinger equation, Phys. Rev. A, 34, 2, 1200 (1986) · doi:10.1103/PhysRevA.34.1200 |
| [25] | Okamoto, H.; Ohkitani, K., On the role of the convection term in the equations of motion of incompressible fluid, J. Phys. Soc. Jpn., 74, 10, 2737-2742 (2005) · Zbl 1083.76007 · doi:10.1143/JPSJ.74.2737 |
| [26] | Okamoto, H.; Sakajo, T.; Wunsch, M., On a generalization of the Constantin-Lax-Majda equation, Nonlinearity, 21, 10, 2447 (2008) · Zbl 1221.35300 · doi:10.1088/0951-7715/21/10/013 |
| [27] | Schochet, S., Explicit solutions of the viscous model vorticity equation, Commun. Pure Appl. Math., 39, 4, 531-537 (1986) · Zbl 0623.76012 · doi:10.1002/cpa.3160390404 |
| [28] | Silvestre, L.; Vicol, V., On a transport equation with nonlocal drift, Trans. Am. Math. Soc., 368, 9, 6159-6188 (2016) · Zbl 1334.35254 · doi:10.1090/tran6651 |
| [29] | Sverak, V.: A video lecture on “Small scale dynamics in fluid motion”. https://scgp.stonybrook.edu/video_portal/video.php?id=5281 |
| [30] | Wunsch, M., The generalized Constantin-Lax-Majda equation revisited, Commun. Math. Sci., 9, 3, 929-936 (2011) · Zbl 1272.35049 · doi:10.4310/CMS.2011.v9.n3.a12 |
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