Abstract
The author proposes a two-dimensional generalization of Constantin-Lax-Majda model. Some results about singular solutions are given. This model might be the first step toward the singular solutions of the Euler equations. Along the same line (vorticity formulation), the author presents some further model equations. He possibly models various aspects of difficulties related with the singular solutions of the Euler and Navier-Stokes equations. Some discussions on the possible connection between turbulence and the singular solutions of the Navier-Stokes equations are made.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Choi, K., Hou, T. Y., Kiselev, A., et al., On the finite-time blowup of a one-dimensional model for the three-dimensional axisymmetric Euler equations, Comm. Pure Appl. Math., 70, 2017, 2218–2243.
Constantin, P., Lax, P. and Majda, A., A simple one-dimensional model for the three-dimensional vorticity, Comm. Pure Appl. Math., 38, 1985, 715–724.
Cordoba, A., Cordoba, D. and Fontelos, M. A., Formation of singularities for a transport equation with nonlocal velocity, Annals of Mathematics, 162, 2001, 1377–1389.
De Gregorio, S., A partial differential equation arising in a 1D model for the 3D vorticity equation, Math. Methods Appl. Sci., 19, 1996, 1233–1255.
Dong, H., Du, D. and Li, D., Finite time singularities and global well-posedness for fractal Burgers equations, Indiana University Mathematics Journal, 58, 2009, 807–822.
Du, D., Three Regularity Results Related with the Navier-Stokes Equations, Ph.D. Thesis, University of Minnesota, 2005.
Du, D. and Lv, J., Blow-up for a semi-linear advection-diffusion system with energy conservation, Chin. Ann. Math. Ser. B., 30, 2009, 433–446.
Jia, H., Stewart, S. and Sverak, V., On the De Gregorio Modification of the Constantin Lax Majda Model, Arch. Ration. Mech. Anal., 231, 2019, 1269–1304.
Grafakos, L., Oliveira e Silva, D., et al., Some problems in harmonic analysis, arXiv: 1701.06637, 2017.
Kress, R., Potentialtheoretische Randwertprobleme bei Tensorfeldern beliebiger Dimension und beliebigen Rangs (in German), Arch. Ration. Mech. Anal., 47, 1972, 59–80.
Majda, A. J. and Bertozzi, A. L., Vorticity and Incompressible Flow, Cambridge Texts in Applied Mathematics, 27, Cambridge University Press, Cambridge, 2002.
Nečas, J., Růžička, M. and Šverák, V., On Leray’s self-similar solutions of the Navier-Stokes equations, Acta Math., 323, 1996, 283–294.
Plecháč, P. and Šverák, V., Singular and regular solutions of a nonlinear parabolic system, Nonlinearity, 16, 2003, 2083–2097.
Schochet, S., Explicit solutions of the viscous model vorticity equation, Comm. Pure Appl. Math., 39, 1986, 531–537.
Serrin, J., On the interior regularity of weak solutions of the Navier-Stokes equations, Arch. Ration. Mech. Anal., 9, 1962, 187–195.
Tsai, T.-P., On Leray’s self-similar solutions of the Navier-Stokes equations satisfying local energy estimates, Arch. Ration. Mech. Anal., 143, 1998, 29–51.
Acknowledgement
The author would like to thank Hongjie Dong and Vladimir Šverák for valuable comments. The author would also thank to Yipeng Shi for wonderful discussions on turbulence.
Author information
Authors and Affiliations
Corresponding author
Additional information
This work was supported by the National Natural Science Foundation of China (No. 11571066).
Rights and permissions
About this article
Cite this article
Du, D. On Some Model Equations of Euler and Navier-Stokes Equations. Chin. Ann. Math. Ser. B 42, 281–290 (2021). https://doi.org/10.1007/s11401-021-0257-6
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11401-021-0257-6