American Institute of Mathematical Sciences

Advanced
November  2007, 18(4): 637-642. doi: 10.3934/dcds.2007.18.637

Nonexistence of locally self-similar blow-up for the 3D incompressible Navier-Stokes equations

Thomas Y. Hou 1, and  Ruo Li 2,

1. 

Department of Applied and Computational Mathematics, California Institute of Technology, Pasadena, CA 91125, United States
2. 

LMAM&School of Mathematical Sciences, Peking University, Beijing, 100871, China

Received  March 2007 Published  May 2007

We study locally self-similar solutions of the three dimensional incompressible Navier-Stokes equations. The locally self-similar solutions we consider here are different from the global self-similar solutions. The self-similar scaling is only valid in an inner core region that shrinks to a point dynamically as the time, $t$, approaches a possible singularity time, $T$. The solution outside the inner core region is assumed to be regular, but it does not satisfy self-similar scaling. Under the assumption that the dynamically rescaled velocity profile converges to a limiting profile as $t \rightarrow T$ in $L^p$ for some $p \in (3,\infty )$, we prove that such a locally self-similar blow-up is not possible. We also obtain a simple but useful non-blowup criterion for the 3D Euler equations.
Keywords: locally self-similar, Euler equations, blow-up., Navier-Stokes equations.
Mathematics Subject Classification: Primary: 76D03, 76D05; Secondary: 76B0.
Citation: Thomas Y. Hou, Ruo Li. Nonexistence of locally self-similar blow-up for the 3D incompressible Navier-Stokes equations. Discrete & Continuous Dynamical Systems - A, 2007, 18 (4) : 637-642. doi: 10.3934/dcds.2007.18.637
[1]

Dongho Chae, Kyungkeun Kang, Jihoon Lee. Notes on the asymptotically self-similar singularities in the Euler and the Navier-Stokes equations. Discrete & Continuous Dynamical Systems - A, 2009, 25 (4) : 1181-1193. doi: 10.3934/dcds.2009.25.1181
[2]

Zoran Grujić. Regularity of forward-in-time self-similar solutions to the 3D Navier-Stokes equations. Discrete & Continuous Dynamical Systems - A, 2006, 14 (4) : 837-843. doi: 10.3934/dcds.2006.14.837
[3]

Alberto Bressan, Wen Shen. A posteriori error estimates for self-similar solutions to the Euler equations. Discrete & Continuous Dynamical Systems - A, 2019, 0 (0) : 0-0. doi: 10.3934/dcds.2020168
[4]

Weronika Biedrzycka, Marta Tyran-Kamińska. Self-similar solutions of fragmentation equations revisited. Discrete & Continuous Dynamical Systems - B, 2018, 23 (1) : 13-27. doi: 10.3934/dcdsb.2018002
[5]

Qiaolin He. Numerical simulation and self-similar analysis of singular solutions of Prandtl equations. Discrete & Continuous Dynamical Systems - B, 2010, 13 (1) : 101-116. doi: 10.3934/dcdsb.2010.13.101
[6]

F. Berezovskaya, G. Karev. Bifurcations of self-similar solutions of the Fokker-Plank equations. Conference Publications, 2005, 2005 (Special) : 91-99. doi: 10.3934/proc.2005.2005.91
[7]

Hyungjin Huh. Self-similar solutions to nonlinear Dirac equations and an application to nonuniqueness. Evolution Equations & Control Theory, 2018, 7 (1) : 53-60. doi: 10.3934/eect.2018003
[8]

Yi Zhou, Zhen Lei. Logarithmically improved criteria for Euler and Navier-Stokes equations. Communications on Pure & Applied Analysis, 2013, 12 (6) : 2715-2719. doi: 10.3934/cpaa.2013.12.2715
[9]

Michele Coti Zelati. Remarks on the approximation of the Navier-Stokes equations via the implicit Euler scheme. Communications on Pure & Applied Analysis, 2013, 12 (6) : 2829-2838. doi: 10.3934/cpaa.2013.12.2829
[10]

Carlo Morosi, Livio Pizzocchero. On the constants in a Kato inequality for the Euler and Navier-Stokes equations. Communications on Pure & Applied Analysis, 2012, 11 (2) : 557-586. doi: 10.3934/cpaa.2012.11.557
[11]

Pavel I. Plotnikov, Jan Sokolowski. Compressible Navier-Stokes equations. Conference Publications, 2009, 2009 (Special) : 602-611. doi: 10.3934/proc.2009.2009.602
[12]

Jan W. Cholewa, Tomasz Dlotko. Fractional Navier-Stokes equations. Discrete & Continuous Dynamical Systems - B, 2018, 23 (8) : 2967-2988. doi: 10.3934/dcdsb.2017149
[13]

Hamid Bellout, Jiří Neustupa, Patrick Penel. On a $\nu$-continuous family of strong solutions to the Euler or Navier-Stokes equations with the Navier-Type boundary condition. Discrete & Continuous Dynamical Systems - A, 2010, 27 (4) : 1353-1373. doi: 10.3934/dcds.2010.27.1353
[14]

Jochen Merker, Aleš Matas. Positivity of self-similar solutions of doubly nonlinear reaction-diffusion equations. Conference Publications, 2015, 2015 (special) : 817-825. doi: 10.3934/proc.2015.0817
[15]

Hideo Kubo, Kotaro Tsugawa. Global solutions and self-similar solutions of the coupled system of semilinear wave equations in three space dimensions. Discrete & Continuous Dynamical Systems - A, 2003, 9 (2) : 471-482. doi: 10.3934/dcds.2003.9.471
[16]

Jian Su, Yinnian He. The almost unconditional convergence of the Euler implicit/explicit scheme for the three dimensional nonstationary Navier-Stokes equations. Discrete & Continuous Dynamical Systems - B, 2017, 22 (9) : 3421-3438. doi: 10.3934/dcdsb.2017173
[17]

Jeongho Kim, Weiyuan Zou. Solvability and blow-up criterion of the thermomechanical Cucker-Smale-Navier-Stokes equations in the whole domain. Kinetic & Related Models, 2020, 13 (3) : 623-651. doi: 10.3934/krm.2020021
[18]

Hermenegildo Borges de Oliveira. Anisotropically diffused and damped Navier-Stokes equations. Conference Publications, 2015, 2015 (special) : 349-358. doi: 10.3934/proc.2015.0349
[19]

Hyukjin Kwean. Kwak transformation and Navier-Stokes equations. Communications on Pure & Applied Analysis, 2004, 3 (3) : 433-446. doi: 10.3934/cpaa.2004.3.433
[20]

Vittorino Pata. On the regularity of solutions to the Navier-Stokes equations. Communications on Pure & Applied Analysis, 2012, 11 (2) : 747-761. doi: 10.3934/cpaa.2012.11.747

2018 Impact Factor: 1.143

Tools

Metrics

  • PDF downloads (29)
  • HTML views (0)
  • Cited by (18)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences