A Liouville Theorem for the Axially-symmetric Navier-Stokes Equations

By Zhen Lei, Qi S. Zhang. 2010

78 citations recorded by the source corpus.

Let $v(x, t)= v^r e_r + v^\theta e_\theta + v^z e_z$ be a solution to the three-dimensional incompressible axially-symmetric Navier-Stokes equations. Denote by $b = v^r e_r + v^z e_z$ the radial-axial vector field. Under a general scaling invariant condition on $b$, we prove that the quantity $\Gamma = r v^\theta$ is H\"older continuous at $r = 0$, $t = 0$. As an application, we give a partial proof of a conjecture on Liouville property by Koch-Nadirashvili-Seregin-Sverak in \cite{KNSS} and Seregin-Sverak in \cite{SS}. As another application, we prove that if $b \in L^\infty([0, T], BMO^{-1})$, then $v$ is regular. This provides an answer to an open question raised by Koch and Tataru in \cite{KochTataru} about the uniqueness and regularity of Navier-Stokes equations in the axially-symmetric case.

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/6d7534c550eca3e45d78d07b2aad8eeb623363a2

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In Applied Mathematics

1. Zhen Lei
2. Qi S. Zhang
3. Xiaodong Cao
4. A. Naber
5. H. Hein
6. Renming Song
7. Bing Wang
8. A. Palmieri

In Mathematical Physics

1. Qi S. Zhang
2. Zhen Lei
3. A. Palmieri
4. Michael Reissig
5. Xiaodong Cao
6. Yi Zhou
7. Chi Li
8. M. DʼAbbicco

Citations in the corpus

• Criticality of the Axially Symmetric Navier-Stokes Equations (2015, methodology): “See, for instance, [1, 2, 3, 4, 5, 7, 8, 9, 10, 13, 14, 15, 18, 20, 21, 22, 23, 24, 27] and many others.”