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An example of finitetime singularities in the 3d Euler equations
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He, Xinyu (2007) An example of finitetime singularities in the 3d Euler equations. Journal of Mathematical Fluid Mechanics, Vol.9 (No.3). pp. 398410. doi:10.1007/s0002100502053
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Official URL: http://dx.doi.org/10.1007/s0002100502053
Abstract
Let Omega = R3\(B) over bar _1(0) be the exterior of the closed unit ball. Consider the selfsimilar Euler system
alpha u+beta y.del u+u.del u+del p=0, div u=0 in Omega.
Setting alpha=beta=1/2 gives the limiting case of Leray's selfsimilar NavierStokes equations. Assuming smoothness and smallness of the boundary data on partial derivative Omega, we prove that this system has a unique solution (u,p) epsilon C1(Omega;R(3)xR), vanishing at infinity, precisely
u(y)down arrow 0 as vertical bar y vertical bar up arrow infinity, with u = O(vertical bar y vertical bar(1)), del u=O(vertical bar y vertical bar(2)).
The selfsimilarity transformation is v(x,t)=u(y)/(t*t)(alpha), y=x/(t*t)(beta), where v(x,t) is a solution to the Euler equations. The existence of smooth function u(y) implies that the solution v(x,t) blows up at (x*,t*), x*=0, t*<+infinity. This isolated singularity has bounded energy with unbounded L2norm of curl v.
Item Type:  Journal Article  

Subjects:  Q Science > QA Mathematics T Technology > TJ Mechanical engineering and machinery Q Science > QC Physics 

Divisions:  Faculty of Science > Mathematics  
Journal or Publication Title:  Journal of Mathematical Fluid Mechanics  
Publisher:  Birkhaeuser Verlag AG  
ISSN:  14226928  
Official Date:  August 2007  
Dates: 


Volume:  Vol.9  
Number:  No.3  
Number of Pages:  13  
Page Range:  pp. 398410  
DOI:  10.1007/s0002100502053  
Status:  Peer Reviewed  
Publication Status:  Published  
Access rights to Published version:  Restricted or Subscription Access 
Data sourced from Thomson Reuters' Web of Knowledge
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