The Library
An example of finitetime singularities in the 3d Euler equations
Tools
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
Research output not available from this repository, contact author.
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
Request changes or add full text files to a record
Repository staff actions (login required)
View Item 