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An example of finite-time singularities in the 3d Euler equations

Birkhaeuser Verlag AG
Publication Date
  • Qa Mathematics
  • Tj Mechanical Engineering And Machinery
  • Qc Physics


Let Omega = R-3\(B) over bar _1(0) be the exterior of the closed unit ball. Consider the self-similar 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 self-similar Navier-Stokes 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 C-1(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 self-similarity 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 L-2-norm of curl v.

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