Abstract.
Let \(\Omega = {\user2{\mathbb{R}}}^{3} \backslash \overline{B} _{1} (0)\) be the exterior of the closed unit ball. Consider the self-similar Euler system
Setting α = β = 1/2 gives the limiting case of Leray’s self-similar Navier–Stokes equations. Assuming smoothness and smallness of the boundary data on ∂Ω, we prove that this system has a unique solution \((u,p) \in \user1{\mathcal{C}}^1 (\Omega ;\user2{\mathbb{R}}^3 \times\user2{\mathbb{R}}) \), vanishing at infinity, precisely
The self-similarity transformation is v(x, t) = u(y)/(t* − t)α, y = x/(t* − t)β, 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* < + ∞. This isolated singularity has bounded energy with unbounded L 2 − norm of curl v.
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Communicated by Y. Giga
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He, X. An Example of Finite-time Singularities in the 3d Euler Equations. J. math. fluid mech. 9, 398–410 (2007). https://doi.org/10.1007/s00021-005-0205-3
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DOI: https://doi.org/10.1007/s00021-005-0205-3