Abstract
In this paper we rule out the possibility of asymptotically self-similar singularities for both of the 3D Euler and the 3D Navier–Stokes equations. The notion means that the local in time classical solutions of the equations develop self-similar profiles as t goes to the possible time of singularity T. For the Euler equations we consider the case where the vorticity converges to the corresponding self-similar voriticity profile in the sense of the critical Besov space norm, \(\dot{B}^0_{\infty, 1}(\mathbb{R}^3)\). For the Navier–Stokes equations the convergence of the velocity to the self-similar singularity is in L q(B(z,r)) for some \(q\in [2, \infty)\), where the ball of radius r is shrinking toward a possible singularity point z at the order of \(\sqrt{T-t}\) as t approaches to T. In the \(L^q (\mathbb{R}^3)\) convergence case with \(q\in [3, \infty)\) we present a simple alternative proof of the similar result in Hou and Li in arXiv-preprint, math.AP/0603126.
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This work was supported partially by KRF Grant(MOEHRD, Basic Research Promotion Fund) and the KOSEF Grant no. R01-2005-000-10077-0.
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Chae, D. Nonexistence of asymptotically self-similar singularities in the Euler and the Navier–Stokes equations. Math. Ann. 338, 435–449 (2007). https://doi.org/10.1007/s00208-007-0082-6
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DOI: https://doi.org/10.1007/s00208-007-0082-6