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On the incompressible Euler equations and the blow-up problem. (English) Zbl 1173.35587
Kozono, Hideo (ed.) et al., Asymptotic analysis and singularities. Hyperbolic and dispersive PDEs and fluid mechanics. Papers of the 14th International Research Institute of the Mathematical Society of Japan (MSJ), Sendai, Japan, July, 18–27, 2005. Tokyo: Mathematical Society of Japan (ISBN 978-4-931469-40-2/hbk). Advanced Studies in Pure Mathematics 47-1, 1-30 (2006).
The author reviews some results on the Cauchy problem for Euler and Navier-Stokes equations These are the local existence of the classical solution, finite time blow-up, effects of the vortex stretching term depletion (regularity criteria) and the deformation tensor (a priori estimates and blow-up criteria). A few model problems of the Euler equations are considered: 2-D and 1-D quasi-geostrophic equations, the Bousinesq equations, the modified Euler equations. Finally, the result of the nonexistence of the self-similar blow-up is presented.
For the entire collection see [Zbl 1130.35003].

MSC:
35L65 Hyperbolic conservation laws
35L45 Initial value problems for first-order hyperbolic systems
35L67 Shocks and singularities for hyperbolic equations
35A07 Local existence and uniqueness theorems (PDE) (MSC2000)
35Q35 PDEs in connection with fluid mechanics
76B03 Existence, uniqueness, and regularity theory for incompressible inviscid fluids
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