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Remarks on the lifespan of the solutions to some models of incompressible fluid mechanics. (English) Zbl 1283.35080
Summary: We give lower bounds for the lifespan of a solution to the inviscid Boussinesq system. In dimension two, we point out that it tends to infinity when the initial (relative) temperature tends to zero. This is, to the best of our knowledge, the first result of this kind for the inviscid Boussinesq system. In passing, we provide continuation criteria (of independent interest) in the \( N\)-dimensional case. In the second part of the paper, our method is adapted to handle the axisymmetric incompressible Euler equations with swirl.

MSC:
35Q35 PDEs in connection with fluid mechanics
76B03 Existence, uniqueness, and regularity theory for incompressible inviscid fluids
76B70 Stratification effects in inviscid fluids
35A01 Existence problems for PDEs: global existence, local existence, non-existence
35Q31 Euler equations
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