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The lifespan of smooth solutions to the three-dimensional compressible Euler equations and the incompressible limit. (English) Zbl 0736.35087
Estimates for the lifespan of classical solutions to the three- dimensional compressible isentropic Euler equations with initial data representing a perturbation of order $$\varepsilon$$ from a constant state are derived. It is shown that the lifespan $$T_ \varepsilon$$ satisfies a lower bound of the form $$T_ \varepsilon>\exp(C/\varepsilon)$$ provided that the initial data is irrotational. In the general case, the lower bound $$T_ \varepsilon>C/\varepsilon$$ is obtained along with uniform bounds which allow, after a change of scale, to pass easily to the incompressible limit on a uniform time interval. Results on the location and nature of potential singularities are also given.
Reviewer: T.C.Sideris

##### MSC:
 35Q35 PDEs in connection with fluid mechanics 76N10 Existence, uniqueness, and regularity theory for compressible fluids and gas dynamics 35L60 First-order nonlinear hyperbolic equations
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