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On the Cauchy problem for the two-component Euler-Poincaré equations. (English) Zbl 1303.35064

Summary: In the paper, we first use the energy method to establish the local well-posedness as well as blow-up criteria for the Cauchy problem on the two-component Euler-Poincaré equations in multi-dimensional space. In the case of dimensions 2 and 3, we show that for a large class of smooth initial data with some concentration property, the corresponding solutions blow up in finite time by using Constantin-Escher Lemma and Littlewood-Paley decomposition theory. Then for the one-component case, a more precise blow-up estimate and a global existence result are also established by using similar methods. Next, we investigate the zero density limit and the zero dispersion limit. At the end, we also briefly demonstrate a Liouville type theorem for the stationary weak solution.

MSC:

35Q31 Euler equations
35Q35 PDEs in connection with fluid mechanics
35G25 Initial value problems for nonlinear higher-order PDEs
35B44 Blow-up in context of PDEs
42B25 Maximal functions, Littlewood-Paley theory
35D30 Weak solutions to PDEs
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