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On global large energy solutions to the viscous shallow water equations. (English) Zbl 1451.35145

Summary: By exploring the smooth effect of the heat flows and the weighted-Chemin-Lerner technique, we obtain the global solutions of large energy to the viscous shallow water equations with initial data in the critical Besov spaces, which improves the previous small energy type arguments [Q. Chen et al., SIAM J. Math. Anal. 40, No. 2, 443–474 (2008; Zbl 1169.35048); W. Wang and C.-J. Xu, Rev. Mat. Iberoam. 21, No. 1, 1–24 (2005; Zbl 1095.35037)]. Moreover, the method used here is quiet different from [loc. cit.].

MSC:

35Q35 PDEs in connection with fluid mechanics
76N10 Existence, uniqueness, and regularity theory for compressible fluids and gas dynamics
76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction
35B40 Asymptotic behavior of solutions to PDEs
35B45 A priori estimates in context of PDEs
42B25 Maximal functions, Littlewood-Paley theory
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