On the Cauchy problem for a class of shallow water wave equations with (k + 1)-order nonlinearities☆
On the Cauchy problem for a class of shallow water wave equations with (k + 1)-order nonlinearities☆
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DOI：
10.1016/j.jmaa.2016.07.056
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发表时间：
2017
影响因子：
1.3
通讯作者：
Lei Zhang;Bin Liu
中科院分区：
文献类型：
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作者：
Lei Zhang;Bin Liu
在Besov空间<$t u− <$t <$x 2 u= uk <$x 3 u+ B uk − 1 <$x u <$x 2 u−（b+ 1）uk <$x u中考虑一类具有（k+ 1）阶非线性项的浅水波方程的Cauchy问题，其中包括Camassa-Holm，Degasperis-Procesi和Novikov方程作为特例.首先，利用输运方程和Littlewood-Paley理论，得到了非齐次Besov空间B，rs（s> max <${1+ 1 p，3 2}且p，r∈[1，+∞]）中方程的局部适定性.其次，我们考虑了局部适定性在B 2，rs的临界指标s= 3 2，并表明，该解决方案连续依赖于初始数据。第三，得到了方程强解的爆破准则和守恒性。最后，利用一个新的Ovsyannikov定理，我们研究了解的Gevrey正则性和解析性.此外，我们得到了一个下界的寿命和连续性的数据到解决方案的映射。
This paper considers the Cauchy problem for a class of shallow water wave equations with (k+ 1)-order nonlinearities in the Besov spaces∂ t u−∂ t∂ x 2 u= u k∂ x 3 u+ b u k− 1∂ x u∂ x 2 u−(b+ 1) u k∂ x u, which involves the Camassa–Holm, the Degasperis–Procesi and the Novikov equations as special cases. Firstly, by means of the transport equation and the Littlewood–Paley theory, we obtain the local well-posedness of the equations in the nonhomogeneous Besov space B p, r s (s> max{1+ 1 p, 3 2} and p, r∈[1,+∞]). Secondly, we consider the local well-posedness in B 2, r s with the critical index s= 3 2, and show that the solutions continuously depend on the initial data. Thirdly, the blow-up criteria and the conservative property for the strong solutions are derived. Finally, with the help of a new Ovsyannikov theorem, we investigate the Gevrey regularity and analyticity of the solutions. Moreover, we get a lower bound of the lifespan and the continuity of the data-to-solution mapping.