On the Cauchy problem for a generalized two-component shallow water wave system with fractional higher-order inertia operators

Huijun He; Zhaoyang Yin

doi:10.3934/dcds.2017062

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2017, Volume 37, Issue 3: 1509-1537. Doi: 10.3934/dcds.2017062

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On the Cauchy problem for a generalized two-component shallow water wave system with fractional higher-order inertia operators

Huijun He^1, and
Zhaoyang Yin^1,2, ,

1.
Department of Mathematics, Sun Yat-sen University, Guangzhou 510275, China
2.
Faculty of Information Technology, Macau University of Science and Technology, Macau, China

¹Corresponding author
¹Corresponding author

Received: May 31, 2016

Revised: August 31, 2016

Published: May 2017

This work was partially supported by NNSFC (No.11671407 and No.11271382), FDCT (No. 098/2013/A3), Guangdong Special Support Program (No. 8-2015), and the key project of NSF of Guangdong province (No. 2016A030311004).

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Abstract

In this paper, we mainly consider the Cauchy problem for a generalized two-component shallow water wave system with fractional higher-order inertia operators: $m=(1-\partial_x^2)^su, s>1$. By Littlewood-Paley theory and transport equation theory, we first establish the local well-posedness of the generalized b-equation with fractional higher-order inertia operators which is the subsystem of the generalized two-component water wave system. Then we prove the local well-posedness of the generalized two-component water wave system with fractional higher-order inertia operators. Next, we present the blow-up criteria for these systems. Moreover, we obtain some global existence results for these systems.

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Mathematics Subject Classification: Primary:35G25;Secondary:35L05, 35Q53.

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