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Singularity formation and global existence of classical solutions for one dimensional rotating shallow water system

Cheng, Bin, Qu, P and Xie, C (2018) Singularity formation and global existence of classical solutions for one dimensional rotating shallow water system SIMA (SIAM Journal on Mathematical Analysis), 50 (3). pp. 2486-2508.

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Abstract

We study classical solutions of one dimensional rotating shallow water system which plays an important role in geophysical fluid dynamics. The main results contain two contrasting aspects. First, when the solution crosses certain thresholds, we prove finite-time singularity formation for the classical solutions by studying the weighted gradients of Riemann invariants and utilizing conservation of physical energy. In fact, the singularity formation will take place for a large class of C 1 initial data whose gradients and physical energy can be arbitrarily small and higher order derivatives should be large. Second, when the initial data have constant potential vorticity, global existence of small classical solutions is established via studying an equivalent form of a quasilinear Klein-Gordon equation satisfying certain null conditions. In this global existence result, the smallness condition is in terms of the higher order Sobolev norms of the initial data.

Item Type: Article
Divisions : Faculty of Engineering and Physical Sciences > Mathematics
Authors :
NameEmailORCID
Cheng, Binb.cheng@surrey.ac.uk
Qu, P
Xie, C
Date : 8 May 2018
Identification Number : 10.1137/17M1130101
Copyright Disclaimer : © 2018, Society for Industrial and Applied Mathematics
Uncontrolled Keywords : Rotating shallow water system, formation of singularity, Riemann invariants, global existence, Klein-Gordon equation
Depositing User : Melanie Hughes
Date Deposited : 01 Mar 2018 12:18
Last Modified : 30 May 2018 11:26
URI: http://epubs.surrey.ac.uk/id/eprint/845925

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