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Axisymmetric solutions of the shallow water equations with bottom topography

Axisymmetric solutions of the shallow water equations with bottom topography We construct a family of self-similar and bounded weak solutions for all positive time to the two-dimensional shallow water equations with the bottom topography a quadratic function with respect to the reciprocal of radius under axisymmetry assumptions. The source term generated from the bottom topography leads to a different structure in solutions with that obtained in hyperbolic conservative system. The key point is that the steady-state solution is used to deal with the singularity near the origin in the reduced system under axisymmetric transformation. The obtained results may apply to other systems with similar structures. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Research in the Mathematical Sciences Springer Journals

Axisymmetric solutions of the shallow water equations with bottom topography

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Publisher
Springer Journals
Copyright
Copyright © The Author(s), under exclusive licence to Springer Nature Switzerland AG 2022
eISSN
2197-9847
DOI
10.1007/s40687-022-00312-8
Publisher site
See Article on Publisher Site

Abstract

We construct a family of self-similar and bounded weak solutions for all positive time to the two-dimensional shallow water equations with the bottom topography a quadratic function with respect to the reciprocal of radius under axisymmetry assumptions. The source term generated from the bottom topography leads to a different structure in solutions with that obtained in hyperbolic conservative system. The key point is that the steady-state solution is used to deal with the singularity near the origin in the reduced system under axisymmetric transformation. The obtained results may apply to other systems with similar structures.

Journal

Research in the Mathematical SciencesSpringer Journals

Published: Mar 1, 2022

Keywords: Shallow water equations; Axisymmetric solutions; Source term; Steady-state solution; Rankine–Hugoniot condition

References