Delta shock wave and wave interactions in a thin film of a perfectly soluble anti-surfactant solution

Anupam Sen; T. Raja Sekhar

doi:10.3934/cpaa.2020115

Article Contents

2020, Volume 19, Issue 5: 2641-2653. Doi: 10.3934/cpaa.2020115

This issue Previous Article Local Lipschitz regularity for functions satisfying a time-dependent dynamic programming principle Next Article Existence of weak solution for mean curvature flow with transport term and forcing term

Delta shock wave and wave interactions in a thin film of a perfectly soluble anti-surfactant solution

Anupam Sen and
T. Raja Sekhar^,

Department of Mathematics, Indian Institute of Technology Kharagpur, Kharagpur-2, India

^* Corresponding author
^* Corresponding author

Received: January 31, 2019

Revised: August 31, 2019

Published: March 2020

Research support from University Grant Commission, Government of India (Sr. No. 21215409 47, Ref No: 20=12=2015(ⅱ)EU-V) is gratefully acknowledged by the first author

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Abstract

We study the interactions between classical elementary waves and delta shock wave in quasilinear hyperbolic system of conservation laws. This governing system describes a thin film of a perfectly soluble anti-surfactant solution in the limit of large capillary and P$ \acute{e} $clet numbers. This system is one of the example of non-strictly hyperbolic system whose Riemann solution consists of delta shock wave as well as classical elementary waves such as shock waves, rarefaction waves and contact discontinuities. The global structure of the perturbed Riemann solutions are constructed and analyzed case by case when delta shock wave is involved.

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Mathematics Subject Classification: Primary: 35L45, 35L65, 58J45; Secondary: 35Q35, 35L67.

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Figure 1. Wave curves in $ (h, b) $ phase plane

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Figure 2. $ J+R $ when $ 0<h_lb_l<h_rb_r $

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Figure 3. $ J+S $ when $ 0<h_rb_r<h_lb_l $

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Figure 4. $ \delta{S} $ when $ b_{l, r}>0 $, $ h_l>0 $ and $ h_r = 0 $

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Figure 5. When $ 0<h_mb_m<h_lb_l $ and $ h_r = 0 $

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Figure 6. When $ 0<h_lb_l<h_mb_m $ and $ h_r = 0 $

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