Characterization of shock profiles in terms of stabilization mechanisms in systems with weak dissipations

2016 Fiscal Year Final Research Report

• Project/Area Number: 26800076
• Research Category: Grant-in-Aid for Young Scientists (B)
• Allocation Type: Multi-year Fund
• Research Field: Mathematical analysis
• Research Institution: Tokyo University of Marine Science and Technology (2015-2016); Waseda University (2014)
• Principal Investigator: Ohnawa Masashi 東京海洋大学, 学術研究院, 准教授 (10443243)
• Project Period (FY): 2014-04-01 – 2017-03-31
• Keywords: 不連続衝撃波 / 漸近安定性 / 爆発 / 不連続点の合体

Outline of Final Research Achievements

In the last two decades, many systems with weaker dissipation than diffusion are found to admit discontinuous traveling wave solutions if certain parameters exceed threshold values. Examples include radiating gas systems, saturating dissipation in the Burgers type equations, and hyperbolic systems with relaxation. This study aims to explain variations in the shock profiles in terms of stabilization mechanisms. We have proved the following facts for a simplified radiating gas system called Hamer's model. i) In all subcritical cases, radiation dominates over convection and the solution converges to a traveling wave solution without blowing up. ii) There are arbitrary small perturbations around the critical shock wave which cause blow up of the solution in a finite time. iii) In the supercritical cases, the existence of a discontinuity which satisfies the entropy condition helps convection to recover stability even in the L-infinity topology.

Free Research Field

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