Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

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Shock reflection for the damped $ P$-system


Authors: Ling Hsiao and Hailiang Li
Journal: Quart. Appl. Math. 60 (2002), 437-460
MSC: Primary 35L60; Secondary 35B40, 35L55, 35L67, 76L05, 76S05
DOI: https://doi.org/10.1090/qam/1914435
MathSciNet review: MR1914435
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Abstract: The global existence and the asymptotic behavior of the weak entropy solution, the piecewise smooth solution with one shock discontinuity, on a strip domain is investigated in the present paper. We show that, for small smooth initial data and boundary value with only one small jump at $ \left( x, t \right) = \left( 0, 0 \right)$, the piecewise smooth solution with one shock discontinuity exists globally in time. The shock discontinuity begins from $ \left( x, t \right) = \left( 0, 0 \right)$, moves forward and reflects in a finite time at the boundary $ x = 1$ to form a 1-shock, which goes backward and reflects at $ x = 0$ also in a finite time to create a new 2-shock. The shock strength decays exponentially and never disappears in finite time. As $ t \to \infty $, this solution converges to a constant state determined by the initial and the boundary conditions.


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DOI: https://doi.org/10.1090/qam/1914435
Article copyright: © Copyright 2002 American Mathematical Society


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