Vanishing viscosity approach to a system of conservation laws admitting $\delta''$ waves

K. T. Joseph; Manas R. Sahoo

doi:10.3934/cpaa.2013.12.2091

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2013, Volume 12, Issue 5: 2091-2118. Doi: 10.3934/cpaa.2013.12.2091

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Vanishing viscosity approach to a system of conservation laws admitting $\delta''$ waves

K. T. Joseph^1, and
Manas R. Sahoo^1,

1.
TIFR Centre for Applicable Mathematics, P.B.NO. 6503, Sharada Nagar, Chikkabommasandra, Bangalore 560065

Received: April 30, 2012

Revised: July 31, 2012

Published: August 2013

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Abstract

We construct solution of Riemann problem for a system of four conservation laws admitting $\delta$, $\delta'$ and $\delta''$-waves, using vanishing viscosity method. The system considered here is an extension of a system studied in [9] and [12] and admits more singular solutions. We extend the weak formulation of [12] to the present case. For the rarefaction case, the limit is not yet fully understood, the limit given in [12] is not correct and it does not satisfy the inviscid system. In fact we show that the limit of the third component contains $\delta$ measure and the fourth component contains the measure $\delta$ and its derivative, for a special Riemann data. We also solve Riemann type initial-boundary value problem in the quarter plane.

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Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.

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References

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