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

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September 2013, 12(5): 2091-2118. doi: 10.3934/cpaa.2013.12.2091

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, India, India

Received May 2012 Revised August 2012 Published January 2013

Abstract
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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.

Keywords: $\delta$-waves, conservation laws, vanishing viscosity..

Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C3.

Citation: K. T. Joseph, Manas R. Sahoo. Vanishing viscosity approach to a system of conservation laws admitting $\delta''$ waves. Communications on Pure and Applied Analysis, 2013, 12 (5) : 2091-2118. doi: 10.3934/cpaa.2013.12.2091

References:

[1]	C. Bardos, A. Y. Leroux and J. C. Nedelec, First order quasilinear equation with boundary conditions, Comm. Part. Diff. Eqn., 4 (1979), 1017-1034. doi: 10.1080/03605307908820117.
[2]	J. D. Cole, On a quasi-linear parabolic equation occurring in aerodynamics, Quart. Appl. Math, 9 (1951), 225-236.
[3]	J. F. Colombeau, "New Generalized Functions and Multiplication of Distributions,'' Amsterdam:North Holland, 1984.
[4]	J. F. Colombeau, "New Generalized Functions and Multiplication of Distributions: A Graduate Course, Application to Theoretical and Numerical Solutions of Partial Differential Equations,'' Lyon, 1993.
[5]	J. F. Colombeau and A. Heibig, Generalized solutions to Cauchy problems, Monatsh. Math., 117 (1994), 33-49. doi: 10.1007/BF01299310.
[6]	E. Hopf, The Partial differential equation $u_t+u u_x = \nu u_{x x}$, Comm. Pure Appl. Math., 3 (1950), 201-230. doi: 10.1002/cpa.3160030302.
[7]	K. T. Joseph, A Riemann problem whose viscosity solution contain $\delta$- measures, Asym. Anal., 7 (1993), 105-120 . doi: 10.3233/ASY.19937203.
[8]	K. T. Joseph and A. S. Vasudeva Murthy, Hopf-Cole transformation to some systems of partial differential equations, NoDEA Nonlinear Diff. Eq. Appl., 8 (2001), 173-193 . doi: 10.1007/PL00001444.
[9]	K. T. Joseph, Explicit generalized solutions to a system of conservation laws, Proc. Indian Acad. Sci. Math., 109 (1999), 401-409. doi: 10.1007/BF02838000.
[10]	P. D. Lax, Hyperbolic systems of conservation laws II, Comm.Pure Appl. Math., 10 (1957), 537-566. doi: 10.1002/cpa.3160100406.
[11]	P. G. LeFloch, An existence and uniqueness result for two non-strictly hyperbolic systems in Nonlinear evolution equations that change type, (eds)Barbarae Le Keyfitz and Michael Shearer, IMA, (Springer-Verlag), 27 (1990), 126-139. doi: 10.1007/978-1-4613-9049-7_10.
[12]	V. M. Shelkovich, The Riemann problem admitting $\delta - \delta'$ - shocks, and vacuum states (the vanishing viscosity approach), J. Differential Equations, 231 (2006), 459-500. doi: 10.1016/j.jde.2006.08.003.

show all references

References:

[1]	C. Bardos, A. Y. Leroux and J. C. Nedelec, First order quasilinear equation with boundary conditions, Comm. Part. Diff. Eqn., 4 (1979), 1017-1034. doi: 10.1080/03605307908820117.
[2]	J. D. Cole, On a quasi-linear parabolic equation occurring in aerodynamics, Quart. Appl. Math, 9 (1951), 225-236.
[3]	J. F. Colombeau, "New Generalized Functions and Multiplication of Distributions,'' Amsterdam:North Holland, 1984.
[4]	J. F. Colombeau, "New Generalized Functions and Multiplication of Distributions: A Graduate Course, Application to Theoretical and Numerical Solutions of Partial Differential Equations,'' Lyon, 1993.
[5]	J. F. Colombeau and A. Heibig, Generalized solutions to Cauchy problems, Monatsh. Math., 117 (1994), 33-49. doi: 10.1007/BF01299310.
[6]	E. Hopf, The Partial differential equation $u_t+u u_x = \nu u_{x x}$, Comm. Pure Appl. Math., 3 (1950), 201-230. doi: 10.1002/cpa.3160030302.
[7]	K. T. Joseph, A Riemann problem whose viscosity solution contain $\delta$- measures, Asym. Anal., 7 (1993), 105-120 . doi: 10.3233/ASY.19937203.
[8]	K. T. Joseph and A. S. Vasudeva Murthy, Hopf-Cole transformation to some systems of partial differential equations, NoDEA Nonlinear Diff. Eq. Appl., 8 (2001), 173-193 . doi: 10.1007/PL00001444.
[9]	K. T. Joseph, Explicit generalized solutions to a system of conservation laws, Proc. Indian Acad. Sci. Math., 109 (1999), 401-409. doi: 10.1007/BF02838000.
[10]	P. D. Lax, Hyperbolic systems of conservation laws II, Comm.Pure Appl. Math., 10 (1957), 537-566. doi: 10.1002/cpa.3160100406.
[11]	P. G. LeFloch, An existence and uniqueness result for two non-strictly hyperbolic systems in Nonlinear evolution equations that change type, (eds)Barbarae Le Keyfitz and Michael Shearer, IMA, (Springer-Verlag), 27 (1990), 126-139. doi: 10.1007/978-1-4613-9049-7_10.
[12]	V. M. Shelkovich, The Riemann problem admitting $\delta - \delta'$ - shocks, and vacuum states (the vanishing viscosity approach), J. Differential Equations, 231 (2006), 459-500. doi: 10.1016/j.jde.2006.08.003.

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