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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

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 & 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.  Google Scholar

[2]

J. D. Cole, On a quasi-linear parabolic equation occurring in aerodynamics, Quart. Appl. Math, 9 (1951), 225-236. Google Scholar

[3]

J. F. Colombeau, "New Generalized Functions and Multiplication of Distributions,'' Amsterdam:North Holland, 1984.  Google Scholar

[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. Google Scholar

[5]

J. F. Colombeau and A. Heibig, Generalized solutions to Cauchy problems, Monatsh. Math., 117 (1994), 33-49. doi: 10.1007/BF01299310.  Google Scholar

[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.  Google Scholar

[7]

K. T. Joseph, A Riemann problem whose viscosity solution contain $\delta$- measures, Asym. Anal., 7 (1993), 105-120 . doi: 10.3233/ASY.19937203.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

P. D. Lax, Hyperbolic systems of conservation laws II, Comm.Pure Appl. Math., 10 (1957), 537-566. doi: 10.1002/cpa.3160100406.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[2]

J. D. Cole, On a quasi-linear parabolic equation occurring in aerodynamics, Quart. Appl. Math, 9 (1951), 225-236. Google Scholar

[3]

J. F. Colombeau, "New Generalized Functions and Multiplication of Distributions,'' Amsterdam:North Holland, 1984.  Google Scholar

[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. Google Scholar

[5]

J. F. Colombeau and A. Heibig, Generalized solutions to Cauchy problems, Monatsh. Math., 117 (1994), 33-49. doi: 10.1007/BF01299310.  Google Scholar

[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.  Google Scholar

[7]

K. T. Joseph, A Riemann problem whose viscosity solution contain $\delta$- measures, Asym. Anal., 7 (1993), 105-120 . doi: 10.3233/ASY.19937203.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

P. D. Lax, Hyperbolic systems of conservation laws II, Comm.Pure Appl. Math., 10 (1957), 537-566. doi: 10.1002/cpa.3160100406.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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