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Vanishing viscosity approach to a system of conservation laws admitting $\delta''$ waves
1. | TIFR Centre for Applicable Mathematics, P.B.NO. 6503, Sharada Nagar, Chikkabommasandra, Bangalore 560065, India, India |
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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