American Institute of Mathematical Sciences

Advanced
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

[1]

Boris Andreianov, Kenneth H. Karlsen, Nils H. Risebro. On vanishing viscosity approximation of conservation laws with discontinuous flux. Networks & Heterogeneous Media, 2010, 5 (3) : 617-633. doi: 10.3934/nhm.2010.5.617
[2]

Boris P. Andreianov, Giuseppe Maria Coclite, Carlotta Donadello. Well-posedness for vanishing viscosity solutions of scalar conservation laws on a network. Discrete & Continuous Dynamical Systems, 2017, 37 (11) : 5913-5942. doi: 10.3934/dcds.2017257
[3]

Giuseppe Maria Coclite, Lorenzo di Ruvo, Jan Ernest, Siddhartha Mishra. Convergence of vanishing capillarity approximations for scalar conservation laws with discontinuous fluxes. Networks & Heterogeneous Media, 2013, 8 (4) : 969-984. doi: 10.3934/nhm.2013.8.969
[4]

Wen Shen. Traveling waves for conservation laws with nonlocal flux for traffic flow on rough roads. Networks & Heterogeneous Media, 2019, 14 (4) : 709-732. doi: 10.3934/nhm.2019028
[5]

K. T. Joseph, Philippe G. LeFloch. Boundary layers in weak solutions of hyperbolic conservation laws II. self-similar vanishing diffusion limits. Communications on Pure & Applied Analysis, 2002, 1 (1) : 51-76. doi: 10.3934/cpaa.2002.1.51
[6]

Qinglong Zhang. Delta waves and vacuum states in the vanishing pressure limit of Riemann solutions to Baer-Nunziato two-phase flow model. Communications on Pure & Applied Analysis, 2021, 20 (9) : 3235-3258. doi: 10.3934/cpaa.2021104
[7]

Stefano Bianchini, Alberto Bressan. A case study in vanishing viscosity. Discrete & Continuous Dynamical Systems, 2001, 7 (3) : 449-476. doi: 10.3934/dcds.2001.7.449
[8]

Umberto Mosco, Maria Agostina Vivaldi. Vanishing viscosity for fractal sets. Discrete & Continuous Dynamical Systems, 2010, 28 (3) : 1207-1235. doi: 10.3934/dcds.2010.28.1207
[9]

Wenjun Wang, Lei Yao. Vanishing viscosity limit to rarefaction waves for the full compressible fluid models of Korteweg type. Communications on Pure & Applied Analysis, 2014, 13 (6) : 2331-2350. doi: 10.3934/cpaa.2014.13.2331
[10]

Avner Friedman. Conservation laws in mathematical biology. Discrete & Continuous Dynamical Systems, 2012, 32 (9) : 3081-3097. doi: 10.3934/dcds.2012.32.3081
[11]

Mauro Garavello. A review of conservation laws on networks. Networks & Heterogeneous Media, 2010, 5 (3) : 565-581. doi: 10.3934/nhm.2010.5.565
[12]

Len G. Margolin, Roy S. Baty. Conservation laws in discrete geometry. Journal of Geometric Mechanics, 2019, 11 (2) : 187-203. doi: 10.3934/jgm.2019010
[13]

Mauro Garavello, Roberto Natalini, Benedetto Piccoli, Andrea Terracina. Conservation laws with discontinuous flux. Networks & Heterogeneous Media, 2007, 2 (1) : 159-179. doi: 10.3934/nhm.2007.2.159
[14]

Tong Yang, Huijiang Zhao. Asymptotics toward strong rarefaction waves for $2\times 2$ systems of viscous conservation laws. Discrete & Continuous Dynamical Systems, 2005, 12 (2) : 251-282. doi: 10.3934/dcds.2005.12.251
[15]

Shuichi Kawashima, Shinya Nishibata, Masataka Nishikawa. Asymptotic stability of stationary waves for two-dimensional viscous conservation laws in half plane. Conference Publications, 2003, 2003 (Special) : 469-476. doi: 10.3934/proc.2003.2003.469
[16]

Hua Chen, Jian-Meng Li, Kelei Wang. On the vanishing viscosity limit of a chemotaxis model. Discrete & Continuous Dynamical Systems, 2020, 40 (3) : 1963-1987. doi: 10.3934/dcds.2020101
[17]

Wen-Xiu Ma. Conservation laws by symmetries and adjoint symmetries. Discrete & Continuous Dynamical Systems - S, 2018, 11 (4) : 707-721. doi: 10.3934/dcdss.2018044
[18]

Tai-Ping Liu, Shih-Hsien Yu. Hyperbolic conservation laws and dynamic systems. Discrete & Continuous Dynamical Systems, 2000, 6 (1) : 143-145. doi: 10.3934/dcds.2000.6.143
[19]

Yanbo Hu, Wancheng Sheng. The Riemann problem of conservation laws in magnetogasdynamics. Communications on Pure & Applied Analysis, 2013, 12 (2) : 755-769. doi: 10.3934/cpaa.2013.12.755
[20]

Stefano Bianchini, Elio Marconi. On the concentration of entropy for scalar conservation laws. Discrete & Continuous Dynamical Systems - S, 2016, 9 (1) : 73-88. doi: 10.3934/dcdss.2016.9.73

2020 Impact Factor: 1.916

Tools

Metrics

  • PDF downloads (67)
  • HTML views (0)
  • Cited by (11)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences