Skip to main content
SpringerLink
Log in

The Riemann problem for the Brio system: a solution containing a Dirac mass obtained via a distributional product

  • Published:
Russian Journal of Mathematical Physics Aims and scope Submit manuscript

Abstract

The system of conservation laws \({u_t} + {\left( {\frac{{{u^2} + {v^2}}}{2}} \right)_x} = 0\), v t + (uv − v)x = 0 with the initial conditions u(x, 0) = l 0 + b 0 H(x), v(x, 0) = k 0 + c 0 H(x), where H is the Heaviside function is studied. This strictly hyperbolic system was introduced by M. Brio in 1988 and provides a simplified model for the magnetohydrodynamics equations. Under certain compatibility conditions for the constants l 0, b 0, k 0, c 0, an explicit solution containing a Dirac mass is given and we prove the uniqueness of this solution within a convenient class of distributions which includes Dirac-delta measures. Our concept of solution is defined within the framework of a distributional product, and it is a consistent extension of the concept of a classical solution. This direct method seems considerably simpler than the weak asymptotic method usually used in the study of delta-shocks emergence in nonlinear conservation laws.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. F. Bouchut and F. James, “Duality Solutions for Pressureless Gases, Monotone Scalar Conservation Laws, and Uniqueness,” Comm. Partial Differential Equations 24, 2173–2190 (1999).

    Article  MathSciNet  MATH  Google Scholar 

  2. Y. Brenier and E. Grenier, “Sticky Particles and Scalar Conservation Laws,” SIAM J. Numer. Anal. 35, 2317–2328 (1988).

    Article  MathSciNet  Google Scholar 

  3. A. Bressan and F. Rampazzo, “On Differential Systems with Vector Valued Impulsive Controls,” Bull. Un. Mat. Ital. 7 2B, 641–656 (1988).

    MathSciNet  Google Scholar 

  4. M. Brio, “Admissibility Conditions for Weak Solutions of Nonstrictly Hyperbolic Systems,” Proc. Int. Conf. on Hyperbolic Problems (Aachen, 1988) (Springer Notes in Mathematics) (Berlin: Springer), pp. 43–50.

    Google Scholar 

  5. G.-Q. Chen and H. Liu, “Formation of d-Shocks and Vacuum States in the Vanishing Pressure Limit of Solutions to Euler Equations for Isentropic Fluids,” SIAM J. Math. Anal. 34 (4), 925–938 (2003).

    Article  MathSciNet  MATH  Google Scholar 

  6. J. F. Colombeau and A. Le Roux, “Multiplication of Distributions in Elasticity and Hydrodynamics,” J. Math. Phys. 29, 315–319 (1988).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  7. Yu. V. Egorov, “On the Theory of Generalized Functions,” Uspekhi Mat. Nauk 45 (5), (275), 3–40 (1990).

    MathSciNet  Google Scholar 

  8. Yu. V. Egorov, “On the Theory of Generalized Functions,” Russian Math. Surveys 45 (5), 1–49 (1990)].

    Article  ADS  MathSciNet  MATH  Google Scholar 

  9. G. Dal Maso, P. G. Lefloch, and F. Murat, “Definition and Weak Stability of Nonconservative Products,” J. Math. Pures Appl. 74 (6), 483–548 (1995).

    MathSciNet  MATH  Google Scholar 

  10. V. G. Danilov, V. P. Maslov, and V. M. Shelkovich, “Algebras of Singularities of Singular Solutions to First-Order Quasi-Linear Strictly Hyperbolic Systems,” Teoret. Mat. Fiz. 114 (1), 3–55 (1998)

    Article  MathSciNet  Google Scholar 

  11. V. G. Danilov, V. P. Maslov, and V. M. Shelkovich, “Algebras of Singularities of Singular Solutions to First-Order Quasi-Linear Strictly Hyperbolic Systems,” Theoret. Math. Phys. 114 (1), 1–42 (1998)].

    Article  MathSciNet  MATH  Google Scholar 

  12. V. G. Danilov and V. M. Shelkovich, “Delta Shock-Wave Type Solution of Hyperbolic Systems of Conservation Laws,” Quart. Appl. Math. 29, 401–427 (2005).

    Article  MathSciNet  Google Scholar 

  13. V. G. Danilov and D. Mitrovic, “Weak Asymptotic of Shock Wave Formation Process,” Nonlinear Anal.: Theory, Methods and Applications 61, 613–635 (2005).

    Article  MathSciNet  MATH  Google Scholar 

  14. V. G. Danilov and V. M. Shelkovich, “Dynamics of Propagation and Interaction of d-Shock Waves in Conservation Law Systems,” J. Differential Equations 211, 333–381 (2005).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  15. B. Hayes and P. G. Lefloch, “Measure-Solution to a Strictly Hyperbolic System of Conservation Laws,” Nonlinearity 9, 1547–1563 (1996).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  16. F. Huang, “Weak Solutions to Pressureless Type System,” Comm. Partial Differential Equations 30, 283–304 (2005).

    Article  MathSciNet  MATH  Google Scholar 

  17. K. T. Joseph, “A Riemann Problem Whose Viscosity Solutions Contains d-measures,” Asymptot. Anal. 7, 105–120 (1993).

    MATH  Google Scholar 

  18. H. Kalisch and D. Mitrovic, “Singular Solutions of a Fully Nonlinear 2 × 2 System of Conservation Laws,” Proc. Edinb. Math. Soc. (2) 55 (3), 711–729 (2012).

    Article  MathSciNet  MATH  Google Scholar 

  19. B. L. Keyfitz and H. C. Kranzer, “Spaces of Weighted Measures for Conservation Laws with Singular Shock Solutions,” J. Differential Equations 118, 420–451 (1995).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  20. C. Korchinski, Solution of a Riemann Problem for a 2×2 System of Conservation Laws Possessing No Classical Weak Solutions (PhD Thesis, Adelphi University, 1977).

    Google Scholar 

  21. R. J. LeVeque, “The Dynamics of Pressureless Dust Clouds and Delta Waves,” J. Hyperbolic Differ. Equ. 1, 315–327 (2004).

    Article  MathSciNet  MATH  Google Scholar 

  22. S. Lojasiewicz, An Introduction to the Theory of Real Functions (John Wiley & Sons Ltd.; 1988).

    MATH  Google Scholar 

  23. V. P. Maslov, “Nonstandard Characteristics in Asymptotical Problems,” Uspekhi Mat. Nauk 38 (6), (234), 3–36 (1983)

    ADS  Google Scholar 

  24. V. P. Maslov, “Nonstandard Characteristics in Asymptotical Problems,” Russian Math. Surveys 38 (6), 1–42 (1983)].

    Article  ADS  MATH  Google Scholar 

  25. V. P. Maslov and G. A. Omel’yanov, “Asymptotic Soliton-Form Solutions of Equations with Small Dispersion,” Uspekhi Mat. Nauk 36 (3), (219), 63–126 (1981)

    MathSciNet  MATH  Google Scholar 

  26. V. P. Maslov and G. A. Omel’yanov, “Asymptotic Soliton-Form Solutions of Equations with Small Dispersion,” Russian Math. Surveys 36 (3), 73–149 (1981)].

    Article  ADS  MathSciNet  MATH  Google Scholar 

  27. M. Nedeljkov, “ShadowWaves: Entropies and Interactions of Delta and Singular Shocks,” Arch. Ration Mech. Anal. 197, 489–537 (2010).

    Article  MathSciNet  MATH  Google Scholar 

  28. M. Nedeljkov, “Unbounded Solutions to Some Systems of Conservation Laws-Split Delta ShockWaves,” Mat. Vesnik 54, 145–149 (2002).

    MathSciNet  MATH  Google Scholar 

  29. M. Nedeljkov and M. Oberguggenberger, “Interactions of Delta Shock Waves in a Strictly Hyperbolic System of Conservation Laws,” J. Math. Anal. Appl. 334, 1143–1157 (2008).

    Article  MathSciNet  Google Scholar 

  30. M. Nedeljkov, “Delta and Singular Delta Locus for One Dimensional Systems of Conservation Laws,” Math. Meth. Appl. Sci. 27, 931–955 (2004).

    Article  MathSciNet  MATH  Google Scholar 

  31. C. O. R. Sarrico, “Distributional Products and Global Solutions for Nonconservative Inviscid Burgers Equation,” J. Math. Anal. Appl. 281, 641–656 (2003).

    Article  MathSciNet  MATH  Google Scholar 

  32. C. O. R. Sarrico, “Collision of Delta-Waves in a Turbulent Model Studied via a Distributional Product,” Nonlinear Anal. 73, 2868–2875 (2010).

    Article  MathSciNet  MATH  Google Scholar 

  33. C. O. R. Sarrico, “New Solutions for the One-Dimensional Nonconservative Inviscid Burgers Equation,” J. Math. Anal. Appl. 317, 496–509 (2006).

    Article  MathSciNet  MATH  Google Scholar 

  34. C. O. R. Sarrico, “About a Family of Distributional Products Important in the Applications,” Port. Math. 45, 295–316 (1988).

    MathSciNet  MATH  Google Scholar 

  35. C. O. R. Sarrico, “Entire Functions of Certain Singular Distributions and the Interaction of Delta Waves in Nonlinear Conservation Laws,” Int. J. Math. Anal. 4 (36), 1765–1778 (2010).

    MathSciNet  MATH  Google Scholar 

  36. C. O. R. Sarrico, “Products of Distributions, Conservation Laws and the Propagation of δ' -Shock Waves,” Chinese. Ann. Math. 33 B(3), 1–18 (2012).

    Article  MathSciNet  Google Scholar 

  37. C. O. R. Sarrico, “Products of Distributions and Singular Travelling Waves as Solutions of Advection-Reaction Equations,” Russ. J. Math. Phys. 19 (2), 244–255 (2012).

    Article  MathSciNet  MATH  Google Scholar 

  38. W. Sheng and T. Zhang, “The Riemann Problem for Transportation Equations in Gas Dynamics,” Mem. Amer. Math. Soc. 137, 1–77 (1999).

    MathSciNet  Google Scholar 

  39. D. Tan, T. Zhang, and Y. Zheng, “Delta Shock Waves as Limits of Vanishing Viscosity for a System of Conservation Laws,” J. Differential Equations 112, 1–32 (1994).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  40. H. Yang and Y. Zhang, “New Developments of Delta Shock Waves and Its Applications in Systems of Conservation Laws,” J. Differential Equations 252, 5951–5993 (2012).

    Article  ADS  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. CMAFCIO, Faculdade de Cincias da Universidade de Lisboa, Campo Grande, 1749-016, Lisboa, Portugal

    C. O. R. Sarrico

Authors
  1. C. O. R. Sarrico
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to C. O. R. Sarrico.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Sarrico, C.O.R. The Riemann problem for the Brio system: a solution containing a Dirac mass obtained via a distributional product. Russ. J. Math. Phys. 22, 518–527 (2015). https://doi.org/10.1134/S1061920815040111

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S1061920815040111

Keywords

Search

Navigation