Vanishing viscosity limit for Riemann solutions to a 2 × 2 hyperbolic system with linear damping

Abstract

In this paper, we propose a time-dependent viscous system and by using the vanishing viscosity method we show the existence of solutions for the Riemann problem to a particular $2 \times 2$ system of conservation laws with linear damping.

Keywords

Nonstrictly hyperbolic system linear damping Riemann problem time-dependent viscous system delta shock wave solution

Get full access to this article

View all access options for this article.

References

G.I.

Barenblatt, Scaling, Self-Similarity, and Intermediate Asymptotics, Cambridge University Press, 1996. doi:10.1017/CBO9781107050242.

D.G.

Crighton, Model equations of nonlinear acoustics, Annu. Rev. Fluid Mech. 11 (1979), 11–33. doi:10.1146/annurev.fl.11.010179.000303.

D.G.

Crighton and

J.F.

Scott, Asymptotic solution of model equations in nonlinear acoustics, Phil. Trans. Koy Sot. A292 (1979), 101–134.

C.M.

Dafermos, Solutions of the Riemann problem for a class of hyperbolic systems of conservation laws by the viscosity method, Arch. Ration. Mech. Anal. 52 (1973), 1–9. doi:10.1007/BF00249087.

De la cruz, Riemann problem for a

2 \times 2

hyperbolic system with linear damping, Acta Appl. Math. 170 (2020), 631–647. doi:10.1007/s10440-020-00350-w.

De la cruz and

Santos, Delta shock waves for a system of Keyfitz–Kranzer type, Z. Angew. Math. Mech. 99 (2019), e201700251.

Doyle and

M.J.

Englefield, Similarity solutions of a generalized Burgers equation, IMA Journal of Applied Mathematics 44 (1990), 145–153. doi:10.1093/imamat/44.2.145.

Ercole, Delta-shock waves as self-similar viscosity limits, Quart. Appl. Math. LVIII(1) (2000), 177–199. doi:10.1090/qam/1739044.

R.N.

Henriksen, Scale Invariance: Self-Similarity of the Physical World, Wiley-VCH, 2015. doi:10.1002/9783527687343.

10.

E.L.

Isaacson and

Temple, Analysis of a singular hyperbolic system of conservation laws, J. Differential Equations 65 (1986), 250–268. doi:10.1016/0022-0396(86)90037-9.

11.

K.T.

Joseph, A Riemann problem whose viscosity solution contain δ-measures, Asymptot. Anal. 7 (1993), 105–120. doi:10.3233/ASY-1993-7203.

12.

P.G.

LeFloch, Entropy weak solutions to nonlinear hyperbolic systems under nonconservative form, Communications in Partial Differential Equation 13 (1988), 669–727. doi:10.1080/03605308808820557.

13.

Y.Q.

Lou and

W.G.

Wang, New self-similar solutions of polytropic gas dynamics, Mon. Not. Roy. Astron. Soc. 372 (2006), 885–900. doi:10.1111/j.1365-2966.2006.10908.x.

14.

Mayil Vaganan and

Senthil Kumaran, Kummer function solutions of damped Burgers equations with time-dependent viscosity by exact linearization, Nonlinear Anal.: Real World Appl. 9 (2008), 2222–2233. doi:10.1016/j.nonrwa.2007.08.001.

15.

A.D.

Polyanin and

V.F.

Zaitsev, Handbook of Nonlinear Partial Differential Equations, Chapman & Hall/CRC, 2003. doi:10.1201/9780203489659.

16.

P.L.

Sachdev, Self-Similarity and Beyond: Exact Solutions of Nonlinear Problems, Monographs and Surveys in Pure and Applied Mathematics 113, Chapman & Hall/CRC, 2000. doi:10.1201/9781420035711.

17.

Sackand and

Schamel, Nonlinear dynamics in expanding plasmas, Phys. Lett. A 110 (1985), 206–212. doi:10.1016/0375-9601(85)90125-2.

18.

J.F.

Scott, The long time asymptotics of solution to the generalized Burgers equation, Proc. Roy. Sot. Land. A 373 (1981), 443–456.

19.

S.F.

Shandarin and

Y.B.

Zeldovich, Large-scale structure of the universe: Turbulence, intermittency, structures in a self-gravitating medium, Rev. Modern Phys. 61 (1989), 185–220. doi:10.1103/RevModPhys.61.185.

20.

Suto and

Silk, Self-similar dynamics of polytropic gaseous spheres, Astrophys. J. 326 (1988), 527–538. doi:10.1086/166114.

21.

Tan,

Zhang and

Zheng, Delta shock waves as limits of vanishing viscosity for hyperbolic systems of conservation laws, J. Differential Equations 112 (1994), 1–32. doi:10.1006/jdeq.1994.1093.

22.

V.A.

Tupciev, On the method of introducing viscosity in the study of problems involving decay of a discontinuity, Dokl. Akad. Nauk SSR 211 (1973), 55–58.

23.

A.I.

Volpert, The space BV and quasilinear equations, Math. Sbornik. 73 (1967), 225–267. doi:10.1070/SM1967v002n02ABEH002340.

24.

Wang and

Zhang, Existence and decay rates of solutions to the generalized Burgers equation, J. Math. Anal. Appl. 284 (2003), 213–235. doi:10.1016/S0022-247X(03)00336-6.

25.

J.H.

Wang and

Zhang, A new viscous regularization of the Riemann problem for Burger’s equation, J. Partial Diff. Eqs. 13 (2000), 253–263.

26.

Yang, Riemann problems for a class of coupled hyperbolic systems of conservation laws, J. Differential Equations 159 (1999), 447–484. doi:10.1006/jdeq.1999.3629.

27.

Yang and

Zhang, New developments of delta shock waves and its applications in systems of conservation laws, J. Differential Equations 252 (2012), 5951–5993. doi:10.1016/j.jde.2012.02.015.

28.

Zhang, Global existence and asymptotic behaviour of the solution of a generalized Burger’s equation with viscosity, Computers Math. Applic. 41 (2001), 589–596. doi:10.1016/S0898-1221(00)00302-3.

29.

Zhang and

Wang, Large-time behavior of smooth solutions to a nonuniformly parabolic equation, Computers Math. Applic. 47 (2004), 353–363. doi:10.1016/S0898-1221(04)90030-2.