Initial boundary value problem for compressible euler equations with damping

Indiana University Mathematics Journal

Volumn 57, Issue 5, 2008, Pages 2257-2282

  Pan, Ronghua ^a   Zhao, Kun ^a  

^a GEORGIA INSTITUTE OF TECHNOLOGY   (United States)

Author keywords

Darcy's law; Hyperbolic conservation laws; Porous medium flows

Indexed keywords

EID: 57449116991     PISSN: 00222518     EISSN: None     Source Type: Journal    
DOI: 10.1512/iumj.2008.57.3366     Document Type: Article

References (49)

1. N. ALIKAKOS and R. ROSTAMIAN, Large time behavior of solutions of Neumann boundary value problem for the porous medium equation, Indiana Univ. Math. J. 30 (1981), 749-785.
2. D.G. ARONSON, The porous media equations, Nonlinear Diffusion Problem (A. Fasano and M. Primicerio, eds.), Lecture Notes in Math., vol. 1224, Springer-Verlag, Berlin, 1986.
3. G.-Q. CHEN and H. FRID, Divergence-measure fields and hyperbolic conservation laws, Arch. Rat. Mech. Anal. 147 (1999), 89-118.
4. C.M. DAFERMOS, A system of hyperbolic conservation laws with frictional damping, Z. Angew. Math. Phys. 46 Special Issue (1995), 294-307.
5. -, Hyperbolic Conservation Laws in Continuum Physics, Springer-Verlag, Berlin, 2000.
6. C.M. DAFERMOS and R.H. PAN, Global BV solutions for the p-system with frictional damping, Comm. Pure Appl. Math. (2006) (submitted).
7. X-X. DING, G.-Q. CHEN, and P.Z. LUO, Convergence of the Lax-Friedrichs scheme for the isentropicgas dynamics. I, Acta Math. Scientia 5 (1985), 415-472.
8. -, Convergence of the Lax-Friedrichs scheme for the isentropic gas dynamics. II, Acta Math. Scientia 6 (1986), 75-120.
9. -, Convergence of the Lax-Friedrichs scheme for the isentropic gas dynamics. III, Acta Math. Scientia 9 (1989), 43-44.
10. -, Convergence of the fraction step Lax-Friedrichs scheme and Godunov scheme for the isentropic system of gas dynamics, Comm. Math. Phys. 121 (1989), 63-84.
11. R.J. DIPERNA, Convergence of the viscosity method for isentropic gas dynamics, Comm. Math. Phys. 91 (1983), 1-30.
12. S. GODUNOV, A Difference method for numerical calculation of discontinuous solutions of the equations of hydrodynamics, Mat. Sb. 47 (1959), 271-360.
13. A. HEIDRICH, Global weak solutions to initial boundary problems for the wave equation with large data, Arch. Rat. Mech. Anal. 126 (1994), 333-368.
14. L. HSIAO, Quasilinear Hyperbolic Systems and Dissipative Mechanisms, World Scientific, Singapore, 1998.
15. L. HSIAO and T.-P. LIU, Convergence to nonlinear diffusion waves for solutions of a system of hyperbolic conservation laws with damping, Comm. Math. Phys. 143 (1992), 599-605.
16. -, Nonlinear diffusive phenomena of nonlinear hyperbolic systems, Chinese Ann. Math. Ser. B 14 (1993), 1-16.
17. L. HSIAO and T. LUO, Nonlinear diffusive phenomena of solutions for the system of compressible adiabaticflow through porous media, J. Differentail Equations 125 (1996), 329-365.
18. L. HSIAO and R.H. PAN, The damped p-system with boundary effects, Contemporary Mathematics 255 (2000), 109-123.
19. -, Initial boundary value problem for the system of compressible adiabaticflow through porous media, J. Differential Equations 159 (1999), 280-305.
20. L. HSIAO and D. SERRE, Global existence of solutions for the system of compressible adiabatic flow through porous media, SIAM J. Math. Anal. 27 (1996), 70-77.
21. F..M. HUANG, P. MARCATI, and R.H. PAN, Convergence to Barenblatt solution for the compressible Euler equations with damping and vacuum, Arch. Ration. Mech. Anal. 176(2005), 1-24.
22. F.M. HUANG and R.H. PAN, Asymptotic behavior of the solutions to the damped compressible Euler equations with vacuum, J. Differential Equations 220 (2006), 207-233.
23. -, Convergence rate for compressible Euler equations with damping and vacuum, Arch. Ration. Mech. Anal. 166 (2003), 359-376.
24. P.L. LIONS, B. PERTHAME, and P.E. SOUGANIDIS, Existence and stability of entropy solutions for the hyperbolic systems of isentropic gas dynamics in Eulerian and Lagrangian coordinates, Comm. Pure Appl. Math. 49 (1996), 599-638.
25. P.L. LIONS, B. PERTHAME, and E. TADMOR, Kinetic formulation of the isentropic gas dynamics and-p-systems, Comm. Math. Phys. 163 (1994), 415-431.
26. T.R LIU, Compressible flow with damping and vacuum, Japan J. Appl. Math 13 (1996), 25-32.
27. T. LIU and T. YANG, Compressible Euler equations with vacuum, J. Differential Equations 140 (1997), 223-237.
28. -, Compressible flow with vacuum and physical singularity (1999) (preprint).
29. M. LUSKIN and B. TEMPLE, The existence of a global weak solution to the nonlinear water-hammar problem, Comm. Pure Appl. Math. 35 (1982), 697-735.
30. P. MARCATI and M. ME, Convergence to nonlinear diffusion waves for solutions of the initial boundary value problem to the hyperbolic conservation laws with damping. Quart. Appl. Math. 58 (2000), 763-783.
31. P. MARCATI and R.H. PAN, On the diffusive profiles for the system of compressible adiabatic flow through porous media, SIAM J. Math. Anal. 33 (2001), 790-826.
32. F. MURAT, Compacité par compensation, Ann. Scuola Norm. Sup. Pisa Sci. Fis. 5 (1978), 489-507.
33. T. NISHIDA, Nonlinear hyperbolic equations and related topics in fluid dynamics, Publ. Math. D'Orsay (1978), 46-53.
34. T. NISHIDA and J. SMOLLER, Mixed problem for nonlinear conservation laws, J. Differential Equations 23 (1977), 244-269.
35. K. NISHIHARA, Convergence rates to nonlinear diffusion waves for solutions of system of hyperbolic conservation laws with damping, J. Differential Equations 131 (1996), 171-188.
36. p-convergence rate to nonlinear diffusion waves for p-system with damping, J. Differential Equations 161 (2000), 191-218.
37. K. NISHIHARA and T. YANG, Boundary effect on asymptotic behavior of solutions to the p-system with damping, J. Diflferentail Equations 156 (1999), 439-458.
38. R.H. PAN, Boundary effects and large time behavior for the system of compressible adiabatic flow through porous media, Michigan Math. J. 49 (2001), 519-540.
39. -, Dairy's law as long time limit of adiabatic porous media flows, J. Differential Equations 220 (2006), 121-146.
40. Y.C. QIU and K.J. ZHANG, On the relaxation limits of the hydrodynamic model for semiconductor devices, Math. Models and Methods in Appl. Sciences 12 (2002), 333-363.
41. D. SERRE and L. XIAO, Asymptotic behavior of large weak entropy solutions of the damped p-system, J. P. Diff. Equa. 10 (1997), 355-368.
42. J.A. SMOLLER, Shock Waves and Reaction-Diffusion Equations, Springer, New York, 1983.
43. S. TAKENO, Initial-boundary value problems for isentropic gas dynamics, Proc. Roy. Soc. Edinburgh 120A (1992).
44. L. TARTAR, Compensated Compactness and Application to Partial Differential Equations, Nonlinear Analysis and Mechanics (R. J. Knops, ed.), Research Notes in Mathematics, vol. 4, Pitman Press, New York, 1979.
45. J.L. VAZQUEZ, The Porous Medium Equation: Mathematical Theory, Oxford Science Publications, Oxford, 2007.
46. B. ZHANG, Convergence of the Godunov scheme for a simplified one-dimensional hydrodynamic model for semiconductor devices, Comm. Math. Phys. 157 (1993), 1-22.
47. H.J. ZHAO, Convergence to strong nonlinear diffusion waves for solutions of p-system with damping, J. Differential Equations 174 (2001), 200-236.
48. Y.S. ZHENG, Global smooth solutions to the adiabatic gas dynamics system with dissipation terms, Chinese Ann. Math. 17A(1996), 155-162.
49. C.J. ZHU, Convergence rates to nonlinear diffusion waves for weak entropy solutions to p-system with damping, Science in China, Ser. A 46 (2003), 562-575.