[Back to Results | New Search]

Student Number 952201025 Author Man-meng Io(姚文銘) Author's Email Address ar_gold@yahoo.com Statistics This thesis had been viewed 1170 times. Download 514 times. Department Mathematics Year 2007 Semester 2 Degree Master Type of Document Master's Thesis Language English Title Classical Solutions to the Perturbed Riemann Problem of Scalar Resonant Balance Law Date of Defense 2008-06-14 Page Count 24 Keyword Characteristic method Conservation laws Lax's method Nonlinear balance laws Perturbed Riemann problems Riemann problems Abstract In this paper we study the classical solutions to the perturbed Riemann problem of some scalar nonlinear balance law in resonant case. The equation with source term is equivalent to a 2×2 nonlinear balance laws as described in [6, 7], and it is a resonant system due to the fact that the speeds of waves in the solution to this 2×2 system coincide. The characteristic method in [8] is applied to construct the classical solutions of perturbed Riemann problem. Moreover, we show that, the pointwise limit of classical solutions, which are deﬁned as the

measurable solutions to the corresponding Riemann problem (with singular source) of perturbed Riemann problem, are self-similar as described in [12].Table of Content 中文摘要 ………………………………………………………………i

英文摘要 ………………………………………………………………ii

Acknowledgement ………………………………………………………iii

目錄 ……………………………………………………………………iv

圖目錄 …………………………………………………………………v

表目錄 ……………………………………………………………………vi

1. Introduction …………………………………………………………2

2. Classical solutions of perturbed Riemann problem …………5

3. Stability of perturbed Riemann solutions …………………17

References ………………………………………………………………23Reference [1] C. Dafermos, Generalized characteristics and the structure of solutions of hyperbolic conservation laws, Ind. Univ. Math. J. 26 (1977), 1097-1119.

[2] C. Dafermos, Solutions of the Riemann problem for a class of conservation laws by the viscosity method, Arch. Ration. Mech. Anal., 52 (1973), 1-9.

[3] G. Dal Maso, P. LeFloch and F. Murat, Deﬁnition and weak stability of nonconservative products, J. Math. Pure. Appl., 74(1995), 483-548.

[4] Ronald J. DiPerna, Measure-Valued Solutions to Conservation Laws, Arch. Ration. Mech. Anal., (1985), 223-270.

[5] J. Glimm, Solutions in the large for nonlinear hyperbolic systems of equations, Comm. Pure Appl. Math., 18(1956), 697-715.

[6] J. M. Hong, An extension of Glimm’s method to inhomogeneous strictly hyperbolic systems of conservation laws by ”weaker than weaker” solutions of the Riemann problem, J. Diﬀ. Equations, 222(2006), 515-549.

[7] J. M. Hong and B. Temple, A Bound on the Total Variation of the Conserved Quantities for Solutions of a General Resonant Nonlinear Balance Law, SIAM J. Appl. Math. 64, No 3, (2004), pp 625-640.

[8] J. M. Hong, Y. Chang and S.-W. Chou, Generalized Solutions to the Riemann Problem of Scalar Balance Law with Singular Source Term, preprint.

[9] E. Isaacson and B. Temple, Convergence of 2 × 2 by Godunov method for a general resonant nonlinear balance law, SIAM J. Appl. Math. 55 (1995), pp 625-640.

[10] K. T. Joseph and P. G. LeFloch, Singular limits for the Riemann problem: general diﬀusion, relaxation, and boundary condition, in ” new analytical approach to multidimensional balance laws”, O. Rozanova ed., Nova Press, 2004.

23[11] S. Kruzkov, First order quasilinear equations with several space variables, Math. USSR Sbornik 10 (1970), 217-273.

[12] P. D. Lax, Hyperbolic system of conservation laws, II, Comm. Pure Appl. Math., 10(1957), 537-566.

[13] P. G. LeFloch, Entropy weak solutions to nonlinear hyperbolic systems under nonconservative form, Comm. Partial Diﬀerential Equations, 13(1988), 669-727.

[14] T. P. Liu, The Riemann problem for general systems of conservation laws, J. Diﬀ. Equations, 18(1975), 218-234.

[15] T. P. Liu, Quaslinear hyperbolic systems, Comm. Math. Phys., 68(1979), 141-172.

[16] C. Mascia and C. Sinestrari, The perturbed Riemann problem for a balance law, Advances in Diﬀerential Equations, 1996-041.

[17] O. A. Oleinik, Discontinuous solutions of nonlinear diﬀerential equations, Amer. Math. Soc. Transl. Ser. 2, 26 (1957), 95-172.

[18] C. Sinestrari, The Riemann problem for an inhomogeneous conservation law without convexity, Siam J. Math. Anal., Vo28, No1, (1997), 109-135.

[19] C. Sinestrari, Asymptotic proﬁle of solutions of conservation laws with source, J. Diﬀ. and Integral Equations, Vo9, No3,(1996), 499-525.

[20] M. Slemrod and A. Tzavaras, A limiting viscosity approach for the Riemann problem in isentropic gas dynamics, Ind. Univ. Math. J. 38 (1989), 1047-1073.

[21] J. Smoller, Shock waves and reaction-dﬀusion equations, Springer, New York, 1983.

[22] A. Tzavaras, Waves interactions and variation estimates for self-similar zero viscosity limits in systems of conservation laws, Arch. Ration. Mech. Anal., 135 (1996), 1-60.

[23] A. Volpert, The space BV and quasilinear equations, Maths. USSR Sbornik 2 (1967), 225-267.Advisor John M. Hong(洪盟凱)

Files approve immediately

952201025.pdf Date of Submission 2008-07-21

Our service phone is (03)422-7151 Ext. 57407,E-mail is also welcomed.