The fundamental solution of a conservation law without convexity
Authors:
Youngsoo Ha and Yong-Jung Kim
Journal:
Quart. Appl. Math. 73 (2015), 661-678
MSC (2000):
Primary 35L67, 35L65, 76L05
DOI:
https://doi.org/10.1090/qam/1397
Published electronically:
September 11, 2015
MathSciNet review:
3432277
Full-text PDF Free Access
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Abstract: The nonnegative fundamental solution of a scalar conservation law is constructed when its flux may have a finite number of inflection points. The constructed solution can be either explicit and implicit depending on the flux. This fundamental solution consists of a series of rarefaction waves, contact discontinuities and a shock. These analytically constructed fundamental solutions are also compared with numerical approximations, which possess the structure of the analytically constructed fundamental solution.
References
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- Burton Wendroff, The Riemann problem for materials with nonconvex equations of state. I. Isentropic flow, J. Math. Anal. Appl. 38 (1972), 454–466. MR 328387, DOI https://doi.org/10.1016/0022-247X%2872%2990103-5
References
- Donald P. Ballou, Solutions to nonlinear hyperbolic Cauchy problems without convexity conditions, Trans. Amer. Math. Soc. 152 (1970), 441–460 (1971). MR 0435615 (55 \#8573)
- A. L. Bertozzi, A. Münch, and M. Shearer, Undercompressive shocks in thin film flows, Phys. D 134 (1999), no. 4, 431–464. MR 1725916 (2001e:76042), DOI https://doi.org/10.1016/S0167-2789%2899%2900134-7
- José A. Carrillo and Juan L. Vázquez, Fine asymptotics for fast diffusion equations, Comm. Partial Differential Equations 28 (2003), no. 5-6, 1023–1056. MR 1986060 (2004a:35118), DOI https://doi.org/10.1081/PDE-120021185
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- Jean Dolbeault and Miguel Escobedo, $L^1$ and $L^\infty$ intermediate asymptotics for scalar conservation laws, Asymptot. Anal. 41 (2005), no. 3-4, 189–213. MR 2127996 (2006i:35233)
- Olivier Glass, An extension of Oleinik’s inequality for general 1D scalar conservation laws, J. Hyperbolic Differ. Equ. 5 (2008), no. 1, 113–165. MR 2405854 (2009c:35292), DOI https://doi.org/10.1142/S0219891608001398
- Youngsoo Ha and Yong Jung Kim, Explicit solutions to a convection-reaction equation and defects of numerical schemes, J. Comput. Phys. 220 (2006), no. 1, 511–531. MR 2281641 (2007k:65119), DOI https://doi.org/10.1016/j.jcp.2006.07.018
- David Hoff, The sharp form of Oleĭnik’s entropy condition in several space variables, Trans. Amer. Math. Soc. 276 (1983), no. 2, 707–714. MR 688972 (84b:35080), DOI https://doi.org/10.2307/1999078
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- Guang-Shan Jiang and Chi-Wang Shu, Efficient implementation of weighted ENO schemes, J. Comput. Phys. 126 (1996), no. 1, 202–228. MR 1391627 (97e:65081), DOI https://doi.org/10.1006/jcph.1996.0130
- Yong Jung Kim, Asymptotic behavior of solutions to scalar conservation laws and optimal convergence orders to $N$-waves, J. Differential Equations 192 (2003), no. 1, 202–224. MR 1987091 (2004e:35147), DOI https://doi.org/10.1016/S0022-0396%2803%2900058-5
- Yong-Jung Kim, Potential comparison and asymptotics in scalar conservation laws without convexity, J. Differential Equations 244 (2008), no. 1, 40–51. MR 2373653 (2009d:35220), DOI https://doi.org/10.1016/j.jde.2006.08.013
- Yong-Jung Kim, A geometric one-sided inequality for zero-viscosity limits, preprint, http://amath.kaist.ac.kr/papers/Kim/32.pdf (2014).
- Yong-Jung Kim and Young-Ran Lee, Structure of the fundamental solution of a nonconvex conservation law, to appear in Proc. Roy. Soc. Edinburgh Sect. A–Math., http://amath.kaist.ac.kr/papers/Kim/17.pdf.
- Yong Jung Kim and Robert J. McCann, Potential theory and optimal convergence rates in fast nonlinear diffusion, J. Math. Pures Appl. (9) 86 (2006), no. 1, 42–67 (English, with English and French summaries). MR 2246356 (2007f:35163), DOI https://doi.org/10.1016/j.matpur.2006.01.002
- S. N. Kružkov, First order quasilinear equations with several independent variables, Mat. Sb. (N.S.) 81 (123) (1970), 228–255 (Russian). MR 0267257 (42 \#2159)
- P. D. Lax, Hyperbolic systems of conservation laws. II, Comm. Pure Appl. Math. 10 (1957), 537–566. MR 0093653 (20 \#176)
- Philippe G. Lefloch and Konstantina Trivisa, Continuous Glimm-type functionals and spreading of rarefaction waves, Commun. Math. Sci. 2 (2004), no. 2, 213–236. MR 2119939 (2005i:35174)
- Tai-Ping Liu and Michel Pierre, Source-solutions and asymptotic behavior in conservation laws, J. Differential Equations 51 (1984), no. 3, 419–441. MR 735207 (85i:35094), DOI https://doi.org/10.1016/0022-0396%2884%2990096-2
- Haim Nessyahu and Eitan Tadmor, Nonoscillatory central differencing for hyperbolic conservation laws, J. Comput. Phys. 87 (1990), no. 2, 408–463. MR 1047564 (91i:65157), DOI https://doi.org/10.1016/0021-9991%2890%2990260-8
- O. A. Oleĭnik, Discontinuous solutions of non-linear differential equations, Uspehi Mat. Nauk (N.S.) 12 (1957), no. 3(75), 3–73 (Russian). MR 0094541 (20 \#1055)
- Tao Tang, Zhen-Huan Teng, and Zhouping Xin, Fractional rate of convergence for viscous approximation to nonconvex conservation laws, SIAM J. Math. Anal. 35 (2003), no. 1, 98–122. MR 2001466 (2004i:35221), DOI https://doi.org/10.1137/S0036141001388993
- C. J. van Duijn, L. A. Peletier, and I. S. Pop, A new class of entropy solutions of the Buckley-Leverett equation, SIAM J. Math. Anal. 39 (2007), no. 2, 507–536 (electronic). MR 2338418 (2008g:35136), DOI https://doi.org/10.1137/05064518X
- Burton Wendroff, The Riemann problem for materials with nonconvex equations of state. I. Isentropic flow, J. Math. Anal. Appl. 38 (1972), 454–466. MR 0328387 (48 \#6729)
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Additional Information
Youngsoo Ha
Affiliation:
Department of Mathematical Sciences, Seoul National University, Gwanakro 1, Gwanak-Gu, Seoul 151-747, Republic of Korea
Email:
youngamath.ha@gmail.com
Yong-Jung Kim
Affiliation:
National Institute of Mathematical Sciences, Daejeon 305-811, Republic of Korea, and Department of Mathematical Sciences, KAIST, Daejeon 305-701, Republic of Korea
MR Author ID:
679734
Email:
yongkim@kaist.edu
Received by editor(s):
January 19, 2014
Published electronically:
September 11, 2015
Additional Notes:
This work was supported by the National Institute of Mathematical Sciences and the National Research Foundation of Korea (NRF-2014M1A7A1A03029872).
Article copyright:
© Copyright 2015
Brown University