An $L^p$ shock admissibility condition for conservation laws
Author:
Hiroki Ohwa
Journal:
Quart. Appl. Math. 80 (2022), 259-275
MSC (2020):
Primary 35L65, 35L67
DOI:
https://doi.org/10.1090/qam/1610
Published electronically:
February 1, 2022
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Abstract: We estimate the $L^p$ ($p>0$) local distance between piecewise constant solutions to the Cauchy problem of conservation laws and propose a shock admissibility condition for having an $L^p$ local contraction of such solutions. Moreover, as an application, we prove that there exist $L^p$ locally contractive solutions on some set of initial functions, to the Cauchy problem of conservation laws with convex or concave flux functions. As a result, for conservation laws with convex or concave flux functions, we see that rarefaction waves have an $L^q$ ($q\geq 1$) local contraction and shock waves have an $L^r$ ($0<r\leq 1$) local contraction.
References
- Alberto Bressan, Hyperbolic systems of conservation laws, Oxford Lecture Series in Mathematics and its Applications, vol. 20, Oxford University Press, Oxford, 2000. The one-dimensional Cauchy problem. MR 1816648
- Alberto Bressan, Tai-Ping Liu, and Tong Yang, $L^1$ stability estimates for $n\times n$ conservation laws, Arch. Ration. Mech. Anal. 149 (1999), no. 1, 1–22. MR 1723032, DOI 10.1007/s002050050165
- Constantine M. Dafermos, Polygonal approximations of solutions of the initial value problem for a conservation law, J. Math. Anal. Appl. 38 (1972), 33–41. MR 303068, DOI 10.1016/0022-247X(72)90114-X
- K. Hinohara, N. Minagawa, H. Ohwa, H. Suzuki and S. Ukita, $L^p$ contractive solutions for scalar conservation laws, submitted.
- Helge Holden and Nils Henrik Risebro, Front tracking for hyperbolic conservation laws, 2nd ed., Applied Mathematical Sciences, vol. 152, Springer, Heidelberg, 2015. MR 3443431, DOI 10.1007/978-3-662-47507-2
- S. N. Kružkov, First order quasilinear equations with several space variables, Math. USSR Sb., 10 (1970), 537–566.
- Tai-Ping Liu and Tong Yang, A new entropy functional for a scalar conservation law, Comm. Pure Appl. Math. 52 (1999), no. 11, 1427–1442. MR 1702712, DOI 10.1002/(SICI)1097-0312(199911)52:11<1427::AID-CPA2>3.0.CO;2-R
- Tai-Ping Liu and Tong Yang, $L_1$ stability for $2\times 2$ systems of hyperbolic conservation laws, J. Amer. Math. Soc. 12 (1999), no. 3, 729–774. MR 1646841, DOI 10.1090/S0894-0347-99-00292-1
- Hiroki Ohwa and Yoshimasa Sasaki, Stability of approximate solutions constructed by the wave front tracking method, J. Math. Anal. Appl. 491 (2020), no. 2, 124385, 18. MR 4123251, DOI 10.1016/j.jmaa.2020.124385
- O. A. Oleĭnik, Uniqueness and stability of the generalized solution of the Cauchy problem for a quasi-linear equation, Uspehi Mat. Nauk 14 (1959), no. 2 (86), 165–170 (Russian). MR 0117408
References
- Alberto Bressan, Hyperbolic systems of conservation laws, Oxford Lecture Series in Mathematics and its Applications, vol. 20, Oxford University Press, Oxford, 2000. The one-dimensional Cauchy problem. MR 1816648
- Alberto Bressan, Tai-Ping Liu, and Tong Yang, $L^1$ stability estimates for $n\times n$ conservation laws, Arch. Ration. Mech. Anal. 149 (1999), no. 1, 1–22. MR 1723032, DOI 10.1007/s002050050165
- Constantine M. Dafermos, Polygonal approximations of solutions of the initial value problem for a conservation law, J. Math. Anal. Appl. 38 (1972), 33–41. MR 303068, DOI 10.1016/0022-247X(72)90114-X
- K. Hinohara, N. Minagawa, H. Ohwa, H. Suzuki and S. Ukita, $L^p$ contractive solutions for scalar conservation laws, submitted.
- Helge Holden and Nils Henrik Risebro, Front tracking for hyperbolic conservation laws, 2nd ed., Applied Mathematical Sciences, vol. 152, Springer, Heidelberg, 2015. MR 3443431, DOI 10.1007/978-3-662-47507-2
- S. N. Kružkov, First order quasilinear equations with several space variables, Math. USSR Sb., 10 (1970), 537–566.
- Tai-Ping Liu and Tong Yang, A new entropy functional for a scalar conservation law, Comm. Pure Appl. Math. 52 (1999), no. 11, 1427–1442. MR 1702712, DOI 10.1002/(SICI)1097-0312(199911)52:11<1427::AID-CPA2>3.0.CO;2-R
- Tai-Ping Liu and Tong Yang, $L_1$ stability for $2\times 2$ systems of hyperbolic conservation laws, J. Amer. Math. Soc. 12 (1999), no. 3, 729–774. MR 1646841, DOI 10.1090/S0894-0347-99-00292-1
- Hiroki Ohwa and Yoshimasa Sasaki, Stability of approximate solutions constructed by the wave front tracking method, J. Math. Anal. Appl. 491 (2020), no. 2, 124385, 18. MR 4123251, DOI 10.1016/j.jmaa.2020.124385
- O. A. Oleĭnik, Uniqueness and stability of the generalized solution of the Cauchy problem for a quasi-linear equation, Uspehi Mat. Nauk 14 (1959), no. 2 (86), 165–170 (Russian). MR 0117408
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Additional Information
Hiroki Ohwa
Affiliation:
Department of Mathematics, Faculty of Science, Niigata University, 8050 Ikarashi 2-no-cho, Nishi-ku, Niigata, 950-2181, Japan
MR Author ID:
879912
Email:
hiroohwa@math.sc.niigata-u.ac.jp
Keywords:
Conservation laws,
Cauchy problem,
$L^p$ locally contractive semigroup,
stability
Received by editor(s):
October 19, 2021
Received by editor(s) in revised form:
December 14, 2021
Published electronically:
February 1, 2022
Article copyright:
© Copyright 2022
Brown University