Decay rates of strong planar rarefaction waves to scalar conservation laws with degenerate viscosity in several space dimensions
Authors:
Jing Chen and Changjiang Zhu
Journal:
Trans. Amer. Math. Soc. 362 (2010), 17971830
MSC (2000):
Primary 35L65, 35K65, 35B40, 35B45
Published electronically:
October 26, 2009
MathSciNet review:
2574878
Fulltext PDF
Abstract 
References 
Similar Articles 
Additional Information
Abstract: This paper is concerned with the decay rates of the solution to the strong planar rarefaction waves for scalar conservation laws with degenerate viscosity in several space dimensions. The analysis is based on the energy method and the decay property of rarefaction waves.
 1.
D. G. Crighton, Model equation of nonlinear acoustics, Annual Rev. Fluid Mech., 11(1979), 1133.
 2.
D.
G. Crighton and J.
F. Scott, Asymptotic solutions of model equations in nonlinear
acoustics, Philos. Trans. Roy. Soc. London Ser. A 292
(1979), no. 1389, 101–134. MR 547939
(80h:76040), 10.1098/rsta.1979.0046
 3.
Eduard
Harabetian, Rarefactions and large time behavior for parabolic
equations and monotone schemes, Comm. Math. Phys. 114
(1988), no. 4, 527–536. MR 929127
(89d:35084)
 4.
Youichi
Hattori and Kenji
Nishihara, A note on the stability of the rarefaction wave of the
Burgers equation, Japan J. Indust. Appl. Math. 8
(1991), no. 1, 85–96. MR 1093830
(91k:35227), 10.1007/BF03167186
 5.
A.
M. Il′in and O.
A. Oleĭnik, Asymptotic behavior of solutions of the Cauchy
problem for some quasilinear equations for large values of the time,
Mat. Sb. (N.S.) 51 (93) (1960), 191–216 (Russian).
MR
0120469 (22 #11222)
 6.
Kazuo
Ito, Asymptotic decay toward the planar rarefaction waves of
solutions for viscous conservation laws in several space dimensions,
Math. Models Methods Appl. Sci. 6 (1996), no. 3,
315–338. MR 1388709
(97e:35106), 10.1142/S0218202596000109
 7.
Shuichi
Kawashima and Akitaka
Matsumura, Asymptotic stability of traveling wave solutions of
systems for onedimensional gas motion, Comm. Math. Phys.
101 (1985), no. 1, 97–127. MR 814544
(87h:35035)
 8.
Sidney
Leibovich and A.
Richard Seebass (eds.), Nonlinear waves, Cornell University
Press, Ithaca, N.Y.London, 1974. Based on a series of seminars held at
Cornell University, Ithaca, N.Y., 1969. MR 0355283
(50 #7759)
 9.
Akitaka
Matsumura and Kenji
Nishihara, Global stability of the rarefaction wave of a
onedimensional model system for compressible viscous gas, Comm. Math.
Phys. 144 (1992), no. 2, 325–335. MR 1152375
(93d:76056)
 10.
K. Nishihara, Asymptotic behaviors of solutions to viscous conservation laws via the energy method, Lecture notes delivered in Summer School in Fudan University, Shanghai, China, 1999.
 11.
Masataka
Nishikawa and Kenji
Nishihara, Asymptotics toward the planar
rarefaction wave for viscous conservation law in two space
dimensions, Trans. Amer. Math. Soc.
352 (2000), no. 3,
1203–1215. MR 1491872
(2000j:35180), 10.1090/S0002994799022904
 12.
P.
L. Sachdev, Nonlinear diffusive waves, Cambridge University
Press, Cambridge, 1987. MR 898438
(88g:35177)
 13.
J.
F. Scott, The long time asymptotics of solutions to the generalized
Burgers equation, Proc. Roy. Soc. London Ser. A 373
(1981), no. 1755, 443–456. MR 603065
(82b:76052), 10.1098/rspa.1981.0003
 14.
Jinghua
Wang and Hui
Zhang, Existence and decay rates of smooth solutions for a
nonuniformly parabolic equation, Proc. Roy. Soc. Edinburgh Sect. A
132 (2002), no. 6, 1477–1491. MR 1950818
(2003j:35192)
 15.
Zhou
Ping Xin, Asymptotic stability of planar
rarefaction waves for viscous conservation laws in several
dimensions, Trans. Amer. Math. Soc.
319 (1990), no. 2,
805–820. MR
970270 (90j:35138), 10.1090/S00029947199009702708
 16.
Yanling
Xu and Mina
Jiang, Asymptotic stability of rarefaction wave for generalized
Burgers equation, Acta Math. Sci. Ser. B Engl. Ed. 25
(2005), no. 1, 119–129. MR 2119344
(2005j:35202)
 17.
Hui
Zhang, Existence of weak solutions for a degenerate generalized
Burgers equation with large initial data, Acta Math. Sci. Ser. B Engl.
Ed. 22 (2002), no. 2, 241–248. MR 1901484
(2003f:35179)
 18.
Huijiang
Zhao, Nonlinear stability of strong planar rarefaction waves for
the relaxation approximation of conservation laws in several space
dimensions, J. Differential Equations 163 (2000),
no. 1, 198–222. MR 1755074
(2001c:35148), 10.1006/jdeq.1999.3722
 19.
Changjiang
Zhu, Asymptotic behavior of solutions for 𝑝system with
relaxation, J. Differential Equations 180 (2002),
no. 2, 273–306. MR 1894014
(2003d:35174), 10.1006/jdeq.2001.4063
 20.
Changjiang
Zhu and Zhian
Wang, Decay rates of solutions to dissipative nonlinear evolution
equations with ellipticity, Z. Angew. Math. Phys. 55
(2004), no. 6, 994–1014. MR 2100527
(2005i:35114), 10.1007/s0003300431179
 1.
 D. G. Crighton, Model equation of nonlinear acoustics, Annual Rev. Fluid Mech., 11(1979), 1133.
 2.
 D. G. Crighton and J. F. Scott, Asymptotic solutions of model equations in nonlinear acoustics, Philos. Trans. R. Soc. London, 292(A)(1979), 101134. MR 547939 (80h:76040)
 3.
 E. Harabetian, Rarefactions and large time behavior for parabolic equations and monotone schemes, Comm. Math. Phys., 114(1988), 527536. MR 929127 (89d:35084)
 4.
 Y. Hattori and K. Nishihara, A note on the stability of the rarefaction wave of the Burgers equation, Japan J. Indust. Appl. Math., 8(1991), 8586. MR 1093830 (91k:35227)
 5.
 A. M. Il'in and O. A. Oleinik, Asymptotic behavior of solutions of the Cauchy problem for certain quasilinear equations for large time (Russian), Mat. Sb., 51(1960), 191216. MR 0120469 (22:11222)
 6.
 K. Ito, Asymptotic decay toward the planar rarefaction waves of solutions for viscous conservation laws in several dimensions, Math. Models Methods Appl. Sci., 6(1996), 315338. MR 1388709 (97e:35106)
 7.
 S. Kawashima and A. Matsumura, Asymptotic stability of travelling wave solutions of systems for onedimensional gas motion, Comm. Math. Phys., 101(1985), 97127. MR 814544 (87h:35035)
 8.
 S. Leibovich and A. R. Seebass, eds., Nonlinear Waves, Cornell University Press, Ithaca, NY, 1974. MR 0355283 (50:7759)
 9.
 A. Matsumara and K. Nishihara, Global stability of the rarefaction wave of a onedimensional model system for compressible viscious gas, Comm. Math. Phys., 144(1992), 325335. MR 1152375 (93d:76056)
 10.
 K. Nishihara, Asymptotic behaviors of solutions to viscous conservation laws via the energy method, Lecture notes delivered in Summer School in Fudan University, Shanghai, China, 1999.
 11.
 M. Nishikawa and K. Nishihara, Asymptotics toward the planar rarefaction wave for viscous conservation law in two space dimensions, Trans. Amer. Math. Soc., 352(2000), 12031215 MR 1491872 (2000j:35180)
 12.
 P. L. Sachdev, Nonlinear Diffusive Waves, Cambridge University Press, Cambridge, 1987. MR 898438 (88g:35177)
 13.
 J. F. Scott, The long time asymptotics of solutions to the generalized Burgers equation, Proc. Roy. Soc. London, Ser. A, 373(1981), 443456. MR 603065 (82b:76052)
 14.
 J. H. Wang and H. Zhang, Existence and decay rates of smooth solutions for a nonuniformly parabolic equation, Proc. Roy. Soc. Edinburgh, Sect. A, 132(2002), 14771491. MR 1950818 (2003j:35192)
 15.
 Z. P. Xin, Asymptotic stability of planar rarefaction waves for viscous conservation laws in several dimensions, Trans. Amer. Math. Soc., 319(1990), 805829. MR 970270 (90j:35138)
 16.
 Y. L. Xu and M. N. Jiang, Asymptotic stability of rarefaction wave for generalized Burgers equation, Acta Math. Sci., 25(B)(2005), 119129. MR 2119344 (2005j:35202)
 17.
 H. Zhang, Existence of weak solutions for a degenerate generalized Burgers equation with large initial data, Acta. Math. Sci, 22(B)(2002), 241248. MR 1901484 (2003f:35179)
 18.
 H. J. Zhao, Nonlinear stability of strong planar rarefaction waves for the relaxation approximation of conservation laws in several space dimensions, J. Differential Equations, 163(2000), 198222. MR 1755074 (2001c:35148)
 19.
 C. J. Zhu, Asymptotic behavior of solutions for system with relaxation, J. Differential Equations, 180(2002), 273306. MR 1894014 (2003d:35174)
 20.
 C. J. Zhu and Z. A. Wang, Decay rates of solutions to dissipative nonlinear evolution equations with ellipticity, Z. Angew. Math. Phys., 55(2004), 121. MR 2100527 (2005i:35114)
Similar Articles
Retrieve articles in Transactions of the American Mathematical Society
with MSC (2000):
35L65,
35K65,
35B40,
35B45
Retrieve articles in all journals
with MSC (2000):
35L65,
35K65,
35B40,
35B45
Additional Information
Jing Chen
Affiliation:
Department of Mathematics, Laboratory of Nonlinear Analysis, Central China Normal University, Wuhan 430079, People’s Republic of China
Changjiang Zhu
Affiliation:
Department of Mathematics, Laboratory of Nonlinear Analysis, Central China Normal University, Wuhan 430079, People’s Republic of China
Email:
cjzhu@mail.ccnu.edu.cn
DOI:
http://dx.doi.org/10.1090/S0002994709046340
Keywords:
Strong planar rarefaction waves,
energy method,
{\it a priori} estimates,
decay rates.
Received by editor(s):
July 28, 2005
Received by editor(s) in revised form:
August 1, 2007
Published electronically:
October 26, 2009
Article copyright:
© Copyright 2009
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
