Remarks on the stability of shock profiles for conservation laws with dissipation
Author:
Robert L. Pego
Journal:
Trans. Amer. Math. Soc. 291 (1985), 353361
MSC:
Primary 35Q20; Secondary 35L67
MathSciNet review:
797065
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Abstract: Two remarks are made. The first is to establish the stability of monotone shock profiles of the KdVBurgers equation, based on an energy method of Goodman. The second remark illustrates, specifically in Burgers' equation, that uniform rates of decay are not to be expected for perturbations of shock profiles in typical norms.
 [1]
J.
L. Bona and M.
E. Schonbek, Travellingwave solutions to the Kortewegde
VriesBurgers equation, Proc. Roy. Soc. Edinburgh Sect. A
101 (1985), no. 34, 207–226. MR 824222
(87k:35208), http://dx.doi.org/10.1017/S0308210500020783
 [2]
J. L. Bona and M. Schonbek, Model equations for long waves with initial data corresponding to bore propagation (in preparation).
 [3]
J.
L. Bona and R.
Smith, The initialvalue problem for the Kortewegde Vries
equation, Philos. Trans. Roy. Soc. London Ser. A 278
(1975), no. 1287, 555–601. MR 0385355
(52 #6219)
 [4]
Avner
Friedman, Partial differential equations of parabolic type,
PrenticeHall, Inc., Englewood Cliffs, N.J., 1964. MR 0181836
(31 #6062)
 [5]
Jonathan
Goodman, Nonlinear asymptotic stability of viscous shock profiles
for conservation laws, Arch. Rational Mech. Anal. 95
(1986), no. 4, 325–344. MR 853782
(88b:35127), http://dx.doi.org/10.1007/BF00276840
 [6]
Eberhard
Hopf, The partial differential equation
𝑢_{𝑡}+𝑢𝑢ₓ=𝜇𝑢ₓₓ,
Comm. Pure Appl. Math. 3 (1950), 201–230. MR 0047234
(13,846c)
 [7]
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)
 [8]
S.
Klainerman and Gustavo
Ponce, Global, small amplitude solutions to nonlinear evolution
equations, Comm. Pure Appl. Math. 36 (1983),
no. 1, 133–141. MR 680085
(84a:35173), http://dx.doi.org/10.1002/cpa.3160360106
 [9]
Andrew
Majda and Robert
L. Pego, Stable viscosity matrices for systems of conservation
laws, J. Differential Equations 56 (1985),
no. 2, 229–262. MR 774165
(86b:35132), http://dx.doi.org/10.1016/00220396(85)90107X
 [10]
Akitaka
Matsumura and Takaaki
Nishida, The initial value problem for the equations of motion of
compressible viscous and heatconductive fluids, Proc. Japan Acad.
Ser. A Math. Sci. 55 (1979), no. 9, 337–342. MR 555060
(83e:35112)
 [11]
L.
A. Peletier, Asymptotic stability of travelling waves,
Instability of continuous systems (IUTAM Sympos., Herrenalb, 1969),
Springer, Berlin, 1971, pp. 418–422. MR 0367423
(51 #3665)
 [12]
D.
H. Sattinger, On the stability of waves of nonlinear parabolic
systems, Advances in Math. 22 (1976), no. 3,
312–355. MR 0435602
(55 #8561)
 [13]
Robert
L. Pego, Stability in systems of conservation laws with
dissipation, Nonlinear systems of partial differential equations in
applied mathematics, Part 1 (Santa Fe, N.M., 1984) Lectures in Appl.
Math., vol. 23, Amer. Math. Soc., Providence, RI, 1986,
pp. 345–357. MR 837685
(88a:35154)
 [1]
 J. Bona and M. Schonbek, Travelingwave solutions of the KortewegdeVriesBurgers' equation, Proc. Roy. Soc. Edinburgh A (to appear). MR 824222 (87k:35208)
 [2]
 J. L. Bona and M. Schonbek, Model equations for long waves with initial data corresponding to bore propagation (in preparation).
 [3]
 J. L. Bona and R. Smith, The initial value problem for the equation, Philos. Trans. Roy. Soc. London Ser. A 278 (1975), 555604. MR 0385355 (52:6219)
 [4]
 A. Friedman, Partial differential equations of parabolic type, PrenticeHall, Englewood Cliffs, N.J., 1964. MR 0181836 (31:6062)
 [5]
 J. Goodman, Nonlinear asymptotic stability of viscous shock profiles for conservation laws, Courant Institute, preprint. MR 853782 (88b:35127)
 [6]
 E. Hopf, The partial differential equation , Comm. Pure Appl. Math. 3 (1950), 201230. MR 0047234 (13:846c)
 [7]
 A. M. Il'in and O. A. Oleinik, Asymptotic behavior of solutions of the Cauchy problem for some quasilinear equations for large time, Mat. Sb. 56 (93) (1960), 191216. (Russian) MR 0120469 (22:11222)
 [8]
 S. Klainerman and G. Ponce, Global small amplitude solutions to nonlinear evolution equations, Comm. Pure Appl. Math. 36 (1983), 133141. MR 680085 (84a:35173)
 [9]
 A. Majda and R. Pego, Stable viscosity matrices for systems of conservation laws, J. Differential Equations (to appear). MR 774165 (86b:35132)
 [10]
 A. Matsumura and T. Nishida, The initial value problem for the equations of motion of compressible viscous and heatconductive fluids, Proc. Japan Acad. Ser. A. Math. Sci. 55 (1979), 337342. MR 555060 (83e:35112)
 [11]
 L. A. Peletier, Asymptotic stability of traveling waves, Instability of Continuous Systems (H. Leipholz, ed.), Springer, New York, 1971, pp. 418422. MR 0367423 (51:3665)
 [12]
 D. H. Sattinger, On the stability of waves of nonlinear parabolic systems, Adv. in Math. 22 (1976), 312355. MR 0435602 (55:8561)
 [13]
 R. L. Pego, Stability in systems of conservation laws with dissipation (Seminar on Nonlinear PDE, Santa Fe, 1984), Lectures in Appl. Math. (to appear). MR 837685 (88a:35154)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947198507970650
PII:
S 00029947(1985)07970650
Article copyright:
© Copyright 1985
American Mathematical Society
