Remarks on the stability of shock profiles for conservation laws with dissipation

Author:
Robert L. Pego

Journal:
Trans. Amer. Math. Soc. **291** (1985), 353-361

MSC:
Primary 35Q20; Secondary 35L67

DOI:
https://doi.org/10.1090/S0002-9947-1985-0797065-0

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 KdV-Burgers 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. Bona and M. Schonbek,*Traveling-wave solutions of the Korteweg-deVries-Burgers' 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), 555-604. MR**0385355 (52:6219)****[4]**A. Friedman,*Partial differential equations of parabolic type*, Prentice-Hall, 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), 201-230. 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), 191-216. (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), 133-141. 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 heat-conductive fluids*, Proc. Japan Acad. Ser. A. Math. Sci.**55**(1979), 337-342. 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. 418-422. MR**0367423 (51:3665)****[12]**D. H. Sattinger,*On the stability of waves of nonlinear parabolic systems*, Adv. in Math.**22**(1976), 312-355. 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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DOI:
https://doi.org/10.1090/S0002-9947-1985-0797065-0

Article copyright:
© Copyright 1985
American Mathematical Society