A note on the stability of the rarefaction wave of the Burgers equation
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This paper is concerned with the asymptotic behavior toward the rarefaction waveu R (x/t) of the solution of the Burgers equation with viscosity. If the initial data are suitably close to constant stateu± atx=±∞, then the solutionu(x, t), roughly speaking, satisfies supR |u −u R | ∼t −1/2 ast → ∞ and, except for the “neighborhoods” of the corners,x=u±t ofu R , sup |u−u R |∼t −1. In the proof the exact forms ofu are available.
- E. Hopf, The partial differential equationu t+uux=μuxx. Comm. Pure Appl. Math.,3 (1950), 201–230. CrossRef
- A.M. Il'in and O.A. Oleinik, Asymptotic behavior of the solutions of the Cauchy problem for certain quasilinear equation for large time (Russian). Mat. Sb.,51 (1960), 191–216.
- S. Kawashima, A. Matsumura and K. Nishihara, Asymptotic behavior of solutions for the equations of a viscous heat-conductive gas. Proc. Japan Acad.,62 (1986), 249–252. CrossRef
- T.P. Liu, Nonlinear stability of shock waves for viscous conservation laws. Mem. Amer. Math. Soc.,329 (1985), 1–108.
- A. Matsumura and K. Nishihara, On the stability of traveling wave solutions of a one-dimensional model system for compressible viscous gas. Japan J. Appl. Math.,2 (1985), 17–25. CrossRef
- A. Matsumura and K. Nishihara, Asymptotics towards the rarefaction waves of the solutions of a one-dimensional model system for compressible viscous gas. Japan J. Appl. Math.,3 (1986), 1–13. CrossRef
- K. Nishihara, A note on the stability of travelling wave solutions of Burgers' equation. Japan J. Appl. Math.,2 (1985), 27–35.
- A note on the stability of the rarefaction wave of the Burgers equation
Japan Journal of Industrial and Applied Mathematics
Volume 8, Issue 1 , pp 85-96
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- rarefaction wave
- Hopf transformation
- asymptotic behavior