Osaka Journal of Mathematics

Large time behavior of solutions to the generalized Burgers equations

Masakazu Kato

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Abstract

We study large time behavior of the solutions to the initial value problem for the generalized Burgers equation. It is known that the solution tends to a self-similar solution to the Burgers equation at the rate $t^{-1} \log t$ in $L^{\infty}$ as $t \to \infty$. The aim of this paper is to show that the rate is optimal under suitable assumptions and to obtain the second asymptotic profile of large time behavior of the solutions.

Article information

Source
Osaka J. Math., Volume 44, Number 4 (2007), 923-943.

Dates
First available in Project Euclid: 7 January 2008

Permanent link to this document
https://projecteuclid.org/euclid.ojm/1199719413

Mathematical Reviews number (MathSciNet)
MR2383818

Zentralblatt MATH identifier
1132.35311

Subjects
Primary: 35B40: Asymptotic behavior of solutions
Secondary: 35L65: Conservation laws

Citation

Kato, Masakazu. Large time behavior of solutions to the generalized Burgers equations. Osaka J. Math. 44 (2007), no. 4, 923--943. https://projecteuclid.org/euclid.ojm/1199719413


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References

  • J.D. Cole: On a quasi-linear parabolic equation occurring in aerodynamics, Quart. Appl. Math. IX (1951), 225--236.
  • N. Hayashi, E.I. Kaikina and P.I. Naumkin: Large time asymptotics for the BBM-Burgers equation, to appear in Ann. Inst. H. Poincaré.
  • N. Hayashi and P.I. Naumkin: Asymptotics for the Korteweg-de Vries-Burgers equation, to appear in Acta Math. Sin. (Engl. Ser.).
  • E. Hopf: The partial differential equation $u_t+uu_x=\mu u_xx$, Comm. Pure Appl. Math. 3 (1950), 201--230.
  • E.I. Kaikina and H.F. Ruiz-Paredes: Second term of asymptotics for KdVB equation with large initial data, Osaka J. Math. 42 (2005), 407--420.
  • S. Kawashima: The asymptotic equivalence of the Broadwell model equation and its Navier-Stokes model equation, Japan. J. Math. (N.S.) 7 (1981), 1--43.
  • S. Kawashima: Large-time behaviour of solutions to hyperbolic-parabolic systems of conservation laws and applications, Proc. Roy. Soc. Edinburgh Sect. A 106 (1987), 169--194.
  • T.-P. Liu: Hyperbolic and Viscous Conservation Laws, CBMS-NSF Regional Conference Series in Applied Mathematics 72, SIAM, Philadelphia, PA, 2000.
  • A. Matsumura and K. Nishihara: Global Solutions of Nonlinear Differential Equa-tions---Mathematical Analysis for Compressible Viscous Fluids, Nippon-Hyoron-Sha, Tokyo, 2004, (in Japanese).
  • T. Nishida: Equations of motion of compressible viscous fluids; in Patterns and Waves (ed. T. Nishida, M. Mimura, H. Fujii), Kinokuniya, Tokyo, North-Holland, Amsterdam, 1986, 97\nobreakdash--128.