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THE INVISCID LIMIT OF THE MODIFIED BENJAMIN–ONO–BURGERS EQUATION

Part of: Equations of mathematical physics and other areas of application

Published online by Cambridge University Press:  06 December 2010

HUA ZHANG  and
YUQIN KE
HUA ZHANG*
Affiliation:
College of Sciences, North China University of Technology, Beijing 100144, PR China (email: zhanghuamaths@163.com)
YUQIN KE
Affiliation:
Faculty of Economics, Guangdong University of Business Studies, Guangzhou 510320, PR China (email: keyuqin@gmail.com)
*
✉For correspondence; e-mail: zhanghuamaths@163.com
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Abstract

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We prove that the modified Benjamin–Ono–Burgers equation is globally well-posed in Hs for s>0. Moreover, we show that the solution of the modified Benjamin–Ono–Burgers equation converges to that of the modified Benjamin–Ono equation in the natural space C([0,T];Hs), s≥1/2, as the dissipative coefficient ϵ goes to zero, provided that the L2 norm of the initial data is sufficiently small.

Keywords

modified Benjamin–Ono–Burgers equationglobal well-posednessinviscid limit

MSC classification

Secondary: 35Q53: KdV-like equations (Korteweg-de Vries)
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 83 , Issue 2 , April 2011 , pp. 301 - 320
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2010

Footnotes

Zhang was partially supported by the Science Research Startup Foundation of North China University of Technology.

References

[1]Bourgain, J., ‘Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations I, II’, Geom. Funct. Anal. 3 (1993), 107–156, 209–262.CrossRefGoogle Scholar
[2]Guo, Z., ‘Local well-posedness and a priori bounds for the modified Benjamin–Ono equation without using a gauge transformation’, arXiv:0807.3764.Google Scholar
[3]Guo, Z., ‘Local well-posedness for dispersion generalized Benjamin–Ono equations in Sobolev spaces’, arXiv:0812.1825v2.Google Scholar
[4]Guo, Z. and Wang, B., ‘Global well-posedness and limit behaviour for the modified finite-depth-fluid equation’, arXiv:0809.2318v1.Google Scholar
[5]Guo, Z. and Wang, B., ‘Global well-posedness and inviscid limit for the Korteweg–de Vries–Burgers equation’, J. Differential Equations 246 (2009), 3864–3901.CrossRefGoogle Scholar
[6]Ionescu, A. D. and Kenig, C. E., ‘Global well-posedness of the Benjamin–Ono equation in low-regularity spaces’, J. Amer. Math. Soc. 20(3) (2007), 753–798.CrossRefGoogle Scholar
[7]Kenig, C., Ponce, G. and Vega, L., ‘Well-posedness and scattering results for the generalized Korteweg–de Vries equation via the contraction principle’, Comm. Pure Appl. Math. 46(4) (1993), 527–620.CrossRefGoogle Scholar
[8]Kenig, C. and Takaoka, H., ‘Global wellposedness of the modified Benjamin–Ono equation with initial data in H 1/2’, Int. Math. Res. Not. (1) (2006), 1–44.CrossRefGoogle Scholar
[9]Molinet, L. and Ribaud, F., ‘On the low regularity of the Korteweg–de Vries–Burgers equation’, Int. Math. Res. Not. 37 (2002), 1979–2005.CrossRefGoogle Scholar
[10]Molinet, L. and Ribaud, F., ‘Well-posedness results for the generalized Benjamin–Ono equation with arbitrary large initial data’, Int. Math. Res. Not. (70) (2004), 3757–3795.Google Scholar
[11]Tao, T., Nonlinear Dispersive Equations, CBMS Regional Conference Series in Mathematics, 106 (Conference Board of the Mathematical Sciences, Washington, DC, 2006), pp. xvi + 373.CrossRefGoogle Scholar
[12]Tao, T., ‘Multilinear weighted convolution of L 2 functions, and applications to nonlinear dispersive equations’, Amer. J. Math. 123 (2001), 839–908.CrossRefGoogle Scholar
[13]Tao, T., ‘Global well-posedness of the Benjamin–Ono equation in H 1(R)’, J. Hyperbolic Differ. Equ. 1 (2004), 27–49.CrossRefGoogle Scholar
[14]Vento, S., ‘Well-posedness and ill-posedness results for dissipative Benjamin–Ono equations’, arXiv:0802.1039v1.Google Scholar
[15]Wang, B., ‘The limit behaviour of solutions for the Cauchy problem of the complex Ginzburg–Landau equation’, Comm. Pure Appl. Math. 53 (2002), 0481–0508.CrossRefGoogle Scholar
[16]Wang, B. and Wang, Y., ‘The inviscid limit of the derivative complex Ginzburg–Landau equation’, J. Math. Pures Appl. 83 (2004), 477–502.Google Scholar