Nagoya Mathematical Journal

Global well-posedness for a system of KdV-type equations with coupled quadratic nonlinearities

Jerry L. Bona, Jonathan Cohen, and Gang Wang

Full-text: Open access

Abstract

In this paper, coupled systems ut+uxxx+P(u,v)x=0, vt+vxxx+Q(u,v)x=0 of Korteweg–de Vries type are considered, where u=u(x,t), v=v(x,t) are real-valued functions and where x,tR. Here, subscripts connote partial differentiation and P(u,v)=Au2+Buv+Cv2andQ(u,v)=Du2+Euv+Fv2 are quadratic polynomials in the variables u and v. Attention is given to the pure initial-value problem in which u(x,t) and v(x,t) are both specified at t=0, namely, u(x,0)=u0(x)andv(x,0)=v0(x), for xR. Under suitable conditions on P and Q, global well-posedness of this problem is established for initial data in the L2-based Sobolev spaces Hs(R)×Hs(R) for any s>3/4.

Article information

Source
Nagoya Math. J., Volume 215 (2014), 67-149.

Dates
First available in Project Euclid: 9 June 2014

Permanent link to this document
https://projecteuclid.org/euclid.nmj/1402319947

Digital Object Identifier
doi:10.1215/00277630-2691901

Mathematical Reviews number (MathSciNet)
MR3263526

Zentralblatt MATH identifier
1372.35271

Subjects
Primary: 35Q35: PDEs in connection with fluid mechanics 35Q51: Soliton-like equations [See also 37K40] 35Q53: KdV-like equations (Korteweg-de Vries) [See also 37K10]
Secondary: 42B35: Function spaces arising in harmonic analysis 42B37: Harmonic analysis and PDE [See also 35-XX] 76B15: Water waves, gravity waves; dispersion and scattering, nonlinear interaction [See also 35Q30] 76B25: Solitary waves [See also 35C11] 86A05: Hydrology, hydrography, oceanography [See also 76Bxx, 76E20, 76Q05, 76Rxx, 76U05] 86A10: Meteorology and atmospheric physics [See also 76Bxx, 76E20, 76N15, 76Q05, 76Rxx, 76U05]

Citation

Bona, Jerry L.; Cohen, Jonathan; Wang, Gang. Global well-posedness for a system of KdV-type equations with coupled quadratic nonlinearities. Nagoya Math. J. 215 (2014), 67--149. doi:10.1215/00277630-2691901. https://projecteuclid.org/euclid.nmj/1402319947


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