Large data local well-posedness for a class of KdV-type equations

Harrop-Griffiths, Benjamin

doi:10.1090/S0002-9947-2014-05882-0

Large data local well-posedness for a class of KdV-type equations
HTML articles powered by AMS MathViewer

by Benjamin Harrop-Griffiths PDF

Trans. Amer. Math. Soc. 367 (2015), 755-773 Request permission

Abstract:

In this article we consider the Cauchy problem with large initial data for an equation of the form \[ (\partial _t+\partial _x^3)u=F(u,u_x,u_{xx})\]where $F$ is a polynomial with no constant or linear terms. Local well-posedness was established in weighted Sobolev spaces by Kenig-Ponce-Vega. In this paper we prove local well-posedness in a translation invariant subspace of $H^s$ by adapting the result of Marzuola-Metcalfe-Tataru on quasilinear Schrödinger equations.

References

Timur Akhunov, Local well-posedness of quasi-linear systems generalizing KdV, Commun. Pure Appl. Anal. 12 (2013), no. 2, 899–921. MR 2982797, DOI 10.3934/cpaa.2013.12.899
Ioan Bejenaru and Daniel Tataru, Large data local solutions for the derivative NLS equation, J. Eur. Math. Soc. (JEMS) 10 (2008), no. 4, 957–985. MR 2443925, DOI 10.4171/JEMS/136
Hongsheng Cai, Dispersive smoothing effects for KdV type equations, J. Differential Equations 136 (1997), no. 2, 191–221. MR 1448822, DOI 10.1006/jdeq.1996.3245
Carlos E. Kenig, Gustavo Ponce, and Luis Vega, On the hierarchy of the generalized KdV equations, Singular limits of dispersive waves (Lyon, 1991) NATO Adv. Sci. Inst. Ser. B: Phys., vol. 320, Plenum, New York, 1994, pp. 347–356. MR 1321214
Carlos E. Kenig, Gustavo Ponce, and Luis Vega, Higher-order nonlinear dispersive equations, Proc. Amer. Math. Soc. 122 (1994), no. 1, 157–166. MR 1195480, DOI 10.1090/S0002-9939-1994-1195480-8
Carlos E. Kenig, Gustavo Ponce, and Luis Vega, Smoothing effects and local existence theory for the generalized nonlinear Schrödinger equations, Invent. Math. 134 (1998), no. 3, 489–545. MR 1660933, DOI 10.1007/s002220050272
Carlos E. Kenig and Gigliola Staffilani, Local well-posedness for higher order nonlinear dispersive systems, J. Fourier Anal. Appl. 3 (1997), no. 4, 417–433. MR 1468372, DOI 10.1007/BF02649104
Felipe Linares and Gustavo Ponce, Introduction to nonlinear dispersive equations, Universitext, Springer, New York, 2009. MR 2492151
Jeremy L. Marzuola, Jason Metcalfe, and Daniel Tataru, Quasilinear Schrödinger equations I: Small data and quadratic interactions, Adv. Math. 231 (2012), no. 2, 1151–1172. MR 2955206, DOI 10.1016/j.aim.2012.06.010
Sigeru Mizohata, On the Cauchy problem, Notes and Reports in Mathematics in Science and Engineering, vol. 3, Academic Press, Inc., Orlando, FL; Science Press Beijing, Beijing, 1985. MR 860041
Didier Pilod, On the Cauchy problem for higher-order nonlinear dispersive equations, J. Differential Equations 245 (2008), no. 8, 2055–2077. MR 2446185, DOI 10.1016/j.jde.2008.07.017
Shigeo Tarama, On the wellposed Cauchy problem for some dispersive equations, J. Math. Soc. Japan 47 (1995), no. 1, 143–158. MR 1304193, DOI 10.2969/jmsj/04710143
Michael E. Taylor, Partial differential equations III. Nonlinear equations, 2nd ed., Applied Mathematical Sciences, vol. 117, Springer, New York, 2011. MR 2744149, DOI 10.1007/978-1-4419-7049-7