Sharp local well-posedness of KdV type equations with dissipative perturbations

Carvajal, Xavier; Panthee, Mahendra

doi:10.1090/qam/1437

Authors: Xavier Carvajal and Mahendra Panthee
Journal: Quart. Appl. Math. 74 (2016), 571-594
MSC (2010): Primary 35A01, 35Q53
DOI: https://doi.org/10.1090/qam/1437
Published electronically: June 16, 2016
MathSciNet review: 3518228
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Abstract: In this work, we study the initial value problems associated to some linear perturbations of the KdV equation. Our focus is on the well-posedness issues for initial data given in the $L^2$-based Sobolev spaces. We derive a bilinear estimate in a space with weight in the time variable and obtain sharp local well-posedness results.

References

Borys Alvarez Samaniego, The Cauchy problem for a nonlocal perturbation of the KdV equation, Differential Integral Equations 16 (2003), no. 10, 1249–1280. MR 2014809
D. J. Benney, Long waves on liquid films, J. Math. and Phys. 45 (1966), 150–155. MR 201125
H. A. Biagioni, J. L. Bona, R. J. Iório Jr., and M. Scialom, On the Korteweg-de Vries-Kuramoto-Sivashinsky equation, Adv. Differential Equations 1 (1996), no. 1, 1–20. MR 1357952
H. A. Biagioni and F. Linares, On the Benney-Lin and Kawahara equations, J. Math. Anal. Appl. 211 (1997), no. 1, 131–152. MR 1460163, DOI 10.1006/jmaa.1997.5438
J. Bourgain, Periodic Korteweg de Vries equation with measures as initial data, Selecta Math. (N.S.) 3 (1997), no. 2, 115–159. MR 1466164, DOI 10.1007/s000290050008
J. Bourgain, Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. II. The KdV-equation, Geom. Funct. Anal. 3 (1993), no. 3, 209–262. MR 1215780, DOI 10.1007/BF01895688
X. Carvajal and M. Panthee, A note on local well-posedness of generalized KdV type equations with dissipative perturbations, arXiv:1305.0511 (2013).
Xavier Carvajal and Mahendra Panthee, Well-posedness of KdV type equations, Electron. J. Differential Equations (2012), No. 40, 15. MR 2900365
Xavier Carvajal and Mahendra Panthee, Well-posedness for some perturbations of the KdV equation with low regularity data, Electron. J. Differential Equations (2008), No. 02, 18. MR 2368889
Xavier Carvajal and Marcia Scialom, On the well-posedness for the generalized Ostrovsky, Stepanyams and Tsimring equation, Nonlinear Anal. 62 (2005), no. 7, 1277–1287. MR 2154109, DOI 10.1016/j.na.2005.04.032
Xavier Carvajal Paredes and Ricardo A. Pastran, Well-posedness for a family of perturbations of the KdV equation in periodic Sobolev spaces of negative order, Commun. Contemp. Math. 15 (2013), no. 6, 1350005, 26. MR 3139405, DOI 10.1142/S0219199713500053
T. Cazenave, An introduction to nonlinear Schrödinger equations, Textos de Metodos Matemáticos 22 (Rio de Janeiro), 1989.
B. I. Cohen, J. A. Krommes, W. M. Tang, and M. N. Rosenbluth, Non-linear saturation of the dissipative trapped-ion mode by mode coupling, Nuclear Fusion 16 9 (1976), 971–992.
Daniel B. Dix, Nonuniqueness and uniqueness in the initial-value problem for Burgers’ equation, SIAM J. Math. Anal. 27 (1996), no. 3, 708–724. MR 1382829, DOI 10.1137/0527038
Amin Esfahani, Sharp well-posedness of the Ostrovsky, Stepanyams and Tsimring equation, Math. Commun. 18 (2013), no. 2, 323–335. MR 3138434
Axel Grünrock, A bilinear Airy-estimate with application to gKdV-3, Differential Integral Equations 18 (2005), no. 12, 1333–1339. MR 2174975
Carlos E. Kenig, Gustavo Ponce, and Luis Vega, On the ill-posedness of some canonical dispersive equations, Duke Math. J. 106 (2001), no. 3, 617–633. MR 1813239, DOI 10.1215/S0012-7094-01-10638-8
Carlos E. Kenig, Gustavo Ponce, and Luis Vega, Well-posedness and scattering results for the generalized Korteweg-de Vries equation via the contraction principle, Comm. Pure Appl. Math. 46 (1993), no. 4, 527–620. MR 1211741, DOI 10.1002/cpa.3160460405
Carlos E. Kenig, Gustavo Ponce, and Luis Vega, A bilinear estimate with applications to the KdV equation, J. Amer. Math. Soc. 9 (1996), no. 2, 573–603. MR 1329387, DOI 10.1090/S0894-0347-96-00200-7
Luc Molinet, Francis Ribaud, and Abdellah Youssfi, Ill-posedness issues for a class of parabolic equations, Proc. Roy. Soc. Edinburgh Sect. A 132 (2002), no. 6, 1407–1416. MR 1950814
Luc Molinet and Francis Ribaud, On the low regularity of the Korteweg-de Vries-Burgers equation, Int. Math. Res. Not. 37 (2002), 1979–2005. MR 1918236, DOI 10.1155/S1073792802112104
L. A. Ostrovsky, Yu. A. Stepanyants, and L. Sh. Tsimring, Radiation instability in a stratified shear flow, Int. J. Non-Linear Mech. 19 (1984), 151–161.
E. Ott and N. Sudan, Damping of solitary waves, Phys. Fluids 13 (1970), no. 6, 1432–1434.
Msanori Otani, Well-posedness of the generalized Benjamin-Ono-Burgers equations in Sobolev spaces of negative order, Osaka J. Math. 43 (2006), no. 4, 935–965. MR 2303557
Didier Pilod, Sharp well-posedness results for the Kuramoto-Velarde equation, Commun. Pure Appl. Anal. 7 (2008), no. 4, 867–881. MR 2393403, DOI 10.3934/cpaa.2008.7.867
Terence Tao, Nonlinear dispersive equations, CBMS Regional Conference Series in Mathematics, vol. 106, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 2006. Local and global analysis. MR 2233925, DOI 10.1090/cbms/106
Terence Tao, Multilinear weighted convolution of $L^2$-functions, and applications to nonlinear dispersive equations, Amer. J. Math. 123 (2001), no. 5, 839–908. MR 1854113
Jeffrey Topper and Takuji Kawahara, Approximate equations for long nonlinear waves on a viscous fluid, J. Phys. Soc. Japan 44 (1978), no. 2, 663–666. MR 489338, DOI 10.1143/JPSJ.44.663
Nickolay Tzvetkov, Remark on the local ill-posedness for KdV equation, C. R. Acad. Sci. Paris Sér. I Math. 329 (1999), no. 12, 1043–1047 (English, with English and French summaries). MR 1735881, DOI 10.1016/S0764-4442(00)88471-2