@article{AIHPC_2008__25_6_1145_0, author = {Bona, Jerry L. and Sun, S. M. and Zhang, Bing-Yu}, title = {Non-homogeneous boundary value problems for the Korteweg-de Vries and the Korteweg-de Vries-Burgers equations in a quarter plane}, journal = {Annales de l'I.H.P. Analyse non lin\'eaire}, volume = {25}, year = {2008}, pages = {1145-1185}, doi = {10.1016/j.anihpc.2007.07.006}, mrnumber = {2466325}, zbl = {1157.35090}, language = {en}, url = {http://dml.mathdoc.fr/item/AIHPC_2008__25_6_1145_0} }
Bona, Jerry L.; Sun, S. M.; Zhang, Bing-Yu. Non-homogeneous boundary value problems for the Korteweg-de Vries and the Korteweg-de Vries-Burgers equations in a quarter plane. Annales de l'I.H.P. Analyse non linéaire, Tome 25 (2008) pp. 1145-1185. doi : 10.1016/j.anihpc.2007.07.006. http://gdmltest.u-ga.fr/item/AIHPC_2008__25_6_1145_0/
[1] The initial-value problem for the generalized Burgers' equation, Differential Integral Equations 9 (1996) 1253-1265. | MR 1409926 | Zbl 0879.35131
,[2] Wave dominated shelves: a model of sand ridge formation by progressive infragravity waves, in: , (Eds.), Shelf Sands and Sandstones, Canadian Society of Petroleum Geologists Memoir, vol. 11, 1986, pp. 163-179.
, ,[3] A mathematical model for long waves generated by wave makers in non-linear dispersive systems, Proc. Cambridge Philos. Soc. 73 (1973) 391. | MR 339651 | Zbl 0261.76007
, ,[4] Generalized Korteweg-de Vries equation in a quarter plane, Contemp. Math. 221 (1999) 59-125. | Zbl 0922.35150
, ,[5] An evaluation of a model equation for water waves, Philos. Trans. Roy. Soc. London Ser. A 302 (1981) 457-510. | MR 633485 | Zbl 0497.76023
, , ,[6] Solutions of the Korteweg-de Vries equation in fractional order Sobolev spaces, Duke Math. J. 43 (1976) 87-99. | MR 393887 | Zbl 0335.35032
, ,[7] The initial-value problem for the Korteweg-de Vries equation, Philos. Trans. Roy. Soc. London Ser. A 278 (1975) 555-601. | MR 385355 | Zbl 0306.35027
, ,[8] A nonhomogeneous boundary-value problem for the Korteweg-de Vries equation in a quarter plane, Trans. Amer. Math. Soc. 354 (2001) 427-490. | MR 1862556 | Zbl 0988.35141
, , ,[9] Forced oscillations of a damped KdV equation in a quarter plane, Commun. Contemp. Math. 5 (2003) 369-400. | MR 1992355 | Zbl 1054.35076
, , ,[10] A nonhomogeneous boundary value problem for the KdV equation posed on a finite domain, Commun. Partial Differential Equations 28 (2003) 1391-1436. | MR 1998942 | Zbl 1057.35049
, , ,[11] Conditional and unconditional well-posedness of non-linear evolution equations, Adv. Differential Equations 9 (2004) 241-265. | MR 2100628 | Zbl 1103.35092
, , ,[12] Boundary smoothing properties of the Korteweg-de Vries equation in a quarter plane and applications, Dynamics Partial Differential Equations 3 (2006) 1-70. | MR 2221746 | Zbl 1152.35099
, , ,[13] The Korteweg-de Vries equation, posed in a quarter plane, SIAM J. Math. Anal. 14 (1983) 1056-1106. | MR 718811 | Zbl 0529.35069
, ,[14] Korteweg-de Vries equation in a quarter plane, continuous dependence results, Differential Integral Equations 2 (1989) 228-250. | MR 984190 | Zbl 0734.35120
, ,[15] Fourier transform restriction phenomena for certain lattice subsets and applications to non-linear evolution equations, part II: the KdV equation, Geom. Funct. Anal. 3 (1993) 209-262. | MR 1215780 | Zbl 0787.35098
,[16] Periodic Korteweg-de Vries equation with measures as initial data, Selecta Math. (N.S.) 3 (1997) 115-159. | MR 1466164 | Zbl 0891.35138
,[17] Solutions of the Korteweg-de Vries equation from irregular data, Duke Math. J. 45 (1978) 149-181. | MR 470533 | Zbl 0372.35022
,[18] Existence and regularity for solutions of the Korteweg-de Vries equation, Arch. Rat. Mech. Anal. 71 (1979) 143-175. | MR 525222 | Zbl 0415.35069
,[19] Asymptotics, frequency modulation, and low regularity ill-posedness for canonical defocusing equations, Amer. J. Math. 125 (2003) 1235-1293. | MR 2018661 | Zbl 1048.35101
, , ,[20] The generalized Korteweg-de Vries equation on the half line, Comm. Partial Differential Equations 27 (2002) 2187-2266. | MR 1944029 | Zbl 1041.35064
, ,[21] Sharp global well-posedness for KdV and modified KdV on R and T, J. Amer. Math. Soc. 16 (2003) 705-749. | MR 1969209 | Zbl 1025.35025
, , , , ,[22] Local smoothing properties of dispersive equations, J. Amer. Math. Soc. 1 (1988) 413-446. | MR 928265 | Zbl 0667.35061
, ,[23] Gain of regularity for equations of the Korteweg-de Vries type, Ann. Inst. H. Poincaré 9 (1992) 147-186. | Numdam | MR 1160847 | Zbl 0764.35021
, , ,[24] The dissipation of non-linear dispersive waves: the case of asymptotically weak non-linearity, Comm. Partial Differential Equations 17 (1992) 1665-1693. | MR 1187625 | Zbl 0762.35110
,[25] Nonuniqueness and uniqueness in the initial-value problem for Burgers' equation, SIAM J. Math. Anal. 27 (1996) 708-724. | MR 1382829 | Zbl 0852.35123
,[26] The Cauchy problem and the mixed problem in the half strip for equations of Korteweg-de Vries type, Dinamika Sploshn. Sredy 162 (1983) 152-158, (in Russian). | MR 809894 | Zbl 0569.35066
,[27] A mixed problem in a semistrip for the Korteweg-de Vries equation and its generalizations, Dinamika Sploshn. Sredy 258 (1988) 54-94, (in Russian). English translation in, Trans. Moscow Math. Soc. 51 (1989) 53-91. | MR 983632 | Zbl 0706.35120
,[28] Mixed problems for the Korteweg-de Vries equation, Sb. Math. 190 (1999) 903-935. | MR 1719565 | Zbl 0933.35167
,[29] An initial boundary-value problem in a half-strip for the Korteweg-de Vries equation in fractional-order Sobolev spaces, Comm. Partial Differential Equations 29 (2004) 1653-1695. | MR 2105984 | Zbl 1080.35117
,[30] Soliton generation for initial-boundary value problems, Phys. Rev. Lett. 68 (1992) 3117-3120. | MR 1163545 | Zbl 0969.35537
, ,[31] An initial-boundary value problem for the Korteweg-de Vries equation, Math. Comput. Simulation 37 (1994) 293-321. | MR 1308105 | Zbl 0832.35125
, ,[32] Integrable equations on the half-infinite line. Solitons in science and engineering: theory and applications, Chaos Solitons Fractals 5 (1995) 2367-2376. | MR 1368225 | Zbl 1080.35531
, ,[33] The solution of certain initial boundary-value problems for the linearized Korteweg-de Vries equation, Proc. Roy. Soc. London Ser. A 454 (1998) 645-657. | MR 1638297 | Zbl 0914.35122
, ,[34] The initial-boundary value problem for the Korteweg-de Vries equation, Comm. Partial Differential Equations 31 (2006) 1151-1190. | MR 2254610 | Zbl 1111.35062
,[35] A non-linear equation incorporating damping and dispersion, J. Fluid Mech. 42 (1970) 49-60. | MR 268546 | Zbl 0213.54904
,[36] Well-posedness of KDV on , Duke Math. J. 135 (2006) 327-360. | Zbl 1106.35081
, ,[37] On the Korteweg-de Vries equation, Manuscripta Math. 29 (1979) 89-99. | Zbl 0415.35070
,[38] The Cauchy problem for the Korteweg-de Vries equation, in: Pitman Research Notes in Math., vol. 53, 1979, pp. 293-307. | MR 631399 | Zbl 0542.35064
,[39] On the Cauchy problem for the (generalized) Korteweg-de Vries equations, in: Advances in Mathematics Supplementary Studies, Studies Appl. Math., vol. 8, 1983, pp. 93-128. | MR 759907 | Zbl 0549.34001
,[40] On the (generalized) Korteweg-de Vries equation, Duke Math. J. 59 (1989) 585-610. | MR 1046740 | Zbl 0795.35105
, , ,[41] Well-posedness of the initial value problem for the KdV equation, J. Amer. Math. Soc. 4 (1991) 323-347. | MR 1086966 | Zbl 0737.35102
, , ,[42] Well-posedness and scattering results for the generalized Korteweg-de Vries equations via the contraction principle, Comm. Pure Appl. Math. 46 (1993) 527-620. | MR 1211741 | Zbl 0808.35128
, , ,[43] The Cauchy problem for the Korteweg-de Vries equation in Sobolev spaces of negative indices, Duke Math. J. 71 (1993) 1-21. | MR 1230283 | Zbl 0787.35090
, , ,[44] A bilinear estimate with applications to the KdV equation, J. Amer. Math. Soc. 9 (1996) 573-603. | MR 1329387 | Zbl 0848.35114
, , ,[45] On the ill-posedness of some canonical dispersive equations, Duke Math. J. 106 (2001) 617-633. | MR 1813239 | Zbl 1034.35145
, , ,[46] Generalized solutions of the Cauchy problem for the Korteweg-de Vries equation, Mat. Sb. (N.S.) 120 (1983) 396-425, (in Russian). English translation in, Math. USSR Sb. 48 (1984) 391-421. | MR 691986 | Zbl 0549.35104
, ,[47] On the low regularity of the Korteweg-de Vries-Burgers equation, Int. Math. Res. Notices 37 (2002) 1979-2005. | MR 1918236 | Zbl 1031.35126
, ,[48] Smoothing properties of solutions of the Korteweg-de Vries equation on a periodic domain with point dissipation, J. Math. Anal. Appl. 190 (1995) 449-488. | MR 1318405 | Zbl 0845.35111
, ,[49] Classical solutions of the Korteweg-de Vries equation for non-smooth initial data via inverse scattering, Comm. Partial Differential Equations 10 (1985) 29-98. | MR 773211 | Zbl 0562.35085
,[50] Remarks on the Korteweg-de Vries equation, Israel J. Math. 24 (1976) 78-87. | MR 454425 | Zbl 0334.35062
, ,[51] On the Korteweg-de Vries equation: Existence and uniqueness, J. Math. Anal. Appl. 29 (1970) 569-579. | MR 410135 | Zbl 0179.43101
,[52] Non linèaire et régularité, J. Funct. Anal. 9 (1972) 469-489. | MR 310619 | Zbl 0241.46035
,[53] Sur un problème non linéaire, J. Math. Pures Appl. 48 (1969) 159-172. | MR 261183 | Zbl 0187.03902
,[54] Initial boundary value problems for the Korteweg-de Vries equation, J. Differential Equations 25 (1977) 288-309. | MR 487097 | Zbl 0372.35070
,[55] Remark on the local ill-posedness for the KdV equation, C. R. Acad. Sci. Paris Sér. I Math. 329 (1999) 1043-1047. | MR 1735881 | Zbl 0941.35097
,[56] Analyticity of solutions for the generalized Korteweg-de Vries equation with respect to their initial datum, SIAM J. Math. Anal. 26 (1995) 1488-1513. | MR 1356456 | Zbl 0837.35135
,