Bilinear estimates related to the KP equations
Journées équations aux dérivées partielles (2000), article no. 19, 12 p.

We survey some recent results for the KP-II equation. We also give an idea for treating the “bad frequency interactions” of the bilinear estimates in the Fourier transform restriction spaces related to the KP-I equation.

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     title = {Bilinear estimates related to the {KP} equations},
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     publisher = {Universit\'e de Nantes},
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     mrnumber = {2001f:35358},
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Tzvetkov, Nikolay. Bilinear estimates related to the KP equations. Journées équations aux dérivées partielles (2000), article  no. 19, 12 p. http://www.numdam.org/item/JEDP_2000____A19_0/

[1] J. Bourgain, Fourier transform restriction phenomena for certain lattice subsets and application to nonlinear evolution equations I. Schrödinger equations, GAFA 3 (1993), 107-156, II. The KdV equation, GAFA 3 (1993), 209-262. | Zbl

[2] J. Bourgain, On the Cauchy problem for the Kadomtsev-Petviashvili equation, GAFA 3 (1993), 315-341. | MR | Zbl

[3] J. Bourgain, Refinements of Strichartz inequality and applications to 2D-NLS with critical nonlinearity, IMRN 5 (1998), 145-171. | MR | Zbl

[4] J. Colliander, C. E. Kenig, G. Staffilani, Local well-posedness and regularity properties of solutions of the generalized Kadomtsev-Petviashvili equations, in preparation.

[5] J. Colliander, C. E. Kenig, G. Staffilani, An Xs,b space approach to local well-posedness of Kadomtsev-Petviashvili-I equation, in preparation.

[6] J. Colliander, G. Staffilani, H. Takaoka, Global well-posedness for KdV below L2, MRL 6 (2000), 755-778. | MR | Zbl

[7] G. Fonseca, F. Linares, G. Ponce, Global well-posedness for the modified Korteweg- de Vries equation, Comm. PDE 24 (1999), 683-705. | MR | Zbl

[8] J. Ginibre, Le problème de Cauchy pour des EDP semi-linéaires périodiques en variables d'espace, Séminaire Bourbaki, 796 (1995). | Numdam | Zbl

[9] P. Isaza, J. Mejia, Problemas de Cauchy local y global para la ecuacion de Kadomtsev-Petviashvili (KP-II) en espacios de Sobolev con indices negativos, Preprint.

[10] M. Keel, T. Tao Local and global well-posedness of wave maps on ℝ1+1 with rough data, IMRN, 21 (1998), 1117-1156. | MR | Zbl

[11] C. E. Kenig, G. Ponce, L. Vega, Well-posedness of the initial value problem for the Korteweg-de Vries equation, J. AMS, 4 (1991), 323-347. | MR | Zbl

[12] C. E. Kenig, G. Ponce, L. Vega, The Cauchy problem for the Korteweg-de Vries equation in Sobolev spaces of negative indices, Duke Math. J., 71 (1993), 1-21. | MR | Zbl

[13] C. E. Kenig, G. Ponce, L. Vega, A bilinear estimate with applications to the KdV equations, J. AMS, 9 (1996), 573-603. | MR | Zbl

[14] C. E. Kenig, G. Ponce, L. Vega, Global well-posedness for semilinear wave equations, Preprint. | Zbl

[15] L. Molinet, J. C. Saut, N. Tzvetkov, On the KP-I equation, in preparation.

[16] H. Pecher, Global well-posedness below energy space for the 1D Zakharov system, Preprint. | Zbl

[17] J. C. Saut, Remarks on the generalized Kadomtsev-Petviashvili equations, Indiana Univ. Math. J., 42 (1993), 1017-1029. | MR | Zbl

[18] J. C. Saut, N. Tzvetkov, The Cauchy problem for the fifth order KP equations, J. Math. Pures Appl. (2000), 307-338. | MR | Zbl

[19] M. Schwarz, Periodic solutions of Kadomtsev-Petviashvili, Adv. Math. 66 (1987), 217-233. | MR | Zbl

[20] H. Takaoka, Global well-posedness for the Kadomtsev-Petviashvili II equation, Discrete Contin. Dynam. Systems 6 (2000), 483-499. | MR | Zbl

[21] H. Takaoka, N. Tzvetkov, On the local regularity of Kadomtsev-Petviashvili-II equation, Preprint, Orsay 1999. | Zbl

[22] H. Takaoka, N. Tzvetkov, On 2D dispersive models, in preparation.

[23] N. Tzvetkov, Global low regularity solutions for Kadomtsev-Petviashvili equation, Diff. Int. Eq. (to appear). | Zbl