, we have 1/2 derivative to spare. Moreover, using the relation between the M i 's, we can always place all (3/2)+ derivatives on the term containing v, thus we can sum by using Cauchy-Schwarz to

X. Then,

, R defined above is continuous and nondecreasing, and furthermore Bibliographie

T. Alazard and R. Carles, Loss of regularity for supercritical nonlinear Schrödinger equations, Mathematische Annalen, vol.343, issue.2, pp.397-420, 2009.

J. C. Alexander, R. L. Pego, and R. L. Sachs, On the transverse instability of solitary waves in the Kadomtsev-Petviashvili equation, Physics Letters A, vol.226, issue.3, pp.187-192, 1997.

T. B. Benjamin, The Stability of Solitary Waves, Proceedings of the Royal Society of London A : Mathematical, Physical and Engineering Sciences, vol.328, pp.153-183, 1573.

N. Burq, P. Gérard, N. Tzvetkov-;-björn, C. E. Birnir, G. Kenig et al., On the Ill-Posedness of the IVP for the Generalized Korteweg-De Vries and Nonlinear Schrödinger Equations, Journal of the London Mathematical Society, vol.53, issue.3, pp.551-559, 1996.

J. Bourgain, Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations, Geometric and Functional Analysis, vol.3, issue.3, pp.209-262, 1993.

J. Bourgain, On the Cauchy problem for the Kadomstev-Petviashvili equation, Geometric and Functional Analysis, vol.3, issue.4, pp.315-341, 1993.

J. L. Bona and R. Smith, The Initial-Value Problem for the Korteweg-De Vries Equation, Philosophical Transactions of the Royal Society of London A : Mathematical, Physical and Engineering Sciences, vol.278, pp.555-601, 1287.

M. Christ, J. Colliander, and T. Tao, Asymptotics, frequency modulation, and low regularity ill-posedness for canonical defocusing equations, American journal of mathematics, vol.125, issue.6, pp.1235-1293, 2003.

A. De-bouard and J. Saut, Solitary waves of generalized KadomtsevPetviashvili equations, vol.14, pp.211-236, 1997.

Z. Boling-guo, S. Huo, and . Fang, Low regularity for the fifth order Kadomtsev-Petviashvili-I type equation, Journal of Differential Equations, 2017.

J. Ginibre, Le problème de Cauchy pour des EDP semi-linéaires périodiques en variables d'espace, Séminaire Bourbaki, pp.163-187, 1996.

Z. Guo and T. Oh, Non-Existence of Solutions for the Periodic Cubic NLS below L 2 . International Mathematics Research Notices, 2016.

Z. Guo, L. Peng, B. Wang, and Y. Wang, Uniform wellposedness and inviscid limit for the Benjamin-Ono-Burgers equation, Advances in Mathematics, vol.228, issue.2, pp.647-677, 2011.

M. Grillakis, J. Shatah, and W. Strauss, Stability theory of solitary waves in the presence of symmetry, I, Journal of Functional Analysis, vol.74, issue.1, pp.160-197, 1987.

M. Hadac, Well-Posedness for the Kadomtsev-Petviashvili II Equation and Generalisations, Transactions of the American Mathematical Society, vol.360, issue.12, pp.6555-6572, 2008.

M. Hadac, S. Herr, and H. Koch, Well-posedness and scattering for the KP-II equation in a critical space, vol.26, pp.917-941, 2009.

A. D. Ionescu and C. E. Kenig, Local and Global Wellposedness of Periodic KP-I Equations, pp.181-212, 2007.

A. D. Ionescu, C. E. Kenig, and D. Tataru, Global well-posedness of the KP-I initialvalue problem in the energy space, Inventiones mathematicae, vol.173, issue.2, pp.265-304, 2008.

P. Isaza and J. Mejía, Local and global cauchy problems for the Kadomtsev-Petviashvili (KP-II) equation in Sobolev spaces of negative indices, Communications in Partial Differential Equations, vol.26, pp.1027-1054, 2001.

P. Isaza, J. Mejía, and V. Stallbohm, Local solution for the Kadomtsev-Petviashvili equation with periodic conditions. manuscripta mathematica, vol.75, pp.383-393, 1992.

P. Isaza, J. Mejia, and V. Stallbohm, Local Solution for the Kadomtsev-Petviashvili Equation in R 2, Journal of Mathematical Analysis and Applications, vol.196, issue.2, pp.566-587, 1995.

R. , W. Vieira-leite, and . Nunes, On equations of KP-type, Proceedings of the Royal Society of Edinburgh : Section A Mathematics, vol.128, pp.725-743, 1998.

C. E. Kenig, On the local and global well-posedness theory for the KP-I equation. Annales de l'Institut Henri Poincare (C) Non Linear Analysis, vol.21, pp.827-838, 2004.

B. B. Kadomtsev and V. I. Petviashvili, On the Stability of Solitary Waves in Weakly Dispersing Media, Soviet Physics Doklady, p.15, 1970.

T. Kato and G. Ponce, Commutator estimates and the Euler and NavierStokes equations, Communications on Pure and Applied Mathematics, vol.41, issue.7, pp.891-907, 1988.

C. E. Kenig and D. Pilod, Well-posedness for the fifth-order KdV equation in the energy space, Trans. Amer. Math. Soc, vol.367, issue.4, pp.2551-2612, 2015.

C. E. Kenig, G. Ponce, and L. Vega, Well-Posedness of the Initial Value Problem for the Korteweg-de Vries Equation, Journal of the American Mathematical Society, vol.4, issue.2, pp.323-347, 1991.

C. E. Kenig, G. Ponce, and L. Vega, The Cauchy problem for the Korteweg-de Vries equation in Sobolev spaces of negative indices, Duke Math. J, vol.71, issue.1, pp.1-21, 1993.

C. E. Kenig, G. Ponce, and L. Vega, Well-posedness and scattering results for the generalized Korteweg-De Vries equation via the contraction principle, Communications on Pure and Applied Mathematics, vol.46, issue.4, pp.527-620, 1993.

C. Kenig, G. Ponce, and L. Vega, A bilinear estimate with applications to the KdV equation, Journal of the American Mathematical Society, vol.9, issue.2, pp.573-603, 1996.

C. E. Kenig, G. Ponce, and L. Vega, On the ill-posedness of some canonical dispersive equations, Duke Math. J, vol.106, issue.3, pp.617-633, 2001.

H. Koch and N. Tzvetkov, On the local well-posedness of the Benjamin-Ono equation in H s (R), International Mathematics Research Notices, issue.26, pp.1449-1464, 2003.

H. Koch and N. Tzvetkov, Nonlinear wave interactions for the BenjaminOno equation, International Mathematics Research Notices, issue.30, pp.1833-1847, 2005.
URL : https://hal.archives-ouvertes.fr/hal-00019959

T. Kappeler and P. Topalov, Global wellposedness of KdV in H ?1 (T, R), Duke Math. J, vol.135, issue.2, pp.327-360, 2006.

H. Koch and N. Tzvetkov, On finite energy solutions of the KP-I equation, Mathematische Zeitschrift, vol.258, issue.1, pp.55-68, 2008.

D. Lannes, The Water Waves Problem : Mathematical Analysis and Asymptotics, Mathematical Surveys and Monographs, vol.188, 2013.
URL : https://hal.archives-ouvertes.fr/hal-01101991

G. Lebeau, Perte de régularité pour les équations d'ondes sur-critiques, Bull. Soc. Math. Fr, vol.133, issue.1, pp.145-157, 2005.

Y. Liu and J. Wei, Nondegeneracy, Morse Index and Orbital Stability of the Lump Solution to the KP-I Equation, 2017.

J. Li and J. Xiao, Well-posedness of the fifth order Kadomtsev-Petviashvili I equation in anisotropic Sobolev spaces with nonnegative indices, Journal de Mathématiques Pures et Appliquées, vol.90, issue.4, pp.338-352, 2008.

L. Molinet, Global well-posedness in the energy space for the Benjamin-Ono equation on the circle, Mathematische Annalen, vol.337, issue.2, pp.353-383, 2007.

L. Molinet, J. Saut, and N. Tzvetkov, Global well-posedness for the KP-I equation, Mathematische Annalen, vol.324, issue.2, pp.255-275, 2002.
URL : https://hal.archives-ouvertes.fr/hal-00104398

L. Molinet, J. Saut, and N. Tzvetkov, Well-posedness and illposedness results for the Kadomtsev-Petviashvili-I equation, Duke Math. J, vol.115, issue.2, pp.353-384, 2002.

L. Molinet, J. Saut, and N. Tzvetkov, Global well-posedness for the KP-II equation on the background of a non-localized solution. Annales de l'Institut Henri Poincare (C) Non Linear Analysis, vol.28, pp.653-676, 2011.
URL : https://hal.archives-ouvertes.fr/hal-01270936

T. Mizumachi and N. Tzvetkov, Stability of the line soliton of the KP-II equation under periodic transverse perturbations, Mathematische Annalen, vol.352, issue.3, pp.659-690, 2012.

T. Robert, On the Cauchy problem for the periodic fifth-order KP-I equation, 2017.
URL : https://hal.archives-ouvertes.fr/hal-01761459

T. Robert, Global well-posedness of partially periodic kp-i equation in the energy space and application. Annales de l'Institut Henri Poincaré C, Analyse non linéaire, 2018.
URL : https://hal.archives-ouvertes.fr/hal-01761457

J. Rauch and M. Reed, Nonlinear microlocal analysis of semilinear hyperbolic systems in one space dimension, Duke Math. J, vol.49, issue.2, pp.397-475, 1982.

F. Rousset and N. Tzvetkov, Stability and instability of the KdV solitary wave under the KP-I flow, Communications in Mathematical Physics, vol.313, issue.1, pp.155-173, 2012.
URL : https://hal.archives-ouvertes.fr/hal-00707829

J. Saut, Remarks on the Generalized Kadomtsev-Petviashvili Equations. Indiana University Mathematics Journal, vol.42, issue.3, pp.1011-1026, 1993.

J. , C. Saut, and N. Tzvetkov, The cauchy problem for higher-order KP equations, Journal of Differential Equations, vol.153, issue.1, pp.196-222, 1999.

J. , C. Saut, and N. Tzvetkov, The Cauchy problem for the fifth order KP equations, Journal de Mathématiques Pures et Appliquées, vol.79, issue.4, pp.307-338, 2000.

J. , C. Saut, and N. Tzvetkov, On Periodic KP-I Type Equations, Communications in Mathematical Physics, vol.221, pp.451-476, 2001.

T. Tao, Multilinear Weighted Convolution of <tex-math>L 2 </tex-math> Functions, and Applications to Nonlinear Dispersive Equations, American Journal of Mathematics, vol.123, issue.5, pp.839-908, 2001.

T. Tao, Nonlinear dispersive equations : local and global analysis. Number 106, 2006.

M. Michael and . Tom, On a generalized Kadomtsev-Petviashvili equation, Contemporary Mathematics, vol.200, pp.193-210, 1996.

Y. Tsutsumi, L 2 -Solutions for Nonlinear Schrödinger Equations and Nonlinear Groups, Funkcialaj Ekvacioj, vol.30, pp.115-125, 1987.

H. Takaoka and N. Tzvetkov, On the local regularity of the KadomtsevPetviashvili-II equation, International Mathematics Research Notices, issue.2, pp.77-114, 2001.

N. Tzvetkov, Ill-posedness issues for nonlinear dispersive equations. ArXiv Mathematics e-prints, 2004.
URL : https://hal.archives-ouvertes.fr/hal-00108211

S. Ukai, Local solutions of the Kadomtsev-Petviashvili equation, J. Fac. Sci. Univ. Tokyo Sect. IA Math, vol.36, issue.2, pp.193-209, 1989.

Y. Yamazaki, Stability of the line soliton of the Kadomtsev-Petviashvili-I equation with the critical traveling speed, 2017.

Y. Zhang, Local well-posedness of KP-I initial value problem on torus in the Besov space, Communications in Partial Differential Equations, pp.1-26, 2015.