Global well-posedness of the KP-I initial-value problem in the energy space. (English) Zbl 1188.35163

Ionescu, A. D.; Kenig, C. E.; Tataru, D.

Invent. Math. 173, No. 2, 265-304 (2008).

Summary: We prove that the KP-I initial-value problem
\[ \begin{cases} \partial_tu+\partial_x^3u-\partial_x^{-1}\partial_y^2u+\partial_x(u^2/2)=0 &\text{on}\,\mathbb{R}^2_{x,y}\times\mathbb{R}_t;\\ u(0)=\varphi, \end{cases} \]
is globally well-posed in the energy space
\[ \mathbf{E}^1(\mathbb{R}^2)=\big\{\phi:\mathbb{R}^2\to\mathbb{R}: \|\varphi\|_{\mathbf{E}^1(\mathbb{R}^2)}\approx\|\varphi\|_{L^2}+\|\partial_x\varphi\|_{L^2}+\big\|\partial_x^{-1}\partial_y\varphi\big\|_{L^2}<\infty\big\}. \]

MSC:

35Q53	KdV equations (Korteweg-de Vries equations)
35B45	A priori estimates in context of PDEs

Keywords:

KP-I initial-value problem; global estimate; bilinear estimate; energy estimate

References:

[1]	Ablowitz, M.J., Clarkson, P.A.: Solitons, nonlinear evolution equations and inverse scattering. Lond. Math. Soc. Lect. Note Ser., vol. 149. Cambridge University Press, Cambridge (1991) · Zbl 0762.35001
[2]	Ablowitz, M.J., Segur, H.: Solitons and the Inverse Scattering Transform. SIAM Stud. Appl. Math., vol. 4. Soc. Ind. Appl. Math. (SIAM), Philadelphia, Pa. (1981) · Zbl 0472.35002
[3]	Alexander, J.C., Pego, R.L., Sachs, R.L.: On the transverse instability of solitary waves in the Kadomtsev–Petviashvili equation. Phys. Lett., A 226, 187–192 (1997) · Zbl 0962.35505
[4]	Ben-Artzi, M., Saut, J.-C.: Uniform decay estimates for a class of oscillatory integrals and applications. Differ. Integral Equ. 12, 137–145 (1999) · Zbl 1016.35006
[5]	Bona, J.L., Smith, R.: The initial-value problem for the Korteweg–de Vries equation. Philos. Trans. R. Soc. Lond., Ser. A 278, 555–601 (1975) · Zbl 0306.35027
[6]	Bona, J.L., Souganidis, P.E., Strauss, W.A.: Stability and instability of solitary waves of Korteweg–de Vries type. Proc. R. Soc. Lond., Ser. A 411, 395–412 (1987) · Zbl 0648.76005
[7]	De Bouard, A., Saut, J.-C.: Solitary waves of generalized Kadomtsev–Petviashvili equations. Ann. Inst. Henri Poincaré, Anal. Non Linéaire 14, 211–236 (1997) · Zbl 0883.35103
[8]	Bourgain, J.: Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations II. The KdV-equation. Geom. Funct. Anal. 3, 209–262 (1993) · Zbl 0787.35098
[9]	Bourgain, J.: On the Cauchy problem for the Kadomtsev–Petviashvili equation. Geom. Funct. Anal. 3, 315–341 (1993) · Zbl 0787.35086
[10]	Cazenave, T., Lions, P.-L.: Orbital stability of standing waves for some nonlinear Schrödinger equations. Commun. Math. Phys. 85, 549–561 (1982) · Zbl 0513.35007
[11]	Christ, M., Colliander, J., Tao, T.: A priori bounds and weak solutions for the nonlinear Schrödinger equation in Sobolev spaces of negative order. Preprint (2006) · Zbl 1136.35087
[12]	Colliander, J., Ionescu, A.D., Kenig, C.E., Staffilani, G.: Weighted low-regularity solutions of the KP-I initial-value problem. Preprint (2007) · Zbl 1142.35076
[13]	Ionescu, A.D., Kenig, C.E.: Local and global well-posedness of periodic KP-I equations. In: Mathematical Aspects of Nonlinear Dispersive Equations. Ann. Math. Stud., vol. 163, pp.181–211. Princeton Univ. Press, Princeton, NJ (2007) · Zbl 1387.35528
[14]	Iorio, R.J., Nunes, W.V.L.: On equations of KP-type. Proc. R. Soc. Edinb., Sect. A, Math. 128, 725–743 (1998) · Zbl 0911.35103
[15]	Kenig, C.E., Ponce, G., Vega, L.: Well-posedness and scattering results for the generalized Korteweg–de Vries equation via the contraction principle. Commun. Pure Appl. Math. 46, 527–620 (1993) · Zbl 0808.35128
[16]	Kenig, C.E., Ponce, G., Vega, L.: The Cauchy problem for the Korteweg–de Vries equation in Sobolev spaces of negative indices. Duke Math. J. 71, 1–21 (1993) · Zbl 0787.35090
[17]	Kenig, C.E.: On the local and global well-posedness theory for the KP-I equation. Ann. Inst. Henri Poincaré, Anal. Non Linéaire 21, 827–838 (2004) · Zbl 1072.35162
[18]	Kenig, C.E., Martel, Y.: Asymptotic stability of solitons for the Benjamin–Ono equation. Preprint (2008) · Zbl 1247.35133
[19]	Klainerman, S., Machedon, M.: Space-time estimates for null forms and the local existence theorem. Commun. Pure Appl. Math. 46, 1221–1268 (1993) · Zbl 0803.35095
[20]	Koch, H., Tataru, D.: A-priori bounds for the 1-d cubic NLS in negative Sobolev spaces. Preprint (2006) · Zbl 1169.35055
[21]	Martel, Y., Merle, F.: Asymptotic stability of solitons for subcritical generalized KdV equations. Arch. Ration. Mech. Anal. 157, 219–254 (2001) · Zbl 0981.35073
[22]	Martel, Y., Merle, F.: Asymptotic stability of solitons of the subcritical gKdV equations revisited. Nonlinearity 18, 55–80 (2005) · Zbl 1064.35171
[23]	Molinet, L., Saut, J.-C., Tzvetkov, N.: Well-posedness and ill-posedness results for the Kadomtsev–Petviashvili-I equation. Duke Math. J. 115, 353–384 (2002) · Zbl 1033.35103
[24]	Molinet, L., Saut, J.-C., Tzvetkov, N.: Global well-posedness for the KP-I equation. Math. Ann. 324, 255–275 (2002) · Zbl 1008.35060
[25]	Molinet, L., Saut, J.-C., Tzvetkov, N.: Correction: Global well-posedness for the KP-I equation. Math. Ann. 328, 707–710 (2004) · Zbl 1055.35103
[26]	Rousset, F., Tzvetkov, N.: Transverse nonlinear instability for two-dimensional dispersive models. Preprint (2007) · Zbl 1169.35374
[27]	Saut, J.-C.: Remarks on the generalized Kadomtsev–Petviashvili equations. Indiana Univ. Math. J. 42, 1011–1026 (1993) · Zbl 0814.35119
[28]	Takaoka, H., Tzvetkov, N.: On the local regularity of the Kadomtsev–Petviashvili-II equation. Int. Math. Res. Not. 2001, 77–114 (2001) · Zbl 0977.35126
[29]	Tataru, D.: Local and global results for wave maps I. Commun. Partial Differ. Equations 23, 1781–1793 (1998) · Zbl 0914.35083
[30]	Weinstein, M.I.: Lyapunov stability of ground states of nonlinear dispersive evolution equations. Commun. Pure Appl. Math. 39, 51–67 (1986) · Zbl 0594.35005
[31]	Weinstein, M.I.: Existence and dynamic stability of solitary wave solutions of equations arising in long wave propagation. Commun. Partial Differ. Equations 12, 1133–1173 (1987) · Zbl 0657.73040