On the local and global well-posedness theory for the KP-I equation. (English) Zbl 1072.35162
The KP-I equation arises in the water wave theory, as modeling capillary-gravity waves in the presence of strong surface tension effects. The author shows the local well-posedness of the KP-I equation in the space \(Y_s=\{\, \varphi\in L^2(R^2)|\, \|\varphi\|_{L^2}+\|J_x^s\varphi\|_{L^2}+\|\partial^{-1}_x\partial_y\varphi\|_{L^2}<\infty \,\}\) for \(s>3/2\), where \(\hat{J}_x^s f(\xi,\eta)=(1+|\xi|^2)^{s/2}\hat{f}(\xi,\eta)\), improving the local well-posedness result in [J.-C. Saut, Indiana Univ. Math. J. 42, No. 3, 1011–1026 (1993; Zbl 0814.35119)] given by the classical energy estimate. Then using the results [L. Molinet, J.-C. Saut, N.Tzvetkov, Duke Math. J. 115, No. 2, 353–384 (2002; Zbl 1033.35103)] the global well-posedness for KP-I equations in the space
\[
Z_0=\{\, \varphi\in L^2(R^2)|\, \|\varphi\|_{L^2}+\|\partial^{-1}_x\partial_y\varphi\|_{L^2}+ \|\partial^{2}_x \varphi \|_{L^2}+ \|\partial^{-2}_x\partial^2_y\varphi\|_{L^2} <\infty \,\}
\]
is established.
Reviewer: Boris V. Loginov (Ul’yanovsk)
MSC:
| 35Q53 | KdV equations (Korteweg-de Vries equations) |
| 76B45 | Capillarity (surface tension) for incompressible inviscid fluids |
| 35B30 | Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs |
| 76B03 | Existence, uniqueness, and regularity theory for incompressible inviscid fluids |
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