The Cauchy problem for the fifth order KP equations
The fifth order Kadomtsev-Petviashvili (KP) equation \[ (\partial_t u -\partial^5_x u +\alpha \partial^3_x u +uu_x)_x+ \varepsilon u_{yy}=0 \] is studied. Here, \(u=u(t,x,y)\) is a real-valued function and \((t,x,y)\in {\mathbb R}^3\). Moreover, \(\varepsilon=-1\) corresponds to the ``focusing'' case (KP-I type), while \(\varepsilon=1\) corresponds to the defocusing case (KP-II type). The local well-posedness of the fifth order KP-I is proved for initial data in a suitable anisotropic Sobolev space. Further, the authors extend solutions globally in time due to the conservation of the energy. The proof of the local well-posedness result uses the Fourier transform restriction method due to J. Bourgain. In the case of the fifth order KP-II, local and global well-posedness in \(L^2\) are obtained, removing the restriction on the initial data imposed in [same authors, J. Differ. Equations 153, No. 1, 196--222 (1999; Zbl 0927.35098)]. The case of periodic boundary conditions for the higher-order KP-II equation is also studied in the paper. For the problem in two dimensions, the local-in-time existence of solutions is proved for initial data in a suitable Sobolev space. Those solutions exist globally in time for data in \(L^2\). Finally, the local well-posedness of the three-dimensional problem for initial data in Sobolev spaces of low order is proved.
the Fourier transform restriction method
scientific article; zbMATH DE number 1463878
scientific article