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Global well-posedness for the KP-BBM equations. (English) Zbl 1063.35140
The paper is dealing with two-dimensional generalizations of the Benjamin-Bona-Mahony equation (BBM, alias the regularized wave equation), which is considered as an alternative to the Korteweg-de Vries (KdV) equation for weakly nonlinear long surface waves on inviscid fluids. The BBM equation proper is \[ u_t-u_{txx}+u_x+uu_x. \] Unlike the KdV equation per se, it is not integrable, but it has an advantage of giving a more physically appropariate global dispersion relation for linear waves (without the divergence at large wave numbers). The two-dimensional generalizations of the BBM equation are constructed similar to how the KdV equation gives rise to the Kadomtsev-Petviashvili (KP) equation: \[ (u_t-u_{txx}+u_x+uu_x)_x+\gamma u_{xx}=0, \gamma = \pm 1. \] These equations with \(\gamma=-1\) and \(\gamma=+1\) are called, respectively, the BBM-KPI and BBM-KPII equations. It is known that the proof of the well-posedness for the KP equations proper is still an unsolved problem, because of the difficulties produced by their “bad” dispersion relations. Unlike that, the proof is possible just for the BBO-KP equations of both types. The proof is based on a specific Fourier-transform restriction method, and leads to the establishment of the rigorous existence of global solutions to the corresponding initial-value problems. The method was earlier applied to other problems, but the present paper develops its new ingredient, in the form of a necessary frequency analysis. Additionally, the BBO-KPI equation has exact weakly localized lump-soliton solutions, which it actually shares with the usual KPI equation. The methods developed in the paper make it possible to prove the stability of the lump in the BBO-KPI equation.

35Q53 KdV equations (Korteweg-de Vries equations)
35B40 Asymptotic behavior of solutions to PDEs
76B25 Solitary waves for incompressible inviscid fluids
37K40 Soliton theory, asymptotic behavior of solutions of infinite-dimensional Hamiltonian systems
37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
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