TY - JOUR

T1 - Global well-posedness and conservation laws for the water wave interaction equation

AU - Ogawa, Takayoshi

N1 - Funding Information:
The author is grateful to Professors N. Hayashi and G. Ponce and Dr D. Bekiranov for helpful advice and discussions. This work is partially supported by the Japanese Ministry of Education, Science and Culture.

PY - 1997

Y1 - 1997

N2 - Interaction equations of long and short water wave are considered. It is shown that the Cauchy problem for (Equation Presented) is locally well posed in the largest space where the three conservations ∥u(t)∥2 = ∥u0∥∥2, ∥ν(t)∥22 + 2 Im ∫ℝ u(t)∂xū(t) dx = ∥ν0∥22 + 2 Im ∫ℝ u0∂xū0 dx, E(u(t), ν(t)) = E(u0, ν0) can be justified. Here E(u, ν) is the energy functional associated to the system. By these conservation laws, we establish the global well-posedness of the system in the largest class of initial data.

AB - Interaction equations of long and short water wave are considered. It is shown that the Cauchy problem for (Equation Presented) is locally well posed in the largest space where the three conservations ∥u(t)∥2 = ∥u0∥∥2, ∥ν(t)∥22 + 2 Im ∫ℝ u(t)∂xū(t) dx = ∥ν0∥22 + 2 Im ∫ℝ u0∂xū0 dx, E(u(t), ν(t)) = E(u0, ν0) can be justified. Here E(u, ν) is the energy functional associated to the system. By these conservation laws, we establish the global well-posedness of the system in the largest class of initial data.

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U2 - 10.1017/S0308210500023684

DO - 10.1017/S0308210500023684

M3 - Article

AN - SCOPUS:33745931196

VL - 127

SP - 369

EP - 384

JO - Proceedings of the Royal Society of Edinburgh Section A: Mathematics

JF - Proceedings of the Royal Society of Edinburgh Section A: Mathematics

SN - 0308-2105

IS - 2

ER -