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
SN - 0308-2105
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
IS - 2
ER -