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Volume 17, Issue 2
Local Well-posedness of Interaction Equations for Short and Long Dispersive Waves

Zhaohui Huo & Boling Guo

J. Part. Diff. Eq., 17 (2004), pp. 137-151.

Published online: 2004-05

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  • Abstract
The well-posedness of the Cauchy problem for the system {i∂_tu + ∂²_xu = uv + |u|²u, t, x ∈ \mathbb{R}, ∂_tv + ∂_xΗ∂_xv = ∂_x|u|², u(0, x) = u_0(x), v(0, x) = v_0(x), is considered. It is proved that there exists a unique local solution (u(x, t), v(x, t)) ∈ C([0, T);H^s) ×C([0, T);H^{s-\frac{1}{2}}) for any initial data (u_0, v_0) ∈ H^s(\mathbb{R}) ×H^{s-\frac{1}{2}} (\mathbb{R})(s ≥ \frac{1}{4}) and the solution depends continuously on the initial data.
  • Keywords

Short and long dispersive waves the Fourier restriction norm the Smoothing effects

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COPYRIGHT: © Global Science Press

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@Article{JPDE-17-137, author = {Zhaohui Huo and Boling Guo }, title = {Local Well-posedness of Interaction Equations for Short and Long Dispersive Waves}, journal = {Journal of Partial Differential Equations}, year = {2004}, volume = {17}, number = {2}, pages = {137--151}, abstract = { The well-posedness of the Cauchy problem for the system {i∂_tu + ∂²_xu = uv + |u|²u, t, x ∈ \mathbb{R}, ∂_tv + ∂_xΗ∂_xv = ∂_x|u|², u(0, x) = u_0(x), v(0, x) = v_0(x), is considered. It is proved that there exists a unique local solution (u(x, t), v(x, t)) ∈ C([0, T);H^s) ×C([0, T);H^{s-\frac{1}{2}}) for any initial data (u_0, v_0) ∈ H^s(\mathbb{R}) ×H^{s-\frac{1}{2}} (\mathbb{R})(s ≥ \frac{1}{4}) and the solution depends continuously on the initial data.}, issn = {2079-732X}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/jpde/5382.html} }
TY - JOUR T1 - Local Well-posedness of Interaction Equations for Short and Long Dispersive Waves AU - Zhaohui Huo & Boling Guo JO - Journal of Partial Differential Equations VL - 2 SP - 137 EP - 151 PY - 2004 DA - 2004/05 SN - 17 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/jpde/5382.html KW - Short and long dispersive waves KW - the Fourier restriction norm KW - the Smoothing effects AB - The well-posedness of the Cauchy problem for the system {i∂_tu + ∂²_xu = uv + |u|²u, t, x ∈ \mathbb{R}, ∂_tv + ∂_xΗ∂_xv = ∂_x|u|², u(0, x) = u_0(x), v(0, x) = v_0(x), is considered. It is proved that there exists a unique local solution (u(x, t), v(x, t)) ∈ C([0, T);H^s) ×C([0, T);H^{s-\frac{1}{2}}) for any initial data (u_0, v_0) ∈ H^s(\mathbb{R}) ×H^{s-\frac{1}{2}} (\mathbb{R})(s ≥ \frac{1}{4}) and the solution depends continuously on the initial data.
Zhaohui Huo and Boling Guo . (2004). Local Well-posedness of Interaction Equations for Short and Long Dispersive Waves. Journal of Partial Differential Equations. 17 (2). 137-151. doi:
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