Global well-posedness and limit behavior for a higher-order Benjamin-Ono equation

Abstract

In this paper, we prove that the Cauchy problem associated to the following higher-order Benjamin-Ono equation $${\partial }_{t}v-b\mathcal{H}{\partial }_x^{2}v-a\epsilon {\partial }_x^{3}v=cv{\partial }_{x}v-d\epsilon {\partial }_x(v\mathcal{H}{\partial }_{x}v+\mathcal{H}(v{\partial }_{x}v)),$$ is globally well-posed in the energy space $H^1(\mathbb{R})$. Moreover, we study the limit behavior when the small positive parameter $\epsilon$ tends to zero and show that, under a condition on the coefficients $a$, $b$, $c$ and $d$, the solution $v_{\epsilon }$ to this equation converges to the corresponding solution of the Benjamin-Ono equation.