Non-uniform dependence on initial data for the two-component fractional shallow water wave system
This paper deals with the continuity of the following two-component water wave system \[ \begin{cases} m_t +um_x +bmu_x =\alpha u_x -\kappa \rho \rho_x,&\quad t>0,\,x\in\mathbb{R},\\ \rho_t +u\rho_x +(b-1)u_x\rho =0,&\quad t>0,\,x\in\mathbb{R},\\ u(0, x) = u_0(x),\>\rho(0, x) = \rho_ 0(x),&\quad x\in\mathbb{R}, \end{cases} \] where \(m = ((1 - \partial_x^2 )^r u(t,\cdot))(x),\) with \(r\ge 1\), \(\alpha\in\mathbb{R}\), \(b\not= 1\) and \(\kappa > 0\) are constants. Using an argument based on the approximate solutions technique, the authors are able to find two sequences of solutions \((u_{1,\lambda}(t),\,\rho_{1,\lambda} (t))\) and \((u_{-1,\lambda}(t),\,\rho_{-1,\lambda} (t))\) in \(C([0,T];H^{s_1}(\mathbb{R})\times H^{s_2}(\mathbb{R})) \cap C^1([0,T];H^{s_1-1}(\mathbb{R})\times H^{s_2-1}(\mathbb{R}))\) such that \[\Vert(u_{1,\lambda} (t), u_{-1,\lambda} (t))\Vert_{H^{s_1}\times H^{s_1}} =O( 1),\qquad \Vert(\rho_{1,\lambda} (t), \rho_{-1,\lambda} (t))\Vert_{H^{s_2}\times H^{s_2}} =O( 1),\] \[\lim_{\lambda\to\infty} \Vert(u_{1,\lambda} (0)- u_{-1,\lambda} (0), \rho_{1,\lambda} (0)- \rho_{-1,\lambda} (0))\Vert_{H^{s_1}\times H^{s_2}}= 0,\] and for any late time \(0 < t < \min\{T, \pi\}\), \[ \liminf_{\lambda\to\infty} \Vert(u_{1,\lambda} (t)- u_{-1,\lambda} (t), \rho_{1,\lambda} (t)- \rho_{-1,\lambda} (t))\Vert_{H^{s_1}\times H^{s_2}} \ge C | \sin(t)| > 0, \] for some constant \(C>0\).
two-component fractional Camassa-Holm system
scientific article; zbMATH DE number 7187812
scientific article