Approximate solution of the Kuramoto-Shivashinsky equation on an unbounded domain (Q1696645)

Let \(\varepsilon\), \(\gamma\), and \(\nu\) be three positive parameters and consider the solution \(\psi_\varepsilon\) to the one-dimensional Kuramoto-Sivashinky (KS) equation \[ \partial_t \psi_\varepsilon = - \left( 1 + \partial_x^2 \right)^2 \psi_\varepsilon + \varepsilon^2 \nu \psi_\varepsilon + \gamma \psi_\varepsilon \partial_x \psi_\varepsilon\;, \qquad (t,x)\in (0,\infty)\times\mathbb{R}\;, \] with initial condition \(\psi_{0,\varepsilon}\). Let \(A\) be the (complex-valued) solution to the associated modulation equation \[ \partial_T A = 4 \partial_X^2 A + \nu A - \frac{\gamma^2}{9} |A|^2 A\;, \qquad (T,X)\in (0,\infty)\times\mathbb{R}\;, \] with initial condition \(A_0\) and assume that it belongs to \(C([0,T_0];H^{2\alpha,2}(\mathbb{R}))\) for some \(\alpha>1/2\) and \(T_0>0\). If \[ \sup_{x\in\mathbb{R}} \left| \psi_{0,\varepsilon}(x) - \varepsilon A_0(\varepsilon x) e^{ix} - \varepsilon \overline{A_0}(\varepsilon x) e^{-ix} \right| \leq d \varepsilon^{\min\{ \alpha-1/2 , 3/2\}} \] for some \(d>0\), then it is shown that \[ \sup_{(t,x)\in[0,T_0\varepsilon^{-2}]\times \mathbb{R}} \left| \psi_{\varepsilon}(t,x) - \varepsilon A(\varepsilon^2 t,\varepsilon x) e^{ix} - \varepsilon \overline{A}(\varepsilon^2 t, \varepsilon x) e^{-ix} \right| \leq C \varepsilon^{\min\{ \alpha-1/2 , 3/2\}} \] for some \(C>0\) depending on \(d\) and \(\|A\|_{C([0,T_0];H^{2\alpha,2}(\mathbb{R}))}\). The proof relies on a rescaling of the KS equation and semigroup estimates.