A Coupled Majda Biello Type System with Damping

Abstract

  In this paper, we study a Majda–Biello-Type system with weak damping terms and $(u_0, v_0)\in \mathcal{L}^{\sigma} (\mathbb{R})\times \mathcal{L}^{\sigma} (\mathbb{R}) $. \begin{equation*} \left\{\begin{array}{l} u_t+u_{x x x}+\beta_1uu_x+\beta_2 u v_x+\beta_3u_x v+\beta_4vv_x=0, \\ v_t+\alpha v_{x x x}+\beta_5 uu_x+\beta_6u v_x+\beta_7 u_xv+\beta_8vv_x=0. \end{array}\right. \end{equation*} The system involves two functions, $\mathfrak{d}_1(x)$ and  $\mathfrak{d}_2(x)$, which serve as damping coefficients. We first establish local well-posedness by applying the fixed point theorem in Fourier restriction spaces $\mathcal{Y}_{\sigma,\kappa}(\mathbb{R}^{2})\times\mathcal{Y}^{\alpha}_{\sigma,\kappa}(\mathbb{R}^{2})$. Subsequently, using an approximate conservation law, we extend the local well-posedness result to a global one, demonstrating that the radius of analyticity of the solutions remains uniformly bounded below by a fixed positive constant for all time.