Uniform well-posedness and Inviscid limit for the KdV-Burgers and mKdV-Burgers equations on $\mathbb{T}$

Abstract

This article investigates uniform well-posedness and inviscid limit behavior for the periodic Korteweg-de Vries-Burgers (KdV-B) and modified Korteweg-de Vries-Burgers (mKdV-B) equations: \[ \partial_t u + \partial_x^3 u - \varepsilon \partial_x^2 u = \partial_x(u^\alpha), \quad u(0) = \phi, \] where $\alpha = 2, 3$, $\varepsilon \in (0, 1]$ is the diffusion coefficient, and $u : \mathbb{R}^+ \times \mathbb{T} \to \mathbb{R}$ is real-valued. For the KdV-B equation ($\alpha=2$), we establish unconditional uniform global well-posedness in $H^s(\mathbb{T})$ for $s \geq 0$, uniformly for all $\varepsilon \in [0,1]$, without relying on auxiliary function spaces. Furthermore, we prove that for any $s \geq 0$, there exists $T > 0$ such that solutions converge in $C([0,T]; H^s)$ to those of the KdV equation as $\varepsilon \to 0$. For the mKdV-B equation ($\alpha=3$), we establish analogous results--unconditional uniform well-posedness and inviscid limit behavior in $H^s(\mathbb{T})$ for $s \geq 1/2$.