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

TL;DR

This paper proves uniform well-posedness and inviscid limit results for the periodic KdV-Burgers and mKdV-Burgers equations, showing solutions converge to the classical equations as the diffusion parameter vanishes.

Contribution

It establishes the first unconditional uniform well-posedness and inviscid limit results for these equations on the torus, without auxiliary spaces, for specific Sobolev spaces.

Findings

Uniform global well-posedness in H^s for s ≥ 0 (KdV-B) and s ≥ 1/2 (mKdV-B).

Solutions converge to KdV and mKdV solutions as ε → 0.

Results hold uniformly for all ε in [0,1].

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 $s \geq 1/2$.