Uniform well-posedness and Inviscid limit for the KdV-Burgers and mKdV-Burgers equations on $\mathbb{T}$
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866908473159057408 |
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| author | Li, Xintong Li, Yongsheng |
| author_facet | Li, Xintong Li, Yongsheng |
| contents | 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: \[ \partial_t u + \partial_x^3 u - \varepsilon \partial_x^2 u = \partial_x(u^α), \quad u(0) = ϕ, \] where $α= 2, 3$, $\varepsilon \in (0, 1]$ is the diffusion coefficient, and $u : \mathbb{R}^+ \times \mathbb{T} \to \mathbb{R}$ is real-valued.
For the KdV-B equation ($α=2$), 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 ($α=3$), we establish analogous results--unconditional uniform well-posedness and inviscid limit behavior in $H^s(\mathbb{T})$ for $s \geq 1/2$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_06147 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Uniform well-posedness and Inviscid limit for the KdV-Burgers and mKdV-Burgers equations on $\mathbb{T}$ Li, Xintong Li, Yongsheng Analysis of PDEs 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: \[ \partial_t u + \partial_x^3 u - \varepsilon \partial_x^2 u = \partial_x(u^α), \quad u(0) = ϕ, \] where $α= 2, 3$, $\varepsilon \in (0, 1]$ is the diffusion coefficient, and $u : \mathbb{R}^+ \times \mathbb{T} \to \mathbb{R}$ is real-valued. For the KdV-B equation ($α=2$), 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 ($α=3$), we establish analogous results--unconditional uniform well-posedness and inviscid limit behavior in $H^s(\mathbb{T})$ for $s \geq 1/2$. |
| title | Uniform well-posedness and Inviscid limit for the KdV-Burgers and mKdV-Burgers equations on $\mathbb{T}$ |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2501.06147 |