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| Format: | Preprint |
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2021
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| Online Access: | https://arxiv.org/abs/2109.05299 |
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| _version_ | 1866908794546552832 |
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| author | Hu, Bingyang Xu, Dinghua Zhang, Yeyu |
| author_facet | Hu, Bingyang Xu, Dinghua Zhang, Yeyu |
| contents | In this paper, we consider the advective unstable Cahn-Hilliard equation in 2D with shear flow: \begin{equation*}
\begin{cases} u_t+Av_1(y) \partial_x u+\varepsilon Δ^2 u= Δ(a u^3+ b u^2) \quad & \quad \textrm{on} \quad \mathbb T^2; \\ \\ u \ \textrm{periodic} \quad & \quad \textrm{on} \quad \partial \mathbb T^2, \end{cases} \end{equation*} with an initial data $u_0 \in H_0^2(\mathbb T^2)$, where $\mathbb T^2$ is the two-dimensional torus, $A, \varepsilon>0$, $a<0$, $b \in \mathbb R$. Under the assumption that the shear has a finite number of critical points and there are linearly growing modes only in the direction of the shear, we show the $L^2$-energy of the solutions to such problems converges expotentially to zero, if in addition, both $|a|$ and $\left\| \int_{\mathbb T} u_0(x, \cdot ) dx \right\|_{L_y^2}$ are sufficiently small. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2109_05299 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Global Existence for the unstable Cahn-Hilliard equation in 2D with a Shear Flow Hu, Bingyang Xu, Dinghua Zhang, Yeyu Analysis of PDEs 35K25, 35K58, 76E06, 76F25 In this paper, we consider the advective unstable Cahn-Hilliard equation in 2D with shear flow: \begin{equation*} \begin{cases} u_t+Av_1(y) \partial_x u+\varepsilon Δ^2 u= Δ(a u^3+ b u^2) \quad & \quad \textrm{on} \quad \mathbb T^2; \\ \\ u \ \textrm{periodic} \quad & \quad \textrm{on} \quad \partial \mathbb T^2, \end{cases} \end{equation*} with an initial data $u_0 \in H_0^2(\mathbb T^2)$, where $\mathbb T^2$ is the two-dimensional torus, $A, \varepsilon>0$, $a<0$, $b \in \mathbb R$. Under the assumption that the shear has a finite number of critical points and there are linearly growing modes only in the direction of the shear, we show the $L^2$-energy of the solutions to such problems converges expotentially to zero, if in addition, both $|a|$ and $\left\| \int_{\mathbb T} u_0(x, \cdot ) dx \right\|_{L_y^2}$ are sufficiently small. |
| title | Global Existence for the unstable Cahn-Hilliard equation in 2D with a Shear Flow |
| topic | Analysis of PDEs 35K25, 35K58, 76E06, 76F25 |
| url | https://arxiv.org/abs/2109.05299 |