Stochastic Homogenization of Parabolic Equations with Lower-order Terms
Fuente:
arXiv
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866910573615120384 |
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| author | Yang, Man |
| author_facet | Yang, Man |
| contents | The study of homogenization results has long been a central focus in the field of mathematical analysis, particularly for equations without lower-order terms. However, the importance of studying homogenization results for parabolic equations with lower-order terms cannot be understated. In this study, we aim to extend the analysis to homogenization for the general parabolic equation with random coefficients:
\begin{equation*}
\partial_{t}p^ε-\nabla\cdot\left(\mathbf{a}\left( \dfrac{x}ε,\dfrac{t}{ε^2}\right)\nabla p^ε\right)-\mathbf{b}\left( \dfrac{x}ε,\dfrac{t}{ε^2}\right)\nabla p^ε-\mathbf{d}\left( \dfrac{x}ε,\dfrac{t}{ε^2}\right) p^ε=0.
\end{equation*}
Moreover, we establish the Caccioppoli inequality and Meyers estimate for the generalized parabolic equation. By using the generalized Meyers estimate, we get the weak convergence of $p^ε$ in $H^1$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_12204 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Stochastic Homogenization of Parabolic Equations with Lower-order Terms Yang, Man Analysis of PDEs Probability The study of homogenization results has long been a central focus in the field of mathematical analysis, particularly for equations without lower-order terms. However, the importance of studying homogenization results for parabolic equations with lower-order terms cannot be understated. In this study, we aim to extend the analysis to homogenization for the general parabolic equation with random coefficients: \begin{equation*} \partial_{t}p^ε-\nabla\cdot\left(\mathbf{a}\left( \dfrac{x}ε,\dfrac{t}{ε^2}\right)\nabla p^ε\right)-\mathbf{b}\left( \dfrac{x}ε,\dfrac{t}{ε^2}\right)\nabla p^ε-\mathbf{d}\left( \dfrac{x}ε,\dfrac{t}{ε^2}\right) p^ε=0. \end{equation*} Moreover, we establish the Caccioppoli inequality and Meyers estimate for the generalized parabolic equation. By using the generalized Meyers estimate, we get the weak convergence of $p^ε$ in $H^1$. |
| title | Stochastic Homogenization of Parabolic Equations with Lower-order Terms |
| topic | Analysis of PDEs Probability |
| url | https://arxiv.org/abs/2408.12204 |