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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2403.15963 |
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| _version_ | 1866911810817359872 |
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| author | Kosygina, Elena Yilmaz, Atilla |
| author_facet | Kosygina, Elena Yilmaz, Atilla |
| contents | We establish homogenization for nondegenerate viscous Hamilton-Jacobi equations in one space dimension when the diffusion coefficient $a(x,ω) > 0$ and the Hamiltonian $H(p,x,ω)$ are general stationary ergodic processes in $x$. Our result is valid under mild regularity assumptions on $a$ and $H$ plus standard coercivity and growth assumptions (in $p$) on the latter. In particular, we impose neither any additional condition on the law of the media nor any shape restriction on the graph of $p\mapsto H(p,x,ω)$. Our approach consists of two main steps: (i) constructing a suitable candidate $\overline{H}$ for the effective Hamiltonian; (ii) proving homogenization. In the first step, we work with the set $E$ of all points at which $\overline{H}$ is naturally determined by correctors with stationary derivatives. We prove that $E$ is a closed subset of $\mathbb{R}$ that is unbounded from above and below, and, if $E\neq\mathbb{R}$, then $\overline{H}$ can be extended continuously to $\mathbb{R}$ by setting it to be constant on each connected component of $E^c$. In the second step, we use a key bridging lemma, comparison arguments and several general results to verify that homogenization holds with this $\overline{H}$ as the effective Hamiltonian. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_15963 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Homogenization of nonconvex viscous Hamilton-Jacobi equations in stationary ergodic media in one dimension Kosygina, Elena Yilmaz, Atilla Analysis of PDEs Probability 35B27, 35F21, 60G10 We establish homogenization for nondegenerate viscous Hamilton-Jacobi equations in one space dimension when the diffusion coefficient $a(x,ω) > 0$ and the Hamiltonian $H(p,x,ω)$ are general stationary ergodic processes in $x$. Our result is valid under mild regularity assumptions on $a$ and $H$ plus standard coercivity and growth assumptions (in $p$) on the latter. In particular, we impose neither any additional condition on the law of the media nor any shape restriction on the graph of $p\mapsto H(p,x,ω)$. Our approach consists of two main steps: (i) constructing a suitable candidate $\overline{H}$ for the effective Hamiltonian; (ii) proving homogenization. In the first step, we work with the set $E$ of all points at which $\overline{H}$ is naturally determined by correctors with stationary derivatives. We prove that $E$ is a closed subset of $\mathbb{R}$ that is unbounded from above and below, and, if $E\neq\mathbb{R}$, then $\overline{H}$ can be extended continuously to $\mathbb{R}$ by setting it to be constant on each connected component of $E^c$. In the second step, we use a key bridging lemma, comparison arguments and several general results to verify that homogenization holds with this $\overline{H}$ as the effective Hamiltonian. |
| title | Homogenization of nonconvex viscous Hamilton-Jacobi equations in stationary ergodic media in one dimension |
| topic | Analysis of PDEs Probability 35B27, 35F21, 60G10 |
| url | https://arxiv.org/abs/2403.15963 |