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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2403.10458 |
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| _version_ | 1866908315671330816 |
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| author | Granero-Belinchón, Rafael Magliocca, Martina Ortega, Alejandro |
| author_facet | Granero-Belinchón, Rafael Magliocca, Martina Ortega, Alejandro |
| contents | In this paper, we prove a couple of new nonlinear functional inequalities of Sobolev type akin to the logarithmic Sobolev inequality. In particular, one of the inequalities reads $$ \int_{\mathbb{S}^1}\arctan\left(\frac{\partial_x u}{u}\right)\partial_xu \,dx\geq \arctan\left(\|u(t)\|_{\dot{W}^{1,1}(\mathbb{S}^1)}\right)\|u(t)\|_{\dot{W}^{1,1}(\mathbb{S}^1)}. $$ Then, these inequalities are used in the study of the nonlinear \emph{arctan}-fast diffusion equation $$ \partial_t u-\partial_x\arctan\left(\frac{\partial_x u}{u}\right)=0. $$ For this highly nonlinear PDE we establish a number of well-posedness results and qualitative properties. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_10458 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | New functional inequalities with applications to the arctan-fast diffusion equation Granero-Belinchón, Rafael Magliocca, Martina Ortega, Alejandro Analysis of PDEs In this paper, we prove a couple of new nonlinear functional inequalities of Sobolev type akin to the logarithmic Sobolev inequality. In particular, one of the inequalities reads $$ \int_{\mathbb{S}^1}\arctan\left(\frac{\partial_x u}{u}\right)\partial_xu \,dx\geq \arctan\left(\|u(t)\|_{\dot{W}^{1,1}(\mathbb{S}^1)}\right)\|u(t)\|_{\dot{W}^{1,1}(\mathbb{S}^1)}. $$ Then, these inequalities are used in the study of the nonlinear \emph{arctan}-fast diffusion equation $$ \partial_t u-\partial_x\arctan\left(\frac{\partial_x u}{u}\right)=0. $$ For this highly nonlinear PDE we establish a number of well-posedness results and qualitative properties. |
| title | New functional inequalities with applications to the arctan-fast diffusion equation |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2403.10458 |