Well-posedness and $L^1-L^p$ Smoothing Effect of the Porous Media Equation under Poincaré Inequality
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866916009961586688 |
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| author | Sun, Lukang |
| author_facet | Sun, Lukang |
| contents | We study the Cauchy problem for a weighted porous medium equation on $\R$ associated with a Gibbs probability measure $π=e^{-V}$. Under a Poincaré inequality for $π$ and the convexity assumption on $V$, we prove well-posedness and uniqueness of non-negative weak solutions with initial data in $L^1(\R,π)$. We also establish an $L^1$--$L^p$ smoothing effect at every positive time. More precisely, for every admissible $p>1$, we show that the logarithm of the ratio between the $L^p(\R,π)$ norm of the solution and its conserved $L^1(\R,π)$ mass first decays at a super-exponential rate and then decays exponentially to zero. In particular, even if the initial datum belongs only to $L^1(\R,π)$, the solution belongs to $L^p(\R,π)$ for every finite $p>1$ and every $t>0$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_05722 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Well-posedness and $L^1-L^p$ Smoothing Effect of the Porous Media Equation under Poincaré Inequality Sun, Lukang Analysis of PDEs We study the Cauchy problem for a weighted porous medium equation on $\R$ associated with a Gibbs probability measure $π=e^{-V}$. Under a Poincaré inequality for $π$ and the convexity assumption on $V$, we prove well-posedness and uniqueness of non-negative weak solutions with initial data in $L^1(\R,π)$. We also establish an $L^1$--$L^p$ smoothing effect at every positive time. More precisely, for every admissible $p>1$, we show that the logarithm of the ratio between the $L^p(\R,π)$ norm of the solution and its conserved $L^1(\R,π)$ mass first decays at a super-exponential rate and then decays exponentially to zero. In particular, even if the initial datum belongs only to $L^1(\R,π)$, the solution belongs to $L^p(\R,π)$ for every finite $p>1$ and every $t>0$. |
| title | Well-posedness and $L^1-L^p$ Smoothing Effect of the Porous Media Equation under Poincaré Inequality |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2504.05722 |