Local-in-time strong solutions of the homogeneous Landau-Coulomb equation with $L^p$ initial datum
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2023
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| _version_ | 1866913182674583552 |
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| author | Golding, William Loher, Amélie |
| author_facet | Golding, William Loher, Amélie |
| contents | We consider the homogeneous Landau equation with Coulomb potential and general initial data $f_{in} \in L^p$, where $p$ is arbitrarily close to $3/2$. We show the local-in-time existence and uniqueness of smooth solutions for such initial data. The constraint $p > 3/2$ has appeared in several related works and appears to be the minimal integrability assumption achievable with current techniques. We adapt recent ODE methods and conditional regularity results appearing in [arXiv:2303.02281] to deduce new short time $L^p \to L^\infty$ smoothing estimates. These estimates enable us to construct local-in-time smooth solutions for large $L^p$ initial data, and allow us to show directly conditional regularity results for solutions verifying \emph{unweighted} Prodi-Serrin type conditions. As a consequence, we obtain additional stability and uniqueness results for the solutions we construct. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2308_10288 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Local-in-time strong solutions of the homogeneous Landau-Coulomb equation with $L^p$ initial datum Golding, William Loher, Amélie Analysis of PDEs 35B45, 35B60, 35B65, 35K59, 35K55, 82C40, 82D10 We consider the homogeneous Landau equation with Coulomb potential and general initial data $f_{in} \in L^p$, where $p$ is arbitrarily close to $3/2$. We show the local-in-time existence and uniqueness of smooth solutions for such initial data. The constraint $p > 3/2$ has appeared in several related works and appears to be the minimal integrability assumption achievable with current techniques. We adapt recent ODE methods and conditional regularity results appearing in [arXiv:2303.02281] to deduce new short time $L^p \to L^\infty$ smoothing estimates. These estimates enable us to construct local-in-time smooth solutions for large $L^p$ initial data, and allow us to show directly conditional regularity results for solutions verifying \emph{unweighted} Prodi-Serrin type conditions. As a consequence, we obtain additional stability and uniqueness results for the solutions we construct. |
| title | Local-in-time strong solutions of the homogeneous Landau-Coulomb equation with $L^p$ initial datum |
| topic | Analysis of PDEs 35B45, 35B60, 35B65, 35K59, 35K55, 82C40, 82D10 |
| url | https://arxiv.org/abs/2308.10288 |