Velocity averaging under minimal conditions for deterministic and stochastic kinetic equations with irregular drift
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| author | Erceg, Marko Karlsen, Kenneth H. Mitrović, Darko |
| author_facet | Erceg, Marko Karlsen, Kenneth H. Mitrović, Darko |
| contents | This study investigates the $L^1_{\operatorname{loc}}$ compactness of velocity averages of sequences of solutions $\{u_n\}$ for a class of kinetic equations. The equations are examined within both deterministic and stochastic heterogeneous environments. The primary objective is to deduce velocity averaging results under conditions on $u_n$ and the drift ${\mathfrak f}={\mathfrak f}(t,{\boldsymbol x},{\boldsymbol λ})$ that are more lenient than those stipulated in previous studies. The main outcome permits the inclusion of highly irregular drift vectors ${\mathfrak f} \in L^q$ that adhere to a general non-degeneracy condition. Moreover, the sequence $\{u_n\}$ is uniformly bounded in $L^p$ -- for an exponent $p$ allowed to be strictly smaller than $2$ -- under the requirement $\frac{1}{p} + \frac{1}{q} < 1$. Resolving the matter of strong compactness in velocity averages, considering these assumptions, has remained an open problem for a long time. The cornerstone of our work's progress lies in the strategic employment of the broader concept of $H$-distributions, moving beyond the traditional reliance on $H$-measures. Notably, our study represents one of the first significant uses of $H$-distributions in this context. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2311_01234 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Velocity averaging under minimal conditions for deterministic and stochastic kinetic equations with irregular drift Erceg, Marko Karlsen, Kenneth H. Mitrović, Darko Analysis of PDEs Probability 35B65, 35R60, 35Q83, 42B37 This study investigates the $L^1_{\operatorname{loc}}$ compactness of velocity averages of sequences of solutions $\{u_n\}$ for a class of kinetic equations. The equations are examined within both deterministic and stochastic heterogeneous environments. The primary objective is to deduce velocity averaging results under conditions on $u_n$ and the drift ${\mathfrak f}={\mathfrak f}(t,{\boldsymbol x},{\boldsymbol λ})$ that are more lenient than those stipulated in previous studies. The main outcome permits the inclusion of highly irregular drift vectors ${\mathfrak f} \in L^q$ that adhere to a general non-degeneracy condition. Moreover, the sequence $\{u_n\}$ is uniformly bounded in $L^p$ -- for an exponent $p$ allowed to be strictly smaller than $2$ -- under the requirement $\frac{1}{p} + \frac{1}{q} < 1$. Resolving the matter of strong compactness in velocity averages, considering these assumptions, has remained an open problem for a long time. The cornerstone of our work's progress lies in the strategic employment of the broader concept of $H$-distributions, moving beyond the traditional reliance on $H$-measures. Notably, our study represents one of the first significant uses of $H$-distributions in this context. |
| title | Velocity averaging under minimal conditions for deterministic and stochastic kinetic equations with irregular drift |
| topic | Analysis of PDEs Probability 35B65, 35R60, 35Q83, 42B37 |
| url | https://arxiv.org/abs/2311.01234 |