@Article{CMAA-4-2,
author = {Li, Fucai and Ni, Jinkai},
title = {Local Well-Posedness and Weak-Strong Uniqueness to the Incompressible Vlasov-MHD System},
journal = {Communications in Mathematical Analysis and Applications},
year = {2025},
volume = {4},
number = {2},
pages = {202--233},
abstract = {<p style="text-align: justify;">In this paper, we investigate the local well-posedness of strong solutions and weak-strong uniqueness property to the incompressible Vlasov-magnetohydrodynamic (Vlasov-MHD) model in $\mathbb{R}^3_x.$ This model consists of
a Vlasov equation and the incompressible MHD equations, which interact via
the so-called Lorentz force. We first establish the local well-posedness of
a strong solution $(f ,u,B)$ by utilizing the delicate energy method for the iteration sequence of approximate solutions, provided that the initial data $(f_0,u_0,B_0)$ are $H^2$-regular and $f_0(x,v)$ has a compact support in the velocity $v.$ We further
demonstrate the weak-strong uniqueness property of solutions if $f_0(x,v)∈L^1∩L^∞(\mathbb{R}^3_x×\mathbb{R}^3_v),$ and thereby establish a rigorous connection between the strong
and weak solutions to the Vlasov-MHD system. The absence of a dissipation
structure in the Vlasov equation and the presence of the strong trilinear coupling term $((u−v)×B)f$ in the model pose significant challenges in deriving
our results. To address these issues, we employ the method of characteristics
to estimate the size of the support of $f,$ which enables us to overcome the difficulties associated with evaluating the integral $\int_{\mathbb{R}^3} ((u−v)×B)f {\rm d}v.$</p>},
issn = {2790-1939},
doi = {https://doi.org/10.4208/cmaa.2025-0003},
url = {https://global-sci.com/article/91920/local-well-posedness-and-weak-strong-uniqueness-to-the-incompressible-vlasov-mhd-system}
}