Commun. Math. Anal. Appl., 4 (2025), pp. 202-233.
Published online: 2025-06
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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 $\int_{\mathbb{R}^3} ((u−v)×B)f {\rm d}v.$
}, issn = {2790-1939}, doi = {https://doi.org/10.4208/cmaa.2025-0003}, url = {http://global-sci.org/intro/article_detail/cmaa/24122.html} }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 $\int_{\mathbb{R}^3} ((u−v)×B)f {\rm d}v.$