Crippa, Gianluca and Inversi, Marco and Saffirio, Chiara and Stefani, Giorgio. (2023) Existence and stability of weak solutions of the VlasovPoisson system in localized Yudovich spaces. Preprints Fachbereich Mathematik, 2023 (05).

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Abstract
We consider the VlasovPoisson system both in the repulsive (electrostatic potential) and in the attractive (gravitational potential) cases. In our first main theorem, we prove the uniqueness and the quantitative stability of Lagrangian solutions $f=f(t,x,v)$ whose associated spatial density $\rho_f=\rho_f(t,x)$ is potentially unbounded but belongs to suitable uniformlylocalized Yudovich spaces. This requirement imposes a condition of slow growth on the function $p \mapsto \\rho_f(t,\cdot)\_{L^p}$ uniformly in time. Previous works by Loeper, Miot and HoldingMiot have addressed the cases of bounded spatial density, i.e., $\\rho_f(t,\cdot)\_{L^p} \lesssim 1$, and spatial density such that $\\rho_f(t,\cdot)\_{L^p} \sim p^{1/\alpha}$ for $\alpha\in[1,+\infty)$. Our approach is Lagrangian and relies on an explicit estimate of the modulus of continuity of the electric field and on a secondorder Osgood lemma. It also allows for iteratedlogarithmic perturbations of the linear growth condition. In our second main theorem, we complement the aforementioned result by constructing solutions whose spatial density sharply satisfies such iteratedlogarithmic growth. Our approach relies on realvariable techniques and extends the strategy developed for the Euler equations by the first and fourthnamed authors. It also allows for the treatment of more general equations that share the same structure as the VlasovPoisson system. Notably, the uniqueness result and the stability estimates hold for both the classical and the relativistic VlasovPoisson systems.
Faculties and Departments:  05 Faculty of Science > Departement Mathematik und Informatik > Mathematik > Analysis (Crippa) 05 Faculty of Science > Departement Mathematik und Informatik > Mathematik > Mathematical Physics (Saffirio) 12 Special Collections > Preprints Fachbereich Mathematik 

UniBasel Contributors:  Crippa, Gianluca and Inversi, Marco and Saffirio, Chiara 
Item Type:  Preprint 
Publisher:  Universität Basel 
Language:  English 
edoc DOI:  
Last Modified:  09 Jun 2023 08:44 
Deposited On:  09 Jun 2023 08:44 
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