edoc: No conditions. Results ordered -Date Deposited. 2024-08-10T18:41:21ZEPrintshttps://edoc.unibas.ch/images/uni-logo.jpghttps://edoc.unibas.ch/2024-06-19T06:54:55Z2024-06-19T06:54:55Zhttps://edoc.unibas.ch/id/eprint/96519This item is in the repository with the URL: https://edoc.unibas.ch/id/eprint/965192024-06-19T06:54:55ZNormal traces and applications to continuity equations on bounded domainsIn this work, we study several properties of the normal Lebesgue trace of vector fields introduced by the second and third author in [18] in the context of the energy conservation for the Euler equations in Onsager-critical classes. Among several properties, we prove that the normal Lebesgue trace satisfies the Gauss-Green identity and, by providing explicit counterexamples, that it is a notion sitting strictly between the distributional one for measure-divergence vector fields and the strong one for $BV$ functions. These results are then applied to the study of the uniqueness of weak solutions for continuity equations on bounded domains, allowing to remove the assumption in [15] of global $BV$ regularity up to the boundary, at least around the portion of the boundary where the characteristics exit the domain or are tangent. The proof relies on an explicit renormalization formula completely characterized by the boundary datum and the positive part of the normal Lebesgue trace. In the case when the characteristics enter the domain, a counterexample shows that achieving the normal trace in the Lebesgue sense is not enough to prevent non-uniqueness, and thus a $BV$ assumption seems to be necessary for the uniqueness of weak solutions. Gianluca CrippaLuigi De RosaMarco InversiMatteo Nesi2023-06-09T08:44:30Z2023-06-09T08:44:30Zhttps://edoc.unibas.ch/id/eprint/94950This item is in the repository with the URL: https://edoc.unibas.ch/id/eprint/949502023-06-09T08:44:30ZExistence and stability of weak solutions of the Vlasov-Poisson system in localized Yudovich spacesWe consider the Vlasov-Poisson 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 uniformly-localized 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 Holding--Miot 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 second-order Osgood lemma. It also allows for iterated-logarithmic 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 iterated-logarithmic growth. Our approach relies on real-variable techniques and extends the strategy developed for the Euler equations by the first and fourth-named authors. It also allows for the treatment of more general equations that share the same structure as the Vlasov-Poisson system. Notably, the uniqueness result and the stability estimates hold for both the classical and the relativistic Vlasov-Poisson systems. Gianluca CrippaMarco InversiChiara SaffirioGiorgio Stefani