On the well-posedness of a mathematical model for lithium-ion batteries.

We discuss the well-posedness of a mathematical model that is used in the literature for the simulation of lithium-ion batteries. First, a mathematical model based on a macrohomogeneous approach is presented, following previous work. Then it is shown, from a physical and a mathematical point of view, that a boundary condition widely used in the literature is not correct. Although the errors could be just sign typos (which can be explained as carelessness in the use of d/dx versus d/dn, with n the outward unit vector) and authors using this model probably use the correct boundary condition when they solve it in order to do simulations, readers should be aware of the right choice. Therefore, the deduction of the correct boundary condition is done here, and a mathematical study of the well-posedness of the corresponding problem is presented.

Hα emission fluxes and lithium abundances of low-mass stars in the young open cluster IC 4665

As part of a long term effort to understand pre-main sequence Li burning, we have obtained high resolution spectroscopic observations of 14 late type stars (G0-M1) in the young open cluster IC 4665. Most of the stars have Hα filled-in and Li I absorption, as expected for their young age. From the equivalent widths of Hα emission excess (obtained using the spectral subtraction technique) and the the Li i λ6708 feature, we have derived Hα emission fluxes and photospheric Li abundances. The mean Li abundance of IC 4665 solar-type stars is log N(Li) = 3.1; the same as in other young clusters (α Per, Pleiades) and T Tauri stars. Our results support the conclusions from previous works that PMS Li depletion is very small for masses ∼ 1 M_⨀ . Among the IC 4665 late-G and early K-type stars, there is a spread in Li abundances of about one order of magnitude. The Li-poor IC 4665 members have low Hα excess and vsini≤10. Hence, the Li-activity-rotation connection which has been clearly established in the Pleiades also seems to hold in IC 4665. One M-type IC 4665 star that we have observed does not show Li, implying a very efficient Li depletion as observed in α Per stars of the same spectral type. The level of chromospheric activity and Li depletion among the low-mass stars of IC 4665 is similar to that in the Pleiades. In fact, we note that the Li abundance distributions in several young clusters (α Per, Pleiades, IC 2391, IC 4665) and in post T Tauri stars are strikingly similar. This result suggests that Hα emission and Li abundance not well correlated with age for low-mass stars between 20 and 100 Myr old. We argue that a finer age indicator, the ''LL-clock'', would be the luminosity at which the transition between efficient Li depletion and preservation takes place for fully convective objects. The LL-clock could allow in the near future to derive the relative ages of young open clusters, and clarify the study of PMS evolution of cool stars.