IBVPs for Scalar Conservation Laws with Time Discontinuous Fluxes

The initial boundary value problem for a class of scalar non autonomous conservation laws in one space dimension is proved to be well posed and stable with respect to variations in the flux. Targeting applications to traffic, the regularity assumptions on the flow are extended to a merely $\mathbf{L}^{\infty}$ dependence on time. These results ensure, for instance, the well posedness of a class of vehicular traffic models with time dependent speed limits. A traffic management problem is then shown to admit an optimal solution

Non Local Conservation Laws in Bounded Domains

The well posedness for a class of non local systems of conservation laws in a bounded domain is proved and various stability estimates are provided. This construction is motivated by the modelling of crowd dynamics, which also leads to define a non local operator adapted to the presence of a boundary. Numerical integrations show that the resulting model provides qualitatively reasonable solutions

Hyperbolic predators vs parabolic preys

We present a nonlinear predator-prey system consisting of a nonlocal conservation law for predators coupled with a parabolic equation for preys. The drift term in the predators' equation is a nonlocal function of the prey density, so that the movement of predators can be directed towards region with high prey density. Moreover, Lotka-Volterra type right hand sides describe the feeding. A theorem ensuring existence, uniqueness, continuous dependence of weak solutions and various stability estimates is proved, in any space dimension. Numerical integrations show a few qualitative features of the solutions.Comment: 35 pages, 7 figure

Bad prospects for the detection of giant stars' tidal disruption: effect of the ambient medium on bound debris

Most massive galaxies are thought to contain a supermassive black hole in their centre surrounded by a tenuous gas environment, leading to no significant emission. In these quiescent galaxies, tidal disruption events represent a powerful detection method for the central black hole. Following the disruption, the stellar debris evolves into an elongated gas stream, which partly falls back towards the disruption site and accretes onto the black hole producing a luminous flare. Using an analytical treatment, we investigate the interaction between the debris stream and the gas environment of quiescent galaxies. Although we find dynamical effects to be negligible, we demonstrate that Kelvin-Helmholtz instability can lead to the dissolution of the stream into the ambient medium before it reaches the black hole, likely dimming the associated flare. This result is robust against the presence of a typical stellar magnetic field and fast cooling within the stream. Furthermore, we find this effect to be enhanced for disruptions involving more massive black holes and/or giant stars. Consequently, although disruptions of evolved stars have been proposed as a useful probe of black holes with masses $\gtrsim 10^8 \, M_{\odot}$, we argue that the associated flares are likely less luminous than expected.Comment: 8 pages, 6 figures, accepted for publication in MNRA

Ejection and Capture Dynamics in Restricted Three-body Encounters

We study the tidal disruption of binaries by a massive point mass (e.g., the black hole at the Galactic center), and we discuss how the ejection and capture preference between unequal-mass binary members depends on which orbit they approach the massive object. We show that the restricted three-body approximation provides a simple and clear description of the dynamics. The orbit of a binary with mass m around a massive object M should be almost parabolic with an eccentricity of |1 – e| ≲ (m/M)^(1/3) ≪ 1 for a member to be captured, while the other is ejected. Indeed, the energy change of the members obtained for a parabolic orbit can be used to describe non-parabolic cases. If a binary has an encounter velocity much larger than (M/m)^(1/3) times the binary rotation velocity, it would be abruptly disrupted, and the energy change at the encounter can be evaluated in a simple disruption model. We evaluate the probability distributions for the ejection and capture of circular binary members and for the final energies. In principle, for any hyperbolic (elliptic) orbit, the heavier member has more chance to be ejected (captured), because it carries a larger fraction of the orbital energy. However, if the orbital energy is close to zero, the difference between the two members becomes small, and there is practically no ejection and capture preferences. The preference becomes significant when the orbital energy is comparable to the typical energy change at the encounter. We discuss its implications to hypervelocity stars and irregular satellites around giant planets

Streams collision as possible precursor of double tidal disruption events

The rate of tidal disruption events (TDEs) can vary by orders of magnitude depending on the environment and the mechanism that launches the stars towards the black hole’s vicinity. For the largest rates, two disruptions can take place shortly one after the other in a double TDE. In this case, the two debris streams may collide with each other before falling back to the black hole resulting in an electromagnetic emission that is absent from single TDEs. We analytically evaluate the conditions for this streams collision to occur. It requires that the difference in pericentre location between the two disruptions makes up for the time delay between them. In addition, the width of the streams must compensate for the vertical offset induced by the inclination of their orbital planes. If the double TDE happens following the tidal separation of a binary, we find that the streams can collide with a probability as high as 44 per cent⁠. We validate our analytical conditions for streams collision through hydrodynamical simulations and find that the associated shocks heat the gas significantly. If photons are able to rapidly escape, a burst of radiation ensues lasting a few days with a luminosity ∼10^(43(ergs^(−1)⁠, most likely in the optical band. This signal represents a precursor to the main flare of TDEs that could in particular be exploited to determine the efficiency of disc formation from the stellar debris