arXiv subjects
Publications and source records attributed to Elena Rossi.
We prove well posedness and stability in $\mathbf{L}^1$ for a class of mixed hyperbolic-parabolic non linear and non local equations in a bounded domain with no flow along the boundary. While the treatment of boundary conditions for the hyperbolic equation is standard, the extension to $\mathbf{L}^1$ of classical results about parabolic equations with Neumann conditions is here achieved.
Motivated by a vaccination coverage problem, we consider here a zero-sum differential game governed by a differential system consisting of a hyperbolic partial differential equation (PDE) and an ordinary differential equation (ODE). Two players act through their respective controls to influence the evolution of the system with the aim of minimizing their objective functionals $\mathcal F_1$ and $\mathcal F_2$, under the assumption that $\mathcal F_1 +\mathcal F_2 = 0$. First we prove a well posedness and a stability result for the differential system, once the control functions are fixed. Then we introduce the concept of non-anticipating strategies for both players and we consider the associated value functions, which solve two infinite-dimensional Hamilton-Jacobi-Isaacs equations in the viscosity sense.
We prove the well posedness of a class of non linear and non local mixed hyperbolic-parabolic systems in bounded domains, with Dirichlet boundary conditions. In view of control problems, stability estimates on the dependence of solutions on data and parameters are also provided. These equations appear in models devoted to population dynamics or to epidemiology, for instance.
We present a unified framework ensuring well posedness and providing stability estimates to a class of Initial Boundary Value Problems for renewal equations comprising a variety of biological or epidemiological models. This versatility is achieved considering fairly general -- possibly non linear and/or non local -- interaction terms, allowing both low regularity assumptions and independent variables with or without a boundary. In particular, these results also apply, for instance, to a model for the spreading of a Covid like pandemic or other epidemics. Further applications are shown to be covered by the present setting.
In this paper we consider optimal control problems for a parabolic system modeling a therapy, based on oncolytic viruses, for the glioma brain cancer. Using several techniques typical of functional analysis, we prove the global in time well posedness of the control model, the existence of optimal controls for specific objective functionals, which are natural for cancer therapies, and we derive necessary conditions for optimality.
We present a multilane traffic model based on balance laws, where the nonlocal source term is used to describe the lane changing rate. The modelling framework includes the consideration of local and nonlocal flux functions. Based on a Godunov type numerical scheme, we provide BV estimates and a discrete entropy inequality. Together with the $L^1$-contractivity property, we prove existence and uniqueness of weak solutions. Numerical examples show the nonlocal impact compared to local flux functions and local sources.
We develop a time and space dependent predator-prey model. The predators' equation is a non local hyperbolic balance law, while the diffusion of prey obeys a parabolic equation, so that predators "hunt" for prey, while prey diffuse. A control term allows to describe the use of predators as parasitoids to limit the growth of prey-parasites. The general well posedness and stability results here obtained ensure the existence of optimal pest control strategies, as discussed through some numerical integrations. The specific example we have in mind is that of Trichopria drosophilae used to fight against the spreading of Drosophila suzukii.
We study the dynamical evolution of eccentric massive black hole binaries (MBHBs) interacting with unbound stars by means of an extensive set of three body scattering experiments. Compared to previous studies, we extend the investigation down to a MBHB mass ratio of $q=m_2/m_1=10^{-4}$, where $m_1$ and $m_2$ are the masses of the primary and secondary hole respectively. Contrary to a simple extrapolation from higher mass ratios, we find that for $q\lesssim 10^{-3}$ the eccentricity growth rate becomes negative, i.e., the binary {\it circularises} as it shrinks. This behaviour is due to the subset of interacting stars captured in metastable counter-rotating orbits; those stars tend to extract angular momentum from the binary, promoting eccentricity growth for $q>10^{-3}$, but tend to inject angular momentum into the binary driving it towards circularisation for $q<10^{-3}$. The physical origin of this behaviour requires a detailed study of the orbits of this subset of stars and is currently under investigation. Our findings might have important consequences for intermediate MBHs (IMBHs) inspiralling onto MBHs (e.g. a putative $10^3\rm M_{\odot}$ black hole inspiralling onto SgrA$^*$).
We prove the well-posedness of a system of balance laws inspired by [8], describing macro-scopically the traffic flow on a multi-lane road network. Motivated by real applications, we allow for the the presence of space discontinuities both in the speed law and in the number of lanes. This allows to describe a number of realistic situations. Existence of solutions follows from compactness results on a sequence of Godunov's approximations, while $L^1$-stability is obtained by the doubling of variables technique. Some numerical simulations illustrate the behaviour of solutions in sample cases.
In this paper, we focus on finite volume approximation schemes to solve a non-local material flow model in two space dimensions. Based on the numerical discretisation with dimensional splitting, we prove the convergence of the approximate solutions, where the main difficulty arises in the treatment of the discontinuity occurring in the flux function. In particular, we compare a Roe-type scheme to the well-established Lax-Friedrichs method and provide a numerical study highlighting the benefits of the Roe discretisation. Besides, we also prove the L1-Lipschitz continuous dependence on the initial datum, ensuring the uniqueness of the solution.
We consider the initial boundary value problem (IBVP) for a non-local scalar conservation laws in one space dimension. The non-local operator in the flux function is not a mere convolution product, but it is assumed to be aware of boundaries. Introducing an adapted Lax-Friedrichs algorithm, we provide various estimates on the approximate solutions that allow to prove the existence of solutions to the original IBVP. The uniqueness follows from the Lipschitz continuous dependence on initial and boundary data, which is proved exploiting results available for the local IBVP.
We focus on the initial boundary value problem for a general scalar balance law in one space dimension. Under rather general assumptions on the flux and source functions, we prove the well-posedness of this problem and the stability of its solutions with respect to variations in the flux and in the source terms. For both results, the initial and boundary data are required to be bounded functions with bounded total variation. The existence of solutions is obtained from the convergence of a Lax-Friedrichs type algorithm with operator splitting. The stability result follows from an application of Kružkov's doubling of variables technique, together with a careful treatment of the boundary terms.
We prove the stability of entropy weak solutions of a class of scalar conservation laws with non-local flux arising in traffic modelling. We obtain an estimate of the dependence of the solution with respect to the kernel function, the speed and the initial datum. Stability is obtained from the entropy condition through doubling of variable technique. We finally provide some numerical simulations illustrating the dependencies above for some cost functionals derived from traffic flow applications.
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.
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.
We consider four definitions of solution to the initial-boundary value problem for a scalar balance laws in several space dimensions. These definitions are generalised to the same most general framework and then compared. The first aim of this paper is to detail differences and analogies among them. We focus then on the ways the boundary conditions are fulfilled according to each definition, providing also connections among these various modes. The main result is the proof of the equivalence among the presented definitions of solution.
Following the selection of The Gravitational Universe by ESA, and the successful flight of LISA Pathfinder, the LISA Consortium now proposes a 4 year mission in response to ESA's call for missions for L3. The observatory will be based on three arms with six active laser links, between three identical spacecraft in a triangular formation separated by 2.5 million km. LISA is an all-sky monitor and will offer a wide view of a dynamic cosmos using Gravitational Waves as new and unique messengers to unveil The Gravitational Universe. It provides the closest ever view of the infant Universe at TeV energy scales, has known sources in the form of verification binaries in the Milky Way, and can probe the entire Universe, from its smallest scales near the horizons of black holes, all the way to cosmological scales. The LISA mission will scan the entire sky as it follows behind the Earth in its orbit, obtaining both polarisations of the Gravitational Waves simultaneously, and will measure source parameters with astrophysically relevant sensitivity in a band from below $10^{-4}\,$Hz to above $10^{-1}\,$Hz.
We prove the stability with respect to the flux of solutions to initial-boundary value problems for scalar non-autonomous conservation laws in one space dimension. Key estimates are obtained through a careful construction of the solutions.