Long time behavior of fractional impulsive stochastic differential equations with infinite delay

This paper is first devoted to the local and global existence of mild solutions for a class of fractional impulsive stochastic differential equations with infinite delay driven by both K-valued Q-cylindrical Brownian motion and fractional Brownian motion with Hurst parameter H ∈ (1/2, 1). A general framework which provides an effective way to prove the continuous dependence of mild solutions on initial value is established under some appropriate assumptions. Furthermore, it is also proved the exponential decay to zero of solutions to fractional stochastic impulsive differential equations with infinite delay.European Commission (EC). Fondo Europeo de Desarrollo Regional (FEDER)Ministerio de Economía y Competitividad (MINECO). EspañaConsejería de Innovación, Ciencia y Empresa (Junta de Andalucía

Non-linear Schrodinger equations with singular perturbations and with rough magnetic potentials

In this thesis we discuss thoroughly a class of linear and non-linear Schrodinger equations that arise in various physical contexts of modern relevance. First we work in the scenario where the main linear part of the equation is a singular perturbation of a symmetric pseudo-differential operator, which formally amounts to add to it a potential supported on a finite set of points. A detailed discussion on the rigorous realisations and the main properties of such objects is given when the unperturbed pesudo-differential operator is the fractional Laplacian on R^d. We then consider the relevant special case of singular perturbations of the three-dimensional non-fractional Laplacian: we qualify their smoothing and scattering properties, and characterise their fractional powers and induced Sobolev norms. As a consequence, we are able to establish local and global solution theory for a class of singular Schrodinger equations with convolution-type non-linearity. As a second main playground, we consider non-linear Schrodinger equations with time-dependent, rough magnetic fields, and with local and non-local non-linearities. We include magnetic fields for which the corresponding Strichartz estimates are not available. To this aim, we introduce a suitable parabolic regularisation in the magnetic Laplacian: by exploiting the smoothing properties of the heat-Schrodinger propagator and the mass/energy bounds, we are able to construct global solutions for the approximated problem. Finally, through a compactness argument, we can remove the regularisation and deduce the existence of global, finite energy, weak solutions to the original equation

On the analytical aspects of inertial particle motion

In their seminal 1983 paper, M. Maxey and J. Riley introduced an equation for the motion of a sphere through a fluid. Since this equation features the Basset history integral, the popularity of this equation has broadened the use of a certain form of fractional differential equation to study inertial particle motion. In this paper, we give a comprehensive theoretical analysis of the Maxey-Riley equation. In particular, we build on previous local in time existence and uniqueness results to prove that solutions of the Maxey-Riley equation are global in time. In doing so, we also prove that the notion of a maximal solution extends to this equation. We furthermore prove conditions under which solutions are differentiable at the initial time. By considering the derivative of the solution with respect to the initial conditions, we perform a sensitivity analysis and demonstrate that two inertial trajectories can not meet, as well as provide a control on the growth of the distance between a pair of inertial particles. The properties we prove here for the Maxey-Riley equations are also possessed, mutatis mutandis, by a broader class of fractional differential equations of a similar form