Classical limit of the Nelson model

Since the development of quantum mechanics it has been natural to analyze the connection between classical and quantum mechanical descriptions of physical systems. In particular one should expect that in some sense when quantum mechanical effects becomes negligible the system will behave like it is dictated by classical mechanics. One famous relation between classical and quantum theory is due to Ehrenfest. This result was later developed and put on firm mathematical foundations by Hepp. He proved that matrix elements of bounded functions of quantum observables between suitable coherents states (that depend on Planck's constant h) converge to classical values evolving according to the expected classical equations when h goes to zero. His results were later generalized by Ginibre and Velo to bosonic systems with infinite degrees of freedom and scattering theory. In this thesis we study the classical limit of Nelson model, that describes non relativistic particles, whose evolution is dictated by Schrödinger equation, interacting with a scalar relativistic field, whose evolution is dictated by Klein-Gordon equation, by means of a Yukawa-type potential. The classical limit is a mean field and weak coupling limit. We proved that the transition amplitude of a creation or annihilation operator, between suitable coherent states, converges in the classical limit to the solution of the system of differential equations that describes the classical evolution of the theory. The quantum evolution operator converges to the evolution operator of fluctuations around the classical solution. Transition amplitudes of normal ordered products of creation and annihilation operators between coherent states converge to suitable products of the classical solutions. Transition amplitudes of normal ordered products of creation and annihilation operators between fixed particle states converge to an average of products of classical solutions, corresponding to different initial conditions.

New knowledge, diagnostic and control tools to limit the impact of viral nervous necrosis (VNN) in the Mediterranean aquaculture sector

The role of aquaculture in satisfying the global seafood demand is essential. The expansion of the aquaculture sector and the intensification of its activities have enhanced the circulation of infectious agents. Among these, the nervous necrosis virus (NNV) represents the most widespread in the Mediterranean basin. The NNV is responsible for a severe neuropathological condition named viral nervous necrosis (VNN), impacting hugely on fish farms due to the serious disease-associated losses. Therefore, it is fundamental to develop new strategies to limit the impact of VNN in this area, interconnecting several aspects of disease management, diagnosis and prevention. This PhD thesis project, focusing on aquatic animals’ health, deals with these topics. The first two chapters expand the knowledge on VNN epidemiology and distribution, showing the possibility of interspecies transmission, persistent infections and a potential carrier role for invertebrates. The third study expands the horizon of VNN diagnosis, by developing a quick and affordable multiplex RT-PCR able to detect and simultaneously discriminate between NNV variants, reducing considerably the time and costs of genotyping. The fourth study, with the development of a fluorescent in situ hybridization technique and its application to aquatic vertebrates and invertebrates’ tissues, contributes to expand the knowledge on NNV distribution at cellular level, localizing also the replication site of the virus. Finally, the last study dealing with an in vitro evaluation of the NNV susceptibility to a commercial biocide, stress the importance to implement proper disinfectant procedures in fish farms to prevent virus spread and disease outbreaks.

Incompressible limit and well-posedness of PDE models of tissue growth

Both compressible and incompressible porous medium models are used in the literature to describe the mechanical aspects of living tissues. Using a stiff pressure law, it is possible to build a link between these two different representations. In the incompressible limit, compressible models generate free boundary problems where saturation holds in the moving domain. Our work aims at investigating the stiff pressure limit of reaction-advection-porous medium equations motivated by tumor development. Our first study concerns the analysis and numerical simulation of a model including the effect of nutrients. A coupled system of equations describes the cell density and the nutrient concentration and the derivation of the pressure equation in the stiff limit was an open problem for which the strong compactness of the pressure gradient is needed. To establish it, we use two new ideas: an L3-version of the celebrated Aronson-Bénilan estimate, and a sharp uniform L4-bound on the pressure gradient. We further investigate the sharpness of this bound through a finite difference upwind scheme, which we prove to be stable and asymptotic preserving. Our second study is centered around porous medium equations including convective effects. We are able to extend the techniques developed for the nutrient case, hence finding the complementarity relation on the limit pressure. Moreover, we provide an estimate of the convergence rate at the incompressible limit. Finally, we study a multi-species system. In particular, we account for phenotypic heterogeneity, including a structured variable into the problem. In this case, a cross-(degenerate)-diffusion system describes the evolution of the phenotypic distributions. Adapting methods recently developed in the context of two-species systems, we prove existence of weak solutions and we pass to the incompressible limit. Furthermore, we prove new regularity results on the total pressure, which is related to the total density by a power law of state.