Interaction functionals, Glimm approximations and Lagrangian structure of BV solutions for Hyperbolic Systems of Conservation Laws

This thesis is a contribution to the mathematical theory of Hyperbolic Conservation Laws. Three are the main results which we collect in this work. The first and the second result (denoted in the thesis by Theorem A and Theorem B respectively) deal with the following problem. The most comprehensive result about existence, uniqueness and stability of the solution to the Cauchy problem \begin{equation}\tag{$\mathcal C$} \label{E:abstract} \begin{cases} u_t + F(u)_x = 0, \\u(0, x) = \bar u(x), \end{cases} \end{equation} where $F: \R^N \to \R^N$ is strictly hyperbolic, $u = u(t,x) \in \R^N$, $t \geq 0$, $x \in \R$, \TV(\bar u) \ll 1, can be found in [Bianchini, Bressan 2005], where the well-posedness of \eqref{E:abstract} is proved by means of vanishing viscosity approximations. After the paper [Bianchini, Bressan 2005], however, it seemed worthwhile to develop a \emph{purely hyperbolic} theory (based, as in the genuinely nonlinear case, on Glimm or wavefront tracking approximations, and not on vanishing viscosity parabolic approximations) to prove existence, uniqueness and stability results. The reason of this interest can be mainly found in the fact that hyperbolic approximate solutions are much easier to study and to visualize than parabolic ones. Theorems A and B in this thesis are a contribution to this line of research. In particular, Theorem A proves an estimate on the change of the speed of the wavefronts present in a Glimm approximate solution when two of them interact; Theorem B proves the convergence of the Glimm approximate solutions to the weak admissible solution of \eqref{E:abstract} and provides also an estimate on the rate of convergence. Both theorems are proved in the most general setting when no assumption on $F$ is made except the strict hyperbolicity. The third result of the thesis, denoted by Theorem C, deals with the Lagrangian structure of the solution to \eqref{E:abstract}. The notion of Lagrangian flow is a well-established concept in the theory of the transport equation and in the study of some particular system of conservation laws, like the Euler equation. However, as far as we know, the general system of conservations laws \eqref{E:abstract} has never been studied from a Lagrangian point of view. This is exactly the subject of Theorem C, where a Lagrangian representation for the solution to the system \eqref{E:abstract} is explicitly constructed. The main reasons which led us to look for a Lagrangian representation of the solution of \eqref{E:abstract} are two: on one side, this Lagrangian representation provides the continuous counterpart in the exact solution of \eqref{E:abstract} to the well established theory of wavefront approximations; on the other side, it can lead to a deeper understanding of the behavior of the solutions in the general setting, when the characteristic field are not genuinely nonlinear or linearly degenerate

Non-uniqueness and energy dissipation for 2D Euler equations with vorticity in Hardy spaces

We construct by convex integration examples of energy dissipating solutions to the 2D Euler equations on $\mathbb{R}^2$ with vorticity in the real Hardy space $H^p(\mathbb{R}^2)$, for any $2/3<p<1$

Local Nonuniqueness for Stochastic Transport Equations with Deterministic Drift

We study well-posedness for the stochastic transport equation with transport noise, as introduced by Flandoli, Gubinelli and Priola. We consider periodic solutions in $\rho \in L^{\infty}_{t} L_{x}^{p}$ for divergence-free drifts $u \in L^{\infty}_{t} W_{x}^{\theta, \tilde{p}}$ for a large class of parameters. We prove local-in-time pathwise nonuniqueness and compare them to uniqueness results by Beck, Flandoli, Gubinelli and Maurelli, addressing a conjecture made by these authors, in the case of bounded-in-time drifts for a large range of spatial parameters. To this end, we use convex integration techniques to construct velocity fields $u$ for which several solutions $\rho$ exist in the classes mentioned above. The main novelty lies in the ability to construct deterministic drift coefficients, which makes it necessary to consider a convex integration scheme \textit{with a constraint}, which poses a series of technical difficulties.Comment: 38 pages, 2 figures. Comments very welcome! Added nonuniqueness to stochastic transport-diffusion equation and an appendix sketching a proof of uniqueness of the stochastic transport equation in an LPS parameter range. Corrected a few typo

Evaluation of primary stability of single implants placed in fresh extraction sockets: a clinical trial

ABSTRACTDental implants have been used for the last 20 years. With the latest modern developments, however, minimally invasive protocols and immediate implants are currently used. The aim of this study was to evaluate the primary stability of a new implant design. Thirty immediate implants were placed and they all achieved successful osseointegration. Primary stability was reached with all the implants after the first apical threads. Within the limitations of the present study, the immediate implant approach seems to be a predictable treatment option, especially in conjunction with a specifically designed implant system

ARE: Ada Rendering Engine

E' ormai pratica diffusa, nello sviluppo di applicazioni web, l'utilizzo di template e di potenti template engine per automatizzare la generazione dei contenuti da presentare all'utente. Tuttavia a volte la potenza di tali engine è ottenuta mescolando logica e interfaccia, introducendo linguaggi diversi da quelli di descrizione della pagina, o addirittura inventando nuovi linguaggi dedicati.ARE (ADA Rendering Engine) è pensato per gestire l'intero flusso di creazione del contenuto HTML/XHTML dinamico, la selezione del corretto template, CSS, JavaScript e la produzione dell'output separando completamente logica e interfaccia. I templates utilizzati sono puro HTML senza parti in altri linguaggi, e possono quindi essere gestiti e visualizzati autonomamente. Il codice HTML generato è uniforme e parametrizzato.E' composto da due moduli, CORE (Common Output Rendering Engine) e ALE (ADA Layout Engine).Il primo (CORE) viene utilizzato per la generazione OO degli elementi del DOM ed eÌ pensato per aiutare lo sviluppatore nella produzione di codice valido rispetto al DTD utilizzato. CORE genera automaticamente gli elementi del DOM in base al DTD impostato nella configurazioneIl secondo (ALE) viene utilizzato come template engine per selezionare automaticamente in base ad alcuni parametri (modulo, profilo utente, tipologia del nodo, del corso, preferenze di installazione) il template HTML, i CSS e i file JavaScript appropriati. ALE permette di usare templates di default e microtemplates ricorsivi per semplificare il lavoro del grafico.I due moduli possono in ogni caso essere utilizzati indipendentemente l'uno dall'altro. E' possibile generare e renderizzare una pagina HTML utilizzando solo CORE oppure inviare gli oggetti CORE al template engine ALE che provvede a renderizzare la pagina HTML. Viceversa eÌ possibile generare HTML senza utilizzare CORE ed inviarlo al template engine ALECORE è alla prima release ed è già utilizzato all'interno dei progetti ADA e MAKO.Tra gli sviluppi previsti: il completamento della libreria per diverse DTD; la creazione di classi di livello superiori che automatizzino compiti ripetitivi (creazione di form, tabelle, etc)