Existence and uniqueness of maximal regular flows for non-smooth vector fields

In this paper we provide a complete analogy between the Cauchy-Lipschitz and the DiPerna-Lions theories for ODE's, by developing a local version of the DiPerna-Lions theory. More precisely, we prove existence and uniqueness of a maximal regular flow for the DiPerna-Lions theory using only local regularity and summability assumptions on the vector field, in analogy with the classical theory, which uses only local regularity assumptions. We also study the behaviour of the ODE trajectories before the maximal existence time. Unlike the Cauchy-Lipschitz theory, this behaviour crucially depends on the nature of the bounds imposed on the spatial divergence of the vector field. In particular, a global assumption on the divergence is needed to obtain a proper blow-up of the trajectories.Comment: 38 page

Peginterferon maintenance therapy in patients with advanced hepatitis C to prevent hepatocellular carcinoma: The plot thickens

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Reply to: "The "pegylated" story continues – Perhaps because both ends (α2a and α2b) are true?"

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HCV genotype 3: An independent predictor of fibrosis progression in chronic hepatitis C

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Twelve-week posttreatment follow-up predicts a sustained virological response to pegylated interferon and ribavirin therapy

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Nonlinear model order reduction for problems with microstructure using mesh informed neural networks

Many applications in computational physics involve approximating problems with microstructure, characterized by multiple spatial scales in their data. However, these numerical solutions are often computationally expensive due to the need to capture fine details at small scales. As a result, simulating such phenomena becomes unaffordable for many-query applications, such as parametrized systems with multiple scale-dependent features. Traditional projection-based reduced order models (ROMs) fail to resolve these issues, even for second-order elliptic PDEs commonly found in engineering applications. To address this, we propose an alternative nonintrusive strategy to build a ROM, that combines classical proper orthogonal decomposition (POD) with a suitable neural network (NN) model to account for the small scales. Specifically, we employ sparse mesh-informed neural networks (MINNs), which handle both spatial dependencies in the solutions and model parameters simultaneously. We evaluate the performance of this strategy on benchmark problems and then apply it to approximate a real-life problem involving the impact of microcirculation in transport phenomena through the tissue microenvironment

Evaporation of multicomponent fuel droplets in buoyancy driven convection

In this work, the evaporation process of multicomponent fuel droplets is analyzed, both from an experimental and numerical point of view. The droplets are hanged on a thin thermocouple against gravity and evaporated in natural convection regime, following the process by means of high speed shadowgraphs. The experimental analyses were performed hierarchically, starting from pure components (n-dodecane and n-hexadecane), then moving to their mixtures. The numerical modeling is performed with the DropletSMOKE++ code, a comprehensive CFD framework for the simulation of 3D evaporating droplets under gravity conditions. The numerical results present a good agreement with the experimental data, especially if compared with the same cased modeled in microgravity conditions. The difference evaporation rate is analyzed as well as the different surface temperature, highlighting the important role of internal and external convection on the evaporation process