Invariant measures for systems of Kolmogorov equations

In this paper we provide sufficient conditions which guarantee the existence of a system of invariant measures for semigroups associated to systems of parabolic differential equations with unbounded coefficients. We prove that these measures are absolutely continuous with respect to the Lebesgue measure and study some of their main properties. Finally, we show that they characterize the asymptotic behaviour of the semigroup at infinity

A mixed type identification problem related to a phase-field model with memory

In this paper we consider an integro-differential system consisting of a parabolic and a hyperbolic equation related to phase transition models. The first equation is integro-differential and of hyperbolic type. It describes the evolution of the temperature and also accounts for memory effects through a memory kernel k via the Gurtin-Pipkin heat flux law. The latter equation, governing the evolution of the order parameter, is semilinear, parabolic and of the fourth order (in space). We prove a local in time existence result and a global uniqueness result for the identification problem consisting in recovering the memory kernel k appearing in the first equation

LpL^p Maximal regularity for vector-valued Schr\"{o}dinger operators

In this paper we consider the vector-valued Schr\"{o}dinger operator $-\Delta + V$, where the potential term $V$ is a matrix-valued function whose entries belong to $L^1_{\rm loc}(\mathbb{R}^d)$ and, for every $x\in\mathbb{R}^d$, $V(x)$ is a symmetric and nonnegative definite matrix, with non positive off-diagonal terms and with eigenvalues comparable each other. For this class of potential terms we obtain maximal inequality in $L^1(\mathbb{R}^d,\mathbb{R}^m).$ Assuming further that the minimal eigenvalue of $V$ belongs to some reverse H\"older class of order $q\in(1,\infty)\cup\{\infty\}$, we obtain maximal inequality in $L^p(\mathbb{R}^d,\mathbb{R}^m)$, for $p$ in between $1$ and some $q$

Elliptic equations in non-smooth plane domains with an application to a parabolic problem

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Erratum to: A Serpent/OpenFOAM coupling for 3D burnup analysis

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diagnosis of vortex induced vibration of a gravity damper

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A Reduced Order Kalman Filter for Computational Fluid-Dynamics Applications

In the last decade, the importance of numerical simulations for the analysis of complex engineering systems, such as thermo-fluid dynamics in nuclear reactors, has grown exponentially. In spite of the large experimental databases available for validation of mathematical models, in order to identify the most suitable one for the system under investigation, the inverse integration of such data into the CFD model is nowadays an ongoing challenge. In addition, such integration could tackle the problem of propagation of epistemic uncertainties, both in the numerical model and in the experimental data. In this framework, the data-assimilation method allows for the dynamic incorporation of observations within the computational model. Perhaps the most famous among these methods, due to its simple implementation and yet robust nature, is the Kalman filter. Although this approach has found success in fields such as weather forecast and geoscience, its application in Computational Fluid-Dynamics (CFD) is still in its first stages. In this setting, a new algorithm based on the integration between the segregated approach, which is the most common method adopted by CFD applications for the solution of the incompressible Navier-Stokes equations, and a Kalman filter modified for fluid-dynamics problems, while preserving mass conservation of the solution, has already been developed and tested in a previous work. Whereas such method is able to robustly integrate experimental data within the numerical model, its computational cost increases with model complexity. In particular, in high-fidelity realistic scenarios the error covariance matrix for the model, which represents the uncertainties associated with it, becomes dense, thus affecting the efficiency and computational cost of the method. For this reason, due to the promised reduction of computational requirements recently investigated, which combines model reduction and data-assimilation, in this work a combination of reduced order model and mass-conservative Kalman filter within a segregated approach for CFD analysis is proposed. The novelty lies in the peculiar formulation of the Kalman filter and how to construct a low-dimensional manifold to approximate, with sufficient accuracy, the high fidelity model. With respect to literature, in which the full-order Kalman filter is applied to a reduced model, the reduction is performed directly on the integrated model in order to obtain a reduced-order Kalman filter already optimised for fluid-dynamics applications. In order to verify the capabilities of this approach, this reduced-order algorithm has been tested against the lid-driven cavity test case.</jats:p

A mass conservative Kalman filter algorithm for computational thermo-fluid dynamics

This paper studies Kalman filtering applied to Reynolds-Averaged Navier-Stokes (RANS) equations for turbulent flow. The integration of the Kalman estimator is extended to an implicit segregated method and to the thermodynamic analysis of turbulent flow, adding a sub-stepping procedure that ensures mass conservation at each time step and the compatibility among the unknowns involved. The accuracy of the algorithm is verified with respect to the heated lid-driven cavity benchmark, incorporating also temperature observations, comparing the augmented prediction of the Kalman filter with the Computational Fluid-Dynamic solution found on a fine grid