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Luisa Consiglieri

Publications and source records attributed to Luisa Consiglieri.

18 recordsLinked to original sources
arXiv2025

Exact solutions to the cancer laser ablation modeling

The present paper deals with the study of the fluence rate over both healthy and tumor tissues in the presence of focal laser ablation (FLA). We propose new analytical solutions for the coupled partial differential equations (PDE) system, which includes the transport equation modeling the light penetration into biological tissue, the bioheat equation modeling the heat transfer and its respective damage. The present building could be the first step to the knowledge of the mathematical framework for biothermophysical problems, as well as the main key to simplify the numerical calculation due to its no cost. We derive exact solutions and simulate results from them. We discuss the potential physical contributions and present respective conclusions about (1) the validness of the diffusion approximation of the radiative transfer equation; (2) the local behavior of the source of scattered photons; (3) the unsteady-state of the fluence rate; and (4) the boundedness of the critical time of the thermal damage to the cancerous tissue. We also discuss some controversial and diverging hypotheses.

math.AP↗
arXiv2023

On the smallness conditions for a PEMFC single cell problem

The aim of the present paper is to prove whose smallness conditions being necessary in order to get the final result of existence of a solution. In the first part, we present the model for a proton exchange membrane fuel cell (PEMFC) single cell and we clarify the interactions of the different components namely, velocity, pressure, density, temperature and potential. The final mathematical model is a quasilinear elliptic system where the cross effects have a strong interlink. It consists of the Stokes--Darcy system altogether with thermoelectrochemical system under some non-standard interface and boundary conditions. The proof of existence of weak solutions relies on the Tychonof fixed point theorem, by providing some regularity and some smallness conditions. The actual system is divided into two systems of equations and they are separately studied. The novelty of the present work is to establish quantitative estimates for improving the technical hypotheses and, in particular, the smallness conditions in the two-dimensional case. Indeed, the smallness conditions only can be explicit if quantitative estimates are established. To this aim, we establish quantitative estimates for the Poincaré and Sobolev inequalities and for some trilinear terms.

math.AP↗
arXiv2022

On the wellposedness for a fuel cell problem

This paper investigates the existence of weak solutions to two problems set of elliptic equations in adjoining domains, with Beavers--Joseph--Saffman and regularized Butler--Volmer boundary conditions being prescribed on the common interfaces, porous-fluid and membrane, respectively. Mathematically, the modeling tool is the coupled Stokes/Darcy problem, which consists of the Stokes equation on one part of the domain coupled to the Darcy equation, where the flow velocities are small and mainly driven by the pressure gradient in porous medium, completed by the thermoelectrochemical (TEC) system, which consists of the energy equation and the mass transport associated with electrochemical reactions, where the fluxes are given by generalized Fourier, Fick and Ohm laws, by including the Dufour--Soret and Peltier--Seebeck cross effects, in the multidimensional domain. The present model includes macrohomogeneous models for both hydrogen and methanol crossover. The novelty in the presented model lies in the presence of the Joule effect into the Stokes/Darcy-TEC system altogether to the quasilinear character given by temperature dependence of the physical parameters such as the viscosities and the diffusion coefficients, by the concentration-temperature dependence of cross-effects coefficients, and by the pressure dependence of the permeability. The purpose of the present work is to derive quantitative estimates for solutions to explicit smallness conditions on the data. We use fixed point and compactness arguments based on the quantitative estimates of approximated solutions.

math.AP↗
arXiv2022

Analytical solutions in the modeling of the endovenous laser ablation

We model the operative treatment of incompetent truncal veins using endovenous laser ablation (EVLA). Three differential equations, namely the diffusion, the heat and the bioheat equations, are considered in the endovenous-perivenous multidomain, describing the lumen, the vein wall, the tissue pad and the skin. Exact solutions are provided. Our main concern is to accurate the heat source by taking the Beer--Lambert law into account in the irradiance of the incident beam. To accurate the heat transfer at the skin boundary, the Newton law of cooling is considered as a Robin boundary condition. Open problems are presented.

math.AP↗
arXiv2021

Compressible Navier-Stokes-Fourier flows at steady-state

The heat conducting compressible viscous flows are governed by the Navier-Stokes-Fourier (NSF) system. In this paper, we study the NSF system accomplished by the Newton law of cooling for the heat transfer at the boundary. On one part of the boundary, we consider the Navier slip boundary condition, while in the remaining part the inlet and outlet occur. The existence of a weak solution is proved via a new fixed point argument. With this new approach, the weak solvability is possible in Lipschitz domains, by making recourse to \(L^q\)-Neumann problems with \(q>n\).Thus, standard existence results can be applied to auxiliary problems and the claim follows by compactness techniques. Quantitative estimates are established.

math.AP↗
arXiv2018

Weak solutions for multiquasilinear elliptic-parabolic systems. Application to thermoelectrochemical problems

This paper investigates the existence of weak solutions of biquasilinear boundary value problem for a coupled elliptic-parabolic system of divergence form with discontinuous leading coefficients. The mathematical framework addressed in the article considers the presence of an additional nonlinearity in the model which reflects the radiative thermal boundary effects in some applications of interest. The results are obtained via the Rothe-Galerkin method. Only weak assumptions are made on the data and the boundary conditions are allowed to be on a general form. The major contribution of the current paper is the explicit expressions for the constants appeared in the quantitative estimates that are derived. These detailed and explicit estimates may be useful for the study on nonlinear problems that appear in the real world applications. In particular, they clarify the smallness conditions. In conclusion, we illustrate how the above results may be applied to the thermoelectrochemical phenomena in an electrolysis cell. This problem has several applications as for instance to optimize the cell design and operating conditions.

math.AP↗
arXiv2017

Weak solutions for a thermoelectric problem with power-type boundary effects

This paper deals with thermoelectric problems including the Peltier and Seebeck effects. The coupled elliptic and doubly quasilinear parabolic equations for the electric and heat currents are stated, respectively, accomplished with power-type boundary conditions that describe the thermal radiative effects. To verify the existence of weak solutions to this coupled problem (Theorem 1), analytical investigations for abstract multi-quasilinear elliptic-parabolic systems with nonsmooth data are presented (Theorem 2 and 3). They are essentially approximated solutions based on the Rothe method. It consists on introducing time discretized problems, establishing their existence, and then passing to the limit as the time step goes to zero. The proof of the existence of time discretized solutions relies on fixed point and compactness arguments. In this study, we establish quantitative estimates to clarify the smallness conditions.

math.AP↗
arXiv2015

Sufficient conditions to the existence for solutions of a thermoelectrochemical problem

A mathematical model of nonlinear radiation is introduced into a thermoelectrochemical problem, and its qualitative analysis is focused on existence of solutions. The main objective is the nonconstant character of each parameter, that is, the coefficients are assumed to be depend on the spatial variable and the temperature. Making recourse of known estimates of solutions for some auxiliary elliptic and parabolic problems, which are explicitly determined by the Gehring-Giaquinta-Modica theory, we find sufficient smallness conditions on the data to the existence of the original solutions via the Schauder fixed point argument. These conditions may provide useful informations for numerical as well as real applications. We conclude with an example of application, namely the electrolysis of molten sodium chloride.

math.AP↗
arXiv2015

Radiative effects on the thermoelectric problems

There are two main directions in this paper. One is to find sufficient conditions to ensure the existence of weak solutions to thermoelectric problems. At the steady-state, these problems consist by a coupled system of elliptic equations of the divergence form, commonly accomplished with nonlinear radiation-type conditions on at least on a nonempty part of the boundary of a $C^1$ domain. The model under study takes the thermoelectric Peltier and Seebeck effects into account, whose describe the Joule-Thomson effect. The proof method makes recourse of a fixed point argument. To this end, well-determined estimates are our main concern. The paper is in the second direction for the derivation of explicit $W^{1,p}$-estimates $(p>2)$ for solutions of nonlinear radiation-type problems, where the leading coefficient is assumed to be a discontinuous function on the space variable. In particular, the behavior of the leading coefficient is conveniently explicit on the estimate of any solution. This regularity result is sufficiently general to contribute to other problems, in which the dependence on the values of the involved constants is essential, instead of the problem under study only.

math.AP↗
arXiv2014

Explicit estimates on a mixed Neumann-Robin-Cauchy problem

We deal with the existence of weak solutions for a mixed Neumann-Robin-Cauchy problem. The existence results are based on global-in-time estimates of approximating solutions, and the passage to the limit exploits compactness techniques. We investigate explicit estimates for solutions of the parabolic equations with nonhomogeneous boundary conditions and distributional right hand sides. The parabolic equation is of divergence form with discontinuous coefficients. We consider a nonlinear condition on a part of the boundary that the power laws (and the Robin boundary condition) appear as particular cases.

math.AP↗
arXiv2013

Architectural form as space-time cell

The architecture has its basis in a dialectic search of new choices of representation. We deal with the form on the contemporary architecture under two approaches: expression and content. We examine how mathematical principles based on natural growth can be applied in architectural design in order to create a dynamic, rather than static, structure. The dynamic process of a cell and its growth provides the basic structure. We exemplify the impact of the new forms on the new society that already began.

math.HO↗
arXiv2013

Explicit estimates for solutions of nonlinear radiation-type problems

We establish the existence of weak solutions of a nonlinear radiation-type boundary value problem for elliptic equation on divergence form with discontinuous leading coefficient. Quantitative estimates play a crucial role on the real applications. Our objective is the derivation of explicit expressions of the involved constants in the quantitative estimates, the so-called absolute or universal bounds. The dependence on the leading coefficient and on the size of the spatial domain is precise.This work shows that the expressions of those constants are not so elegant as we might expect.

math.AP↗
arXiv2013

Explicit estimates for solutions of mixed elliptic problems

We deal with the existence of quantitative estimates for solutions of mixed problems to an elliptic second order equation in divergence form with discontinuous coefficient. Our concern is to estimate the solutions with explicit constants, for domains in $\mathbb{R}^n$ ($n\geq 2$) of class $C^{0,1}$. The existence of $L^\infty$ and $W^{1,q}$-estimates is assured for $q=2$ and any $q<n/(n-1)$ (depending on the data), whenever the coefficient is only measurable and bounded. The proof method of the quantitative $L^\infty$-estimates is based on the DeGiorgi technique developed by Stampacchia. By using the potential theory, we derive $W^{1,p}$-estimates for different ranges of the exponent $p$ depending on that the coefficient is either Dini-continuous or only measurable and bounded. In this process, we establish new existences of Green functions on such domains. The last but not least concern is to unify (whenever possible) the proofs of the estimates to the extreme Dirichlet and Neumann cases of the mixed problem.

math.AP↗
arXiv2011

A limit model for thermoelectric equations

We analyze the asymptotic behavior corresponding to the arbitrary high conductivity of the heat in the thermoelectric devices. This work deals with a steady-state multidimensional thermistor problem, considering the Joule effect and both spatial and temperature dependent transport coefficients under some real boundary conditions in accordance with the Seebeck-Peltier-Thomson cross-effects. Our first purpose is that the existence of a weak solution holds true under minimal assumptions on the data, as in particular nonsmooth domains. Two existence results are studied under different assumptions on the electrical conductivity. Their proofs are based on a fixed point argument, compactness methods, and existence and regularity theory for elliptic scalar equations. The second purpose is to show the existence of a limit model illustrating the asymptotic situation.

math.AP↗
arXiv2011

Partial regularity for the Navier-Stokes-Fourier system

This paper addresses a nonstationary flow of heat-conductive incompressible Newtonian fluid with temperature-dependent viscosity coupled with linear heat transfer with advection and a viscous heat source term, under Navier/Dirichlet boundary conditions. The partial regularity for the velocity of the fluid is proved to each proper weak solution, that is, for such weak solutions which satisfy some local energy estimates in a similar way to the suitable weak solutions of the Navier-Stokes system. Finally, we study the nature of the set of points in space and time upon which proper weak solutions could be singular.

math.AP↗
arXiv2011

Dynamic bilateral boundary conditions on interfaces

Two boundary value problems for an elliptic equation in divergence form with bounded discontinuous coefficient are studied in a bidomain. On the interface, generalized dynamic boundary conditions such as of the Wentzell-type and Signorini-type transmission are considered in a subdifferential form. Several non-constant coefficients and nonlinearities are the main objective of the present work. Generalized solutions are built via time discretization.

math.AP↗
arXiv2009

Existence of proper weak solutions to the Navier-Stokes-Fourier system

The existence of proper weak solutions of the Dirichlet-Cauchy problem constituted by the Navier-Stokes-Fourier system which characterizes the incompressible homogeneous Newtonian fluids under thermal effects is studied. We call proper weak solutions such weak solutions that verify some local energy inequalities in analogy with the suitable weak solutions for the Navier-Stokes equations. Finally, we deal with some regularity for the temperature.

math.AP↗