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This paper presents a nonlinear reaction-diffusion-fluid system that simulates radiofrequency ablation within cardiac tissue. The model conveys the dynamic evolution of temperature and electric potential in both the fluid and solid regions, along with the evolution of velocity within the solid region. By formulating the system that describes the phenomena across the entire domain, encompassing both solid and fluid phases, we proceed to an analysis of well-posedness, considering a broad class of right-hand side terms. The system involves parameters such as heat conductivity, kinematic viscosity, and electrical conductivity, all of which exhibit nonlinearity contingent upon the temperature variable. The mathematical analysis extends to establishing the existence of a global solution, employing the Faedo-Galerkin method in a three-dimensional space. To enhance the practical applicability of our theoretical results, we complement our study with a series of numerical experiments. We implement the discrete system using the finite element method for spatial discretization and an Euler scheme for temporal discretization. Nonlinear parameters are linearized through decoupling systems, as introduced in our continuous analysis. These experiments are conducted to demonstrate and validate the theoretical findings we have established.

A nonlinear cross-diffusion--fluid system with chemical terms describing the dynamics of predator-prey living in a Newtonian fluid is proposed in this paper. The existence of a weak solution for the proposed macro-scale system is proved based on the Schauder fixed-point theory, a priori estimates, and compactness arguments. The proposed system is derived from the underlying description delivered by a kinetic-fluid theory model by a multiscale approach. Finally, we discuss the computational results for the proposed macro-scale system in two-dimensional space.

This paper deals with the mathematical analysis and numerical simulation of a new nonlinear ablation system modeling radiofrequency ablation phenomena in cardiac tissue, {which incorporates the effects of blood flow on the heat generated when ablation by radiofrequency. The model also considers the effects of viscous energy dissipation. It consists of a coupled thermistor problem and the incompressible Navier--Stokes equations that describe the evolution of temperature, velocity and potential in cardiac tissue.} In addition to Faedo--Galerkin method, we use Schauder's fixed-point theory to prove the existence of the weak solutions in two- and three-dimensional space. Moreover, we prove the uniqueness of the solution under some additional conditions on the data and the solution. Finally, we discuss some numerical results for the validation of the proposed model using the finite element method.

We analyze a system of nonlinear stochastic partial differential equations (SPDEs) of mixed elliptic-parabolic type that models the propagation of electric signals and their effect on the deformation of cardiac tissue. The system governs the dynamics of ionic quantities, intra and extra-cellular potentials, and linearized elasticity equations. We introduce a framework called the active strain decomposition, which factors the material gradient of deformation into an active (electrophysiology-dependent) part and an elastic (passive) part, to capture the coupling between muscle contraction, biochemical reactions, and electric activity. Under the assumption of linearized elastic behavior and a truncation of the nonlinear diffusivities, we propose a stochastic electromechanical bidomain model, and establish the existence of weak solutions for this model. To prove existence through the convergence of approximate solutions, we employ a stochastic compactness method in tandem with an auxiliary non-degenerate system and the Faedo--Galerkin method. We utilize a stochastic adaptation of de Rham's theorem to deduce the weak convergence of the pressure approximations.

This paper is devoted to the mathematical analysis of a nonlinear stochastic reaction-diffusion system modeling predator-prey interactions with prey-taxis and noises. Precisely, we detail the proof of the existence of weak martingale solutions by Faedo-Galerkin approximations and the stochastic compactness method. We prove the nonnegativity of solutions by a stochastic adaptation of the Stampacchia approach. Finally, we prove the uniqueness of the solution via duality technique.

We study the homogenization of a novel microscopic tridomain system, allowing for a more detailed analysis of the properties of cardiac conduction than the classical bidomain and monodomain models. In (Acta Appl.Math. 179 (2022) 1--35), we detail this model in which gap junctions are considered as the connections between adjacent cells in cardiac muscle and could serve as alternative or supporting pathways for cell-to-cell electrical signal propagation. Departing from this microscopic cellular model, we apply the periodic unfolding method to derive the macroscopic tridomain model. Several difficulties prevent the application of unfolding homogenization results, including the degenerate temporal structure of the tridomain equations and a nonlinear dynamic boundary condition on the cellular membrane. To prove the convergence of the nonlinear terms, especially those defined on the microscopic interface, we use the boundary unfolding operator and a Kolmogorov--Riesz compactness's result., Comment: arXiv admin note: text overlap with arXiv:2201.03914

We present a novel microscopic tridomain model describing the electrical activity in cardiac tissue with dynamical gap junctions. The microscopic tridomain system consists of three PDEs modeling the tissue electrical conduction in the intra-and extra-cellular domains, supplemented by a nonlinear ODE system for the dynamics of the ion channels and the gap junctions. We establish the global existence and uniqueness of the weak solutions to our microscopic tridomain model. The global existence of solution, which constitutes the main result of this paper, is proved by means of an approximate non-degenerate system, the Faedo-Galerkin method, and an appropriate compactness argument.

In this paper, we are dealing with a rigorous homogenization result at two different levels for the bidomain model of cardiac electro-physiology. The first level associated with the mesoscopic structure such that the cardiac tissue consists of extracellular and intracellular domains separated by an interface (the sarcolemma). The second one related to the microscopic structure in such a way that the intracellular medium can only be viewed as a periodical layout of unit cells (mitochondria). At the interface between intra-and extracellular media, the fluxes are given by nonlinear functions of ionic and applied currents. A rigorous homogenization process based on unfolding operators is applied to derive the macroscopic (homogenized) model from our meso-microscopic bidomain model. We apply a three-scale unfolding method in the intracellular problem to obtain its homogenized equation at two levels. The first level upscaling of the intracellular structure yields the mesoscopic equation. The second step of the homogenization leads to obtain the intracellular homogenized equation. To prove the convergence of the nonlinear terms, especially those defined on the microscopic interface, we use the boundary unfolding method and a Kolmogorov-Riesz compactness's result. Next, we use the standard unfolding method to homogenize the extracellular problem. Finally, we obtain, at the limit, a reaction-diffusion system on a single domain (the superposition of the intracellular and extracellular media) which contains the homogenized equations depending on three scales. Such a model is widely used for describing the macroscopic behavior of the cardiac tissue, which is recognized to be an important messengers between the cytoplasm (intracellular) and the other extracellular inside the biological cells. Contents 2020 Mathematics Subject Classification. 65N55 and 35A01 and 35B27 and 35K57 and 65M.., Comment: arXiv admin note: substantial text overlap with arXiv:2201.03412

In the present paper, a new three-scale asymptotic homogenization method is proposed to study the electrical behavior of the cardiac tissue structure with multiple heterogeneities at two different levels. The first level is associated with the mesoscopic structure such that the cardiac tissue is composed of extracellular and intracellular domains. The second level is associated with the microscopic structure in such a way the intracellular medium can only be viewed as a periodical layout of unit cells (mitochondria). Then, we define two kinds of local cells that are obtained by upscaling methods. The homogenization method is based on a power series expansion which allows determining the macroscopic (homogenized) bidomain model from the microscopic bidomain problem at each structural level. First, we use the two-scale asymptotic expansion to homogenize the extracellular problem. Then, we apply a three-scale asymptotic expansion in the intracellular problem to obtain its homogenized equation at two levels. The first upscaling level of the intracellular structure yields the mesoscopic equation and the second step of the homogenization leads to obtain the intracellular homogenized equation. Both the mesoscopic and microscopic information is obtained by homogenization to capture local characteristics inside the cardiac tissue structure. Finally, we obtain the macroscopic bidomain model and the heart domain coincides with the intracellular medium and extracellular one, which are two inter-penetrating and superimposed continua connected at each point by the cardiac cellular membrane. The interest of the proposed method comes from the fact that it combines microscopic and mesoscopic characteristics to obtain a macroscopic description of the electrical behavior of the heart.

We present a Virtual Element Method (VEM) for a nonlocal reaction-diffusion system of the cardiac electric field. To this system, we analyze an $H^1(\Omega)$-conforming discretization by means of VEM which can make use of general polygonal meshes. Under standard assumptions on the computational domain, we establish the convergence of the discrete solution by considering a series of a priori estimates and by using a general $L^p$ compactness criterion. Moreover, we obtain optimal order space-time error estimates in the $L^2$ norm. Finally, we report some numerical tests supporting the theoretical results.

The bidomain system of degenerate reaction-diffusion equations is a well-established spatial model of electrical activity in cardiac tissue, with "reaction" linked to the cellular action potential and "diffusion" representing current flow between cells. The purpose of this paper is to introduce a "stochastically forced" version of the bidomain model that accounts for various random effects. We establish the existence of martingale (probabilistic weak) solutions to the stochastic bidomain model. The result is proved by means of an auxiliary nondegenerate system and the Faedo-Galerkin method. To prove convergence of the approximate solutions, we use the stochastic compactness method and Skorokhod-Jakubowski a.s.~representations. Finally, via a pathwise uniqueness result, we conclude that the martingale solutions are pathwise (i.e., probabilistic strong) solutions.

This paper is devoted to the analysis of the uniform null controllability for a family of nonlinear reaction-diffusion systems approximating a parabolic-elliptic system which models the electrical activity of the heart. The uniform, with respect to the degenerating parameter, null controllability of the approximating system by means of a single control is shown. The proof is based on the combination of Carleman estimates and weighted energy inequalities., Comment: 20 pages

A classical model for water-gas flows in porous media is considered. The degenerate coupled system of equations obtained by mass conservation is usually approximated by finite volume schemes in the oil reservoir simulations. The convergence properties of these schemes are only known for incompressible fluids. This chapter deals with construction and convergence analysis of a finite volume scheme for compressible and immiscible flow in porous media. In comparison with incompressible fluid, compressible fluids requires more powerful techniques. We present a new result of convergence in a two or three dimensional porous medium and under the only modification that the density of gas depends on global pressure., Comment: 40 pages, 1 figure

We prove convergence of discrete duality finite volume (DDFV) schemes on distorted meshes for a class of simplified macroscopic bidomain models of the electrical activity in the heart. Both time-implicit and linearised time-implicit schemes are treated. A short description is given of the 3D DDFV meshes and of some of the associated discrete calculus tools. Several numerical tests are presented.

We consider a class of doubly nonlinear degenerate hyperbolic-parabolic equations with homogeneous Dirichlet boundary conditions, for which we first establish the existence and uniqueness of entropy solutions. We then turn to the construction and analysis of discrete duality finite volume schemes (in the spirit of Domelevo and Omn\`es \cite{DomOmnes}) for these problems in two and three spatial dimensions. We derive a series of discrete duality formulas and entropy dissipation inequalities for the schemes. We establish the existence of solutions to the discrete problems, and prove that sequences of approximate solutions generated by the discrete duality finite volume schemes converge strongly to the entropy solution of the continuous problem. The proof revolves around some basic a priori estimates, the discrete duality features, Minty-Browder type arguments, and "hyperbolic" $L^\infty$ weak-$\star$ compactness arguments (i.e., propagation of compactness along the lines of Tartar, DiPerna, ...). Our results cover the case of non-Lipschitz nonlinearities.

This paper presents a finite-volume method, together with fully adaptive multi-resolution scheme to obtain spatial adaptation, and a Runge-Kutta-Fehlberg scheme with a local time-varying step to obtain temporal adaptation, to solve numerically the known "bidominio" equations that model the electrical activity of the tissue in the myocardium. Two simple models are considered for membrane flows and ionic currents. First we define an approximate solution and we verify its convergence to the corresponding weak solution of the continuum problem, obtaining in this way an alternative demonstration that the continuum problem is well-posed. Next we introduce the multiresolution technique and derive an optimal noise reduction threshold. The efficiency and precision of our method is seen in the reduction of machine time, memory usage, and errors in comparison to other methods. ----- En este trabajo se presenta un metodo de volumenes finitos enriquecido con un esquema de multiresolucion completamente adaptativo para obtener adaptatividad espacial, y un esquema Runge-Kutta-Fehlberg con paso temporal de variacion local para obtener adaptatividad temporal, para resolver numericamente las conocidas ecuaciones "bidominio" que modelan la actividad electrica del tejido en el miocardio. Se consideran dos modelos simples para las corrientes de membrana y corrientes ionicas. En primer lugar definimos una solucion aproximada y nos referimos a su convergencia a la correspondiente solucion debil del problema continuo, obteniendo de este modo una demostracion alternativa de que el problema continuo es bien puesto. Luego de introducir la tecnica de multiresolucion, se deriva un umbral optimo para descartar la informacion no significativa, y tanto la eficiencia como la precision de nuestro metodo es vista en terminos de la aceleracion de tiempo de maquina, compresion de memoria computacional y errores en diferentes normas., Comment: Spanish report

We study the well-posedness of triply nonlinear degenerate elliptic-parabolic-hyperbolic problem $$ b(u)_t - {\rm div} \tilde{\mathfrak a}(u,\nabla\phi(u))+\psi(u)=f, \quad u|_{t=0}=u_0 $$ in a bounded domain with homogeneous Dirichlet boundary conditions. The nonlinearities $b,\phi$ and $\psi$ are supposed to be continuous non-decreasing, and the nonlinearity $\tilde{\mathfrak a}$ falls within the Leray-Lions framework. Some restrictions are imposed on the dependence of $\tilde{\mathfrak a}(u,\nabla\phi(u))$ on $u$ and also on the set where $\phi$ degenerates. A model case is $\tilde{\mathfrak a}(u,\nabla\phi(u)) =\tilde{\mathfrak{f}}(b(u),\psi(u),\phi(u))+k(u)\mathfrak{a}_0(\nabla\phi(u)),$ with $\phi$ which is strictly increasing except on a locally finite number of segments, and $\mathfrak{a}_0$ which is of the Leray-Lions kind. We are interested in existence, uniqueness and stability of entropy solutions. If $b=\mathrm{Id}$, we obtain a general continuous dependence result on data $u_0,f$ and nonlinearities $b,\psi,\phi,\tilde{\mathfrak{a}}$. Similar result is shown for the degenerate elliptic problem which corresponds to the case of $b\equiv 0$ and general non-decreasing surjective $\psi$. Existence, uniqueness and continuous dependence on data $u_0,f$ are shown when $[b+\psi](\R)=\R$ and $\phi\circ [b+\psi]^{-1}$ is continuous.

This paper addresses the existence and regularity of weak solutions for a fully parabolic model of chemotaxis, with prevention of overcrowding, that degenerates in a two-sided fashion, including an extra nonlinearity represented by a $p$-Laplacian diffusion term. To prove the existence of weak solutions, a Schauder fixed-point argument is applied to a regularized problem and the compactness method is used to pass to the limit. The local H\"older regularity of weak solutions is established using the method of intrinsic scaling. The results are a contribution to showing, qualitatively, to what extent the properties of the classical Keller-Segel chemotaxis models are preserved in a more general setting. Some numerical examples illustrate the model.

We present a fully adaptive multiresolution scheme for spatially two-dimensional, possibly degenerate reaction-diffusion systems, focusing on combustion models and models of pattern formation and chemotaxis in mathematical biology. Solutions of these equations in these applications exhibit steep gradients, and in the degenerate case, sharp fronts and discontinuities. The multiresolution scheme is based on finite volume discretizations with explicit time stepping. The multiresolution representation of the solution is stored in a graded tree. By a thresholding procedure, namely the elimination of leaves that are smaller than a threshold value, substantial data compression and CPU time reduction is attained. The threshold value is chosen optimally, in the sense that the total error of the adaptive scheme is of the same slope as that of the reference finite volume scheme. Since chemical reactions involve a large range of temporal scales, but are spatially well localized (especially in the combustion model), a locally varying adaptive time stepping strategy is applied. It turns out that local time stepping accelerates the adaptive multiresolution method by a factor of two, while the error remains controlled., Comment: 27 pages, 14 figures

This work deals with the numerical solution of the monodomain and bidomain models of electrical activity of myocardial tissue. The bidomain model is a system consisting of a possibly degenerate parabolic PDE coupled with an elliptic PDE for the transmembrane and extracellular potentials, respectively. This system of two scalar PDEs is supplemented by a time-dependent ODE modeling the evolution of the so-called gating variable. In the simpler sub-case of the monodomain model, the elliptic PDE reduces to an algebraic equation. Two simple models for the membrane and ionic currents are considered, the Mitchell-Schaeffer model and the simpler FitzHugh-Nagumo model. Since typical solutions of the bidomain and monodomain models exhibit wavefronts with steep gradients, we propose a finite volume scheme enriched by a fully adaptive multiresolution method, whose basic purpose is to concentrate computational effort on zones of strong variation of the solution. Time adaptivity is achieved by two alternative devices, namely locally varying time stepping and a Runge-Kutta-Fehlberg-type adaptive time integration. A series of numerical examples demonstrates thatthese methods are efficient and sufficiently accurate to simulate the electrical activity in myocardial tissue with affordable effort. In addition, an optimalthreshold for discarding non-significant information in the multiresolution representation of the solution is derived, and the numerical efficiency and accuracy of the method is measured in terms of CPU time speed-up, memory compression, and errors in different norms., Comment: 25 pages, 41 figures

We analyze a finite volume scheme for nonlocal SIR model, which is a nonlocal reaction-diffusion system modeling an epidemic disease. We establish existence solutions to the finite volume scheme, and show that it converges to a weak solution. The convergence proof is based on deriving series of a priori estimates and using a general $L^p$ compactness criterion., Comment: 30 pages