Discrete and Continuous Dynamical Systems - S
August 2022 , Volume 15 , Issue 8
Issue on Mathematics, Models & Applications: Dedicated to Professor Maurizio Grasselli, on the Occasion of His 60th Birthday
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We prove convergence of suitable subsequences of weak solutions of a diffuse interface model for the two-phase flow of incompressible fluids with different densities with a nonlocal Cahn-Hilliard equation to weak solutions of the corresponding system with a standard "local" Cahn-Hilliard equation. The analysis is done in the case of a sufficiently smooth bounded domain with no-slip boundary condition for the velocity and Neumann boundary conditions for the Cahn-Hilliard equation. The proof is based on the corresponding result in the case of a single Cahn-Hilliard equation and compactness arguments used in the proof of existence of weak solutions for the diffuse interface model.
We consider optimal control of fractional in time (subdiffusive, i.e., for
We consider the
Correction: Bari is added after the zip code 70125 in third author’s address. We apologize for any inconvenience this may cause.
In this work we study three different dissipation mechanisms arising in the so-called Moore-Gibson-Thompson porosity. The three cases correspond to the MGT-porous hyperviscosity (fourth-order term), the MGT-porous viscosity (second-order term) and the MGT-porous weak viscosity (zeroth-order term). For all the cases, we prove that there exists a unique solution to the problem and we analyze the resulting point spectrum. We also show that there is an exponential energy decay for the first case, meanwhile for the second and third case only a polynomial decay is found. Finally, we present some one-dimensional numerical simulations to illustrate the behaviour of the discrete energy for each case.
Boundary feedback stabilization of a critical third–order (in time) semilinear Jordan–Moore–Gibson–Thompson (JMGT) is considered. The word critical here refers to the usual case where media–damping effects are non–existent or non–measurable and therefore cannot be relied upon for stabilization purposes. Motivated by modeling aspects in high-intensity focused ultrasound (HIFU) technology, the boundary feedback under consideration is supported only on a portion of the boundary. At the same time, the remaining part is undissipated and subject to Neumann/Robin boundary conditions. As such, unlike Dirichlet, it fails to satisfy the Lopatinski condition, a fact which compromises tangential regularity on the boundary [
We review space and time discretizations of the Cahn-Hilliard equation which are energy stable. In many cases, we prove that a solution converges to a steady state as time goes to infinity. The proof is based on Lyapunov theory and on a Lojasiewicz type inequality. In a few cases, the convergence result is only partial and this raises some interesting questions. Numerical simulations in two and three space dimensions illustrate the theoretical results. Several perspectives are discussed.
The present work is devoted to modeling and simulation of the carbonation process in concrete. To this aim we introduce some free boundary problems which describe the evolution of calcium carbonate stones under the attack of
Many problems arising in biology display a complex system dynamics at different scales of space and time. For this reason, multiscale mathematical models have attracted a great attention as they enable to take into account phenomena evolving at several characteristic lengths. However, they require advanced model reduction techniques to reduce the computational cost of solving all the scales.
In this work, we present a novel version of the Keller-Segel model of chemotaxis on embedded multiscale geometries, i.e., one-dimensional networks embedded in three-dimensional bulk domains. Applying a model reduction technique based on spatial averaging for geometrical order reduction, we reduce a fully three-dimensional Keller-Segel system to a coupled 3D-1D multiscale model. In the reduced model, the dynamics of the cellular population evolves on a one-dimensional network and its migration is influenced by a three-dimensional chemical signal evolving in the bulk domain. We propose the multiscale version of the Keller-Segel model as a realistic approach to describe the invasion of malignant cancer cells along the collagen fibers that constitute the extracellular matrix. Performing several numerical simulations, we investigate how the invasive abilities of the cells are affected by the topology of the network (i.e., matrix fibers orientation and alignment) as well as by three-dimensional spatial effects. We discuss these results in light of biological evidences.
We consider the gradient flow of a Ginzburg-Landau functional of the type
which is defined for tangent vector fields (here
In this paper we consider a model describing the evolution of a nematic liquid crystal flow with delay external forces. We analyze the evolution of the velocity field
The paper treats the problem of optimal distributed control of a Cahn–Hilliard–Oono system in
We provide a complete stability analysis for the abstract differential system made by an antidamped wave-type equation, coupled with a dissipative heat-type equation
Correction: Sections 7, 8 and 9 are missing from this article. Such sections were present and peer-reviewed in the original submission, but they were mistakenly omitted during the preparation of the final version with the AIMS template. They are added in Correction to “Thermoelasticity with antidissipation” (volume 15, number 8, 2022, 2173-2188).
We analyze the stability properties of a linear thermoelastic Timoshenko-Gurtin-Pipkin system with thermal coupling acting on both the shear force and the bending moment. Under either the mixed Dirichlet-Neumann or else the full Dirichlet boundary conditions, we show that the associated solution semigroup in the history space framework of Dafermos is exponentially stable independently of the values of the structural parameters of the model.
We prove some new rigidity results for minimal graphs over unbounded Euclidean domains. In particular we prove that a positive minimal graph over an open affine half-space, and under the homogenous Dirichlet boundary condition, must be an affine function.
We introduce a new concept of dissipative varifold solution to models of two phase compressible viscous fluids. In contrast with the existing approach based on the Young measure description, the new formulation is variational combining the energy and momentum balance in a single inequality. We show the existence of dissipative varifold solutions for a large class of general viscous fluids with non–linear dependence of the viscous stress on the symmetric velocity gradient.
Our aim in this paper is to analyze a model of glioma where oxygen drives cancer diffusion and proliferation. We prove the global well-posedness of the analytical problem and that, in the longtime, the illness does not disappear. Besides, the tumor dynamics increase the oxygen levels.
Addendum: "This research has been performed within the framework of the grant MIUR-PRIN 2020F3NCPX “Mathematics for industry 4.0 (Math4I4)”." is added under Fund Project. We apologize for any inconvenience this may cause.
We investigate the longtime behavior of the solutions to the Navier-Stokes-Cahn-Hilliard system (also known as Model H) with singular (e.g. Flory-Huggins) potential and non-constant viscosity. We prove that the initial and boundary value problem generates a strongly continuous semigroup on a suitable phase-space. Next, we establish the existence of the global attractor and of exponential attractors, and their regularity properties.
We study the global attractors for the damped 3D Euler–Bardina equations with the regularization parameter
In this paper we study the Timoshenko model over the interval
We study an evolutionary system of Cahn-Hilliard-Darcy type including mass source and transport effects. The system may arise in a number of physical situations related to phase separation phenomena with convection, with the main and most specific application being related to tumoral processes, where the variations of the mass may correspond to growth, or shrinking, of the tumor. We prove existence of weak solutions in the case when the configuration potential for the order parameter
The flow of a homogeneous, incompressible and heat conducting fluid is here described by coupling a quasilinear Navier-Stokes-type equation with the equation of heat diffusion, convection and buoyancy. This model is formulated variationally as a problem of null-minimization.
First we review how De Giorgi's theory of
We study a hydrodynamic phase-field system modeling the deformation of functionalized membranes in incompressible viscous fluids. The governing PDE system consists of the Navier–Stokes equations coupled with a convective sixth-order Cahn–Hilliard type equation driven by the functionalized Cahn–Hilliard free energy, which describes the phase separation process in mixtures with an amphiphilic structure. In the three dimensional case, we prove existence of global weak solutions provided that the initial total energy is finite. Then we establish uniqueness of weak solutions under suitable regularity assumptions that are only imposed on the velocity field or its gradient. Next, we prove existence and uniqueness of local strong solutions for arbitrary regular initial data and derive some blow-up criteria. Finally, we show the eventual regularity of global weak solutions for large time. The results are obtained in a general setting with variable fluid viscosity and diffusion mobility.
We present a new computational model for the numerical simulation of blood flow in the human left heart. To this aim, we use the Navier-Stokes equations in an Arbitrary Lagrangian Eulerian formulation to account for the endocardium motion and we model the cardiac valves by means of the Resistive Immersed Implicit Surface method. To impose a physiological displacement of the domain boundary, we use a 3D cardiac electromechanical model of the left ventricle coupled to a lumped-parameter (0D) closed-loop model of the remaining circulation. We thus obtain a one-way coupled electromechanics-fluid dynamics model in the left ventricle. To extend the left ventricle motion to the endocardium of the left atrium and to that of the ascending aorta, we introduce a preprocessing procedure according to which an harmonic extension of the left ventricle displacement is combined with the motion of the left atrium based on the 0D model. To better match the 3D cardiac fluid flow with the external blood circulation, we couple the 3D Navier-Stokes equations to the 0D circulation model, obtaining a multiscale coupled 3D-0D fluid dynamics model that we solve via a segregated numerical scheme. We carry out numerical simulations for a healthy left heart and we validate our model by showing that meaningful hemodynamic indicators are correctly reproduced.
In the present correction we add the missing sections 7, 8 and 9 to the original paper . Such sections were present and peer-reviewed in the original submission, but they were mistakenly omitted during the preparation of the final version with the AIMS template.
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