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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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Pierluigi Colli, Monica Conti, Alain Miranville, Vittorino Pata and Elisabetta Rocca
2022, 15(8): i-iii doi: 10.3934/dcdss.2022126 +[Abstract](160) +[HTML](100) +[PDF](238.53KB)
Convergence of a nonlocal to a local diffuse interface model for two-phase flow with unmatched densities
Helmut Abels and Yutaka Terasawa
2022, 15(8): 1871-1881 doi: 10.3934/dcdss.2022117 +[Abstract](153) +[HTML](32) +[PDF](365.01KB)

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.

A unified framework for optimal control of fractional in time subdiffusive semilinear PDEs
Harbir Antil, Ciprian G. Gal and Mahamadi Warma
2022, 15(8): 1883-1918 doi: 10.3934/dcdss.2022012 +[Abstract](343) +[HTML](129) +[PDF](627.63KB)

We consider optimal control of fractional in time (subdiffusive, i.e., for \begin{document}$ 0<\gamma <1 $\end{document}) semilinear parabolic PDEs associated with various notions of diffusion operators in an unifying fashion. Under general assumptions on the nonlinearity we \begin{document}$\mathsf{first\;show}$\end{document} the existence and regularity of solutions to the forward and the associated \begin{document}$\mathsf{backward\;(adjoint)}$\end{document} problems. In the second part, we prove existence of optimal \begin{document}$\mathsf{controls }$\end{document} and characterize the associated \begin{document}$\mathsf{first\;order}$\end{document} optimality conditions. Several examples involving fractional in time (and some fractional in space diffusion) equations are described in detail. The most challenging obstacle we overcome is the failure of the semigroup property for the semilinear problem in any scaling of (frequency-domain) Hilbert spaces.

$ C^1 $-VEM for some variants of the Cahn-Hilliard equation: A numerical exploration
Paola F. Antonietti, Simone Scacchi, Giuseppe Vacca and Marco Verani
2022, 15(8): 1919-1939 doi: 10.3934/dcdss.2022038 +[Abstract](321) +[HTML](100) +[PDF](2615.53KB)

We consider the \begin{document}$ C^1 $\end{document}-Virtual Element Method (VEM) for the conforming numerical approximation of some variants of the Cahn-Hilliard equation on polygonal meshes. In particular, we focus on the discretization of the advective Cahn-Hilliard problem and the Cahn-Hilliard inpainting problem. We present the numerical approximation and several numerical results to assess the efficacy of the proposed methodology.


Correction: Bari is added after the zip code 70125 in third author’s address. We apologize for any inconvenience this may cause.

On the time decay for the MGT-type porosity problems
Jacobo Baldonedo, José R. Fernández and Ramón Quintanilla
2022, 15(8): 1941-1955 doi: 10.3934/dcdss.2022009 +[Abstract](347) +[HTML](139) +[PDF](490.81KB)

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 nonlinear JMGT equation with Neumann-undissipated part of the boundary
Marcelo Bongarti and Irena Lasiecka
2022, 15(8): 1957-1985 doi: 10.3934/dcdss.2022107 +[Abstract](126) +[HTML](42) +[PDF](444.59KB)

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 [37]. In such a configuration, the analysis of uniform stabilization from the boundary becomes subtle and requires careful geometric considerations and microlocal analysis estimates. The nonlinear effects in the model demand construction of suitably small solutions which are invariant under the dynamics. The assumed smallness of the initial data is required only at the lowest energy level topology, which is sufficient to construct sufficiently smooth solutions to the nonlinear model.

Convergence to equilibrium for time and space discretizations of the Cahn-Hilliard equation
Matthieu Brachet, Philippe Parnaudeau and Morgan Pierre
2022, 15(8): 1987-2031 doi: 10.3934/dcdss.2022110 +[Abstract](180) +[HTML](63) +[PDF](6815.17KB)

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.

A moving boundary problem for reaction and diffusion processes in concrete: Carbonation advancement and carbonation shrinkage
Gabriella Bretti, Maurizio Ceseri and Roberto Natalini
2022, 15(8): 2033-2052 doi: 10.3934/dcdss.2022092 +[Abstract](171) +[HTML](80) +[PDF](536.46KB)

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 \begin{document}$ {CO}_2 $\end{document} dispersed in the atmosphere, taking into account both the shrinkage of concrete and the influence of humidity on the carbonation process. Indeed, two different regimes are described according to the relative humidity in the environment. Finally, some numerical simulations here presented are in substantial accordance with experimental results taken from literature.

A coupled 3D-1D multiscale Keller-Segel model of chemotaxis and its application to cancer invasion
Federica Bubba, Benoit Perthame, Daniele Cerroni, Pasquale Ciarletta and Paolo Zunino
2022, 15(8): 2053-2086 doi: 10.3934/dcdss.2022044 +[Abstract](416) +[HTML](121) +[PDF](6844.58KB)

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.

Motion of vortices for the extrinsic Ginzburg-Landau flow for vector fields on surfaces
Giacomo Canevari and Antonio Segatti
2022, 15(8): 2087-2116 doi: 10.3934/dcdss.2022116 +[Abstract](104) +[HTML](32) +[PDF](736.73KB)

We consider the gradient flow of a Ginzburg-Landau functional of the type

which is defined for tangent vector fields (here \begin{document}$ D $\end{document} stands for the covariant derivative) on a closed surface \begin{document}$ M\subseteq \mathbb{R}^3 $\end{document} and includes extrinsic effects via the shape operator \begin{document}$ \mathscr{S} $\end{document} induced by the Euclidean embedding of \begin{document}$ M $\end{document}. The functional depends on the small parameter \begin{document}$ \varepsilon>0 $\end{document}. When \begin{document}$ \varepsilon $\end{document} is small it is clear from the structure of the Ginzburg-Landau functional that \begin{document}$ \left| {u} \right|_g $\end{document} "prefers" to be close to \begin{document}$ 1 $\end{document}. However, due to the incompatibility for vector fields on \begin{document}$ M $\end{document} between the Sobolev regularity and the unit norm constraint, when \begin{document}$ \varepsilon $\end{document} is close to \begin{document}$ 0 $\end{document}, it is expected that a finite number of singular points (called vortices) having non-zero index emerges (when the Euler characteristic is non-zero). This intuitive picture has been made precise in the recent work by R. Ignat & R. Jerrard [7]. In this paper we are interested the dynamics of vortices generated by \begin{document}$ F_ \varepsilon^{ \mathrm{extr}} $\end{document}. To this end we study the behavior when \begin{document}$ \varepsilon\to 0 $\end{document} of the solutions of the (properly rescaled) gradient flow of \begin{document}$ F_ \varepsilon^{ \mathrm{extr}} $\end{document}. In the limit \begin{document}$ \varepsilon\to 0 $\end{document} we obtain the effective dynamics of the vortices. The dynamics, as expected, is influenced by both the intrinsic and extrinsic properties of the surface \begin{document}$ M\subseteq \mathbb{R}^3 $\end{document}.

A 3D isothermal model for nematic liquid crystals with delay terms
Tomás Caraballo and Cecilia Cavaterra
2022, 15(8): 2117-2133 doi: 10.3934/dcdss.2022097 +[Abstract](202) +[HTML](60) +[PDF](368.11KB)

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 \begin{document}$ {\boldsymbol u} $\end{document} which is ruled by the 3D incompressible Navier-Stokes system containing a delay term and with a stress tensor expressing the coupling between the transport and the induced terms. The dynamics of the director field \begin{document}$ \boldsymbol{d} $\end{document} is described by a modified Allen-Cahn equation with a suitable penalization of the physical constraint \begin{document}$ | \boldsymbol{d}| = 1 $\end{document}. We prove the existence of global in time weak solutions under appropriate assumptions, which in some cases requires the delay term to be small with respect to the viscosity parameter.

Well-posedness and optimal control for a Cahn–Hilliard–Oono system with control in the mass term
Pierluigi Colli, Gianni Gilardi, Elisabetta Rocca and Jürgen Sprekels
2022, 15(8): 2135-2172 doi: 10.3934/dcdss.2022001 +[Abstract](438) +[HTML](145) +[PDF](640.77KB)

The paper treats the problem of optimal distributed control of a Cahn–Hilliard–Oono system in \begin{document}$ {{\mathbb{R}}}^d $\end{document}, \begin{document}$ 1\leq d\leq 3 $\end{document}, with the control located in the mass term and admitting general potentials that include both the case of a regular potential and the case of some singular potential. The first part of the paper is concerned with the dependence of the phase variable on the control variable. For this purpose, suitable regularity and continuous dependence results are shown. In particular, in the case of a logarithmic potential, we need to prove an ad hoc strict separation property, and for this reason we have to restrict ourselves to the case \begin{document}$ d = 2 $\end{document}. In the rest of the work, we study the necessary first-order optimality conditions, which are proved under suitable compatibility conditions on the initial datum of the phase variable and the time derivative of the control, at least in case of potentials having unbounded domain.

Thermoelasticity with antidissipation
Monica Conti, Lorenzo Liverani and Vittorino Pata
2022, 15(8): 2173-2188 doi: 10.3934/dcdss.2022040 +[Abstract](352) +[HTML](114) +[PDF](427.43KB)

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

where \begin{document}$ A $\end{document} is a strictly positive selfadjoint operator on a Hilbert space, \begin{document}$ \gamma, \kappa>0 $\end{document}, and both the parameters \begin{document}$ \alpha $\end{document} and \begin{document}$ \beta $\end{document} can vary between \begin{document}$ 0 $\end{document} and \begin{document}$ 1 $\end{document}. The asymptotic properties of the associated solution semigroup are determined by the strength of the coupling, as well as the quantitative balance between the antidamping \begin{document}$ \gamma $\end{document} and the damping \begin{document}$ \kappa $\end{document}. Depending on the value of \begin{document}$ (\alpha, \beta) $\end{document} in the unit square, one of the following mutually disjoint situations can occur: either the related semigroup decays exponentially fast, or all the solutions vanish but not uniformly, or there exists a trajectory whose norm blows up exponentially fast as \begin{document}$ t\to\infty $\end{document}.


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).

Exponential stability of Timoshenko-Gurtin-Pipkin systems with full thermal coupling
Filippo Dell'Oro, Marcio A. Jorge Silva and Sandro B. Pinheiro
2022, 15(8): 2189-2207 doi: 10.3934/dcdss.2022050 +[Abstract](240) +[HTML](104) +[PDF](372.11KB)

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.

Some rigidity results for minimal graphs over unbounded Euclidean domains
Alberto Farina
2022, 15(8): 2209-2214 doi: 10.3934/dcdss.2022032 +[Abstract](224) +[HTML](88) +[PDF](261.2KB)

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.

Two phase flows of compressible viscous fluids
Eduard Feireisl and Antonín Novotný
2022, 15(8): 2215-2232 doi: 10.3934/dcdss.2022091 +[Abstract](158) +[HTML](94) +[PDF](325.45KB)

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.

An oxygen driven proliferative-to-invasive transition of glioma cells: An analytical study
Stefania Gatti
2022, 15(8): 2233-2248 doi: 10.3934/dcdss.2022002 +[Abstract](316) +[HTML](124) +[PDF](311.44KB)

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.

Attractors for the Navier-Stokes-Cahn-Hilliard system
Andrea Giorgini and Roger Temam
2022, 15(8): 2249-2274 doi: 10.3934/dcdss.2022118 +[Abstract](170) +[HTML](39) +[PDF](395.78KB)

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.

Trajectory attractors for 3D damped Euler equations and their approximation
Alexei Ilyin, Anna Kostianko and Sergey Zelik
2022, 15(8): 2275-2288 doi: 10.3934/dcdss.2022051 +[Abstract](259) +[HTML](70) +[PDF](348.17KB)

We study the global attractors for the damped 3D Euler–Bardina equations with the regularization parameter \begin{document}$ \alpha>0 $\end{document} and Ekman damping coefficient \begin{document}$ \gamma>0 $\end{document} endowed with periodic boundary conditions as well as their damped Euler limit \begin{document}$ \alpha\to0 $\end{document}. We prove that despite the possible non-uniqueness of solutions of the limit Euler system and even the non-existence of such solutions in the distributional sense, the limit dynamics of the corresponding dissipative solutions introduced by P. Lions can be described in terms of attractors of the properly constructed trajectory dynamical system. Moreover, the convergence of the attractors \begin{document}$ \mathcal A(\alpha) $\end{document} of the regularized system to the limit trajectory attractor \begin{document}$ \mathcal A(0) $\end{document} as \begin{document}$ \alpha\to0 $\end{document} is also established in terms of the upper semicontinuity in the properly defined functional space.

About the stability to Timoshenko system with pointwise dissipation
Jaime E. Muñoz Rivera and Maria Grazia Naso
2022, 15(8): 2289-2303 doi: 10.3934/dcdss.2022078 +[Abstract](227) +[HTML](66) +[PDF](382.82KB)

In this paper we study the Timoshenko model over the interval \begin{document}$ (0, \ell) $\end{document} with pointwise dissipation at \begin{document}$ \xi\in (0, \ell) $\end{document}. We prove that this dissipation produces exponential stability when \begin{document}$ \xi\in \mathbb{Q}\ell $\end{document} and \begin{document}$ \xi\ne \frac{n}{2m+1}\ell $\end{document}, where \begin{document}$ n, m\in \mathbb{N} $\end{document} and \begin{document}$ n $\end{document}, and \begin{document}$ 2m+1 $\end{document} are co-prime.

On the Cahn-Hilliard-Darcy system with mass source and strongly separating potential
Giulio Schimperna
2022, 15(8): 2305-2329 doi: 10.3934/dcdss.2022008 +[Abstract](300) +[HTML](136) +[PDF](381.75KB)

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 \begin{document}$ \varphi $\end{document} is designed in such a way to keep \begin{document}$ \varphi $\end{document} in between the reference interval \begin{document}$ (-1, 1) $\end{document} despite the occurrence of mass source effects. Moreover, in the two-dimensional case, we obtain existence and uniqueness of strong (i.e., more regular) solutions.

$ \Gamma $-compactness and $ \Gamma $-stability of the flow of heat-conducting fluids
Augusto Visintin
2022, 15(8): 2331-2343 doi: 10.3934/dcdss.2022066 +[Abstract](198) +[HTML](70) +[PDF](339.16KB)

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 \begin{document}$ \Gamma $\end{document}-convergence can be used to prove the compactness and the stability of evolutionary problems under nonparametric perturbations. Then we illustrate how this theory can be applied to the our problem of fluid and heat flow, and to more general coupled flows.

Well-posedness of a hydrodynamic phase-field system for functionalized membrane-fluid interaction
Hao Wu and Yuchen Yang
2022, 15(8): 2345-2389 doi: 10.3934/dcdss.2022102 +[Abstract](179) +[HTML](39) +[PDF](480.85KB)

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.

A geometric multiscale model for the numerical simulation of blood flow in the human left heart
Alberto Zingaro, Ivan Fumagalli, Luca Dede, Marco Fedele, Pasquale C. Africa, Antonio F. Corno and Alfio Quarteroni
2022, 15(8): 2391-2427 doi: 10.3934/dcdss.2022052 +[Abstract](772) +[HTML](159) +[PDF](24555.54KB)

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.

Correction to "Thermoelasticity with antidissipation" (volume 15, number 8, 2022, 2173–2188)
Monica Conti, Lorenzo Liverani and Vittorino Pata
2022, 15(8): 2429-2431 doi: 10.3934/dcdss.2022125 +[Abstract](162) +[HTML](33) +[PDF](232.42KB)

In the present correction we add the missing sections 7, 8 and 9 to the original paper [1]. 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.

2021 Impact Factor: 1.865
5 Year Impact Factor: 1.622
2021 CiteScore: 3.6

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