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Published by himalia, 2019-06-05 00:33:43




JUNE 10-14, 2019 - MAIORI (SA) ITALY

XX International
Conference on
Waves and Stability
in Continuous Media

Book of Abstracts



The International Conference on Waves and Stability in Continuous Media (WASCOM), now in its XX edition,
is a biennial international conference on Mathematical Physics.
Since its first edition organized in 1981, this meeting turns out to be an opportunity for interaction of Italian
and foreign researchers interested in stability and wave propagation problems in continuous media.
The conference includes different research fields concerning wave propagation, stability problems and
modelling problems such as shock waves, diffusion processes in biology and in continuum mechanics,
kinetics models, non-equilibrium thermodynamics, stochastic processes, group methods, numerical



The previous conferences were organized in Catania (1981), PORTO ERCOLE (GR)
Cosenza (1983), Bari (1985), Taormina (1987), Sorrento
(1989), Acireale (1991), Bologna (1993), Palermo (1995), SORRENTO (NA) MAIORI (SA) BARI
Bari (1997), Vulcano (1999), Porto Ercole (2001), BRINDISI
Villasimius (2003), Acireale (2005), Scicli (2007), Mondello
(2009), Brindisi (2011), Levico (2013), Cetraro (2015), CETRARO (CS)
Bologna (2017). COSENZA



Main Topics of the Conference

Linear and nonlinear stability in fluid dynamics and solid mechanics
Non-linear wave propagation, discontinuity and shock waves
Rational extended thermodynamics and symmetric hyperbolic systems
Kinetic theories and comparison with continuum model
Numerical applications



Florinda Capone Chairmen Tommaso Ruggeri
(Naples, Italy) (Bologna, Italy)
Salvatore Rionero
(Naples, Italy)

Constantine M. Dafermos Scientific Committee Henri Gouin
(USA) (France)
Laurent Desvillettes
(France) Giuseppe Mulone
Seung-Yeal Ha Tai-Ping Liu
(Korea) (USA and Taiwan) Masaru Sugiyama
Giuseppe Saccomandi Marco Sammartino
(Italy) (Italy)

Organizing Committee

Salvatore Rionero Florinda Capone Roberta De Luca Luigi Frunzo
(Naples, Italy) (Naples, Italy) (Naples, Italy) (Naples, Italy)


Dipartimento di

Matematica e


JUNE 10-14, 2019



Sunday, June 9th

18.00 - 20.00 REGISTRATION (Reginna Palace Hotel, Via Cristoforo Colombo 1)

20.00 Dinner (Hotel Pietra di Luna, Via Gaetano Capone 27)

Monday, June 10th

08.30 - 09.00 REGISTRATION

09.00 - 09.30 Opening

09.30 - 09.55 Constantine M. Dafermos: Uniqueness of Zero Relaxation Limit
10.00 - 10.25 Salvatore Rionero: Hopf Bifurcations in Thermal MHD and Spectrum Instability Driven by Perturbations

to Principal Entries
10.30 - 10.55 Henri Gouin: Membranes and Vesicles
11.00 - 11.25 COFFEE BREAK

11.25 - 11.50 Florinda Capone: Recent Results on the Onset of MHD Convection in Porous Media
11.55 - 12.20 Stéphane Brull: Local Discrete Velocity Grids for Multi-Species Rarefied Flow Simulations
12.25 - 12.50 Maurizio Gentile: Thermal Convection in a Rotating Horizontally Isotropic Porous Medium with LTNE
12.55 - 13.10 Roberta De Luca: Onset of Double-Diffusive Convection in Porous Media with Soret Effect

13.30 Lunch (Reginna Palace Hotel, Via Cristoforo Colombo 1)

15.30 - 15.55 Seung-Yeal Ha: On the Second-Order Extensions of First-Order Collective Models
16.00 - 16.25 Giuseppe Mulone: New Nonlinear Stability Results for Plane Couette and Poiseuille Flows
16.30 - 16.55 Andrea Giacobbe: Inclined Convection in a Porous Brinkman Layer: Linear Instability and Nonlinear

17.00 - 17.25 COFFEE BREAK

17.25 - 17.50 Giancarlo Consolo: Propagation of Magnetic Domain Walls in Magnetostrictive Materials with Different

Crystal Symmetry
17.55 - 18.10 Paolo Falsaperla: New Stability Results for Hydromagnetic Plane Couette Flows
18.15 - 18.30 Monica De Angelis: On Solutions Related to FitzHugh-Rinzel Model

20.00 Dinner (Hotel Pietra di Luna, Via Gaetano Capone 27)



Tuesday, June 11th

09.00 - 09.25 Tai-Ping Liu: On Well-Posedness of Weak Solutions
09.30 - 09.55 Marco Sammartino: Viscous MHD Vorticity-Current Equations with Data in L 1 (R 2 )
10.00 - 10.25 Peter Vadasz: Instability, Weak Turbulence and Chaos in Porous Media
10.30 - 10.55 COFFEE BREAK

10.55 - 11.20 Maria Carmela Lombardo: Coherent Structures in a Chemotaxis Model of Acute Inflammation
11.25 - 11.40 Valeria Giunta: Aggregation Phenomena and Well-Posedness for a Multiple Sclerosis Model
11.45 - 12.10 Vincenzo Sciacca: Up-Wind Difference Approximation and Singularity Formation for a Slow Erosion Model
12.15 - 12.40 Gaetano Fiore: On the Impact of Short Laser Pulses on Cold Diluted Plasmas

13.30 Lunch (Hotel Pietra di Luna, Via Gaetano Capone 27)

15.30 - 15.55 Berardino D'Acunto: Mathematical Modelling of Multispecies Biofilms
16.00 - 16.25 Sebastiano Pennisi: A 16 Moments Model in Relativistic Extended Thermodynamics of Rarefied

Polyatomic Gas
16.30 - 16.55 Francesco Demontis: Reflectionless Solutions for Square Matrix Nonlinear Schroedinger Equation with

Vanishing Boundary Conditions
17.00 - 17.25 COFFEE BREAK

17.25 - 17.50 Massimo Trovato: The Extended Thermodynamics for A.C and D.C. Dynamic High-Field Transport in

17.55 - 18.10 Luigi Frunzo: Mathematical Modeling of Dispersal Phenomenon in Biofilms
18.15 - 18.30 Andrea Trucchia: Uncertainty and Sensitivity Analysis for Bacterial Invasion in Multi-Species Biofilms

20.00 Dinner (Hotel Pietra di Luna, Via Gaetano Capone 27)



Wednesday, June 12th

09.00 - 09.25 Tommaso Ruggeri: From the Relativistic Mixture of Gases to the Relativistic Cucker-Smale Flocking
09.30 - 09.45 Jeongho Kim: From the Relativistic Mixture of Gases to the Relativistic Cucker-Smale Flocking:

Mathematical Analysis
09.50 - 10.15 Francesca Brini: On the Hyperbolicity Property of Extended Thermodynamics Models for Rarefied Gases
10.20 - 10.45 Elvira Barbera: Stationary Flow and Heat Transfer in Extended Thermodynamics
10.50 - 11.15 COFFEE BREAK

11.15 - 11.40 Vittorio Romano: Mathematical Modeling of Charge Transport in Graphene
11.45 - 12.10 Andrea Mentrelli: Comparison of Shock Structure Behaviours for Increasing Order of Closures
12.15 - 12.40 Bruno Buonomo: Optimal Public Health Systems Intervention to Favor Vaccine Propensity for Childhood

12.45 - 13.00 Giovanni Nastasi: Numerical Solutions of the Semiclassical Boltzmann Equation for Bipolar Charge

Transport in Graphene
13.05 - 13.20 Giorgio Martalò: Analysis of Evaporation-Condensation Problems for a Binary Gas Mixture

13.30 Lunch (Hotel Pietra di Luna, Via Gaetano Capone 27)

15.30 - 15.45 Session in honour of Masaru Sugiyama
15.45 - 16.10 Masaru Sugiyama: Rational Extended Thermodynamics of a Rarefied Polyatomic Gas with Relaxation

Processes of Molecular Rotation and Vibration
16.15 - 16.40 Takashi Arima: Dispersion Relation of a Rarefied Polyatomic Gas with Molecular Relaxation Processes

Based on Rational Extended Thermodynamics with 15 Fields
16.45 - 17.10 COFFEE BREAK
17.10 - 17.25 Session in honour of Giuseppe Toscani
17.25 - 17.50 Giuseppe Toscani: Kinetic Modeling of Alcohol Consumption
17.55 - 18.20 Laurent Desvillettes: About a Class of Cross Diffusion Systems Arising in Chemotaxis

20.00 SOCIAL DINNER (Reginna Palace Hotel, Via Cristoforo Colombo 1)



Thursday, June 13th

09.00 - 09.25 Giuseppe Saccomandi: Helmholtz-Type Solitary Solutions in Non-Linear Elastodynamics
09.30 - 09.55 Wendi Wang: Global Analysis of Mathematical Models for Nonlocal Epidemic Diseases
10.00 - 10.25 Natale Manganaro: Generalized Simple Waves for Hyperbolic Systems
10.30 - 10.55 COFFEE BREAK

10.55 - 11.20 Maria Groppi: Consistent BGK Models for Gas Mixtures and Hydrodynamic Equations
11.25 - 11.50 Marzia Bisi: Maxwell-Stefan Equations for a Reactive Mixture of Polyatomic Gases
11.55 - 12.20 Francesco Mainardi: On the Evolution of Fractional Diffusive Waves
12.25 - 12.50 Gaetana Gambino: Spatial Patterns and Bistability in a Cross-Diffusive FitzHugh-Nagumo System
12.55 - 13.10 Gianfranco Rubino: Resonant Turing Patterns in the FitzHugh-Nagumo Model with Cross Diffusion
13.15 - 13.30 Francesco Gargano: Transition to Turbulence in the Weakly Stratified Kolmogorov Flow

13.30 Lunch (Reginna Palace Hotel, Via Cristoforo Colombo 1)

20.00 Dinner (Hotel Pietra di Luna, Via Gaetano Capone 27)

Friday, June 14th

09.00 - 09.25 Shigeru Taniguchi: On the Similarity Solution of Strong Spherical Shock Waves Based on Extended

09.30 - 09.55 Paolo Maremonti: Weak Solutions To The Navier-Stokes Equations With Non Decaying Data
10.00 - 10.25 Raffaele Vitolo: Linearly Degenerate Systems of PDEs and Interacting Waves
10.30 - 10.55 COFFEE BREAK

10.55 - 11.20 Michele Sciacca: Extension of the Auxiliary Equation Method by Means of Hyperelliptic Functions
11.25 - 11.40 Qinghua Xiao: The Riemann Problem of Relativistic Euler Equations
11.45 - 12.00 Maria Rosaria Mattei: Modeling Cell Motility in Biofilms
12.05 - 12.20 Vincenzo Luongo: Biosorption of Heavy Metals in a Nitrifying Biofilm
12.25 - 12.40 Vittorio De Falco: The General Relativistic Poynting-Robertson Effect: Non-Linear Dissipative System in

General Relativity
12.45 - 13.00 Alberto Tenore: Modelling of Ecology in a Phototrophic-Heterotrophic Biofilm
13.05 - 13.20 Daniele Bernardo Panaro: Anaerobic Digestion in Plug-Flow Reactors: a Mathematical Model
13.25 - 13.30 CLOSING

13.30 Lunch (Reginna Palace Hotel, Via Cristoforo Colombo 1)


JUNE 10-14, 2019




Takashi Arima

National Institute of Technology, Tomakomai College (Japan)
[email protected]

Rational extended thermodynamics (ET) [1, 2] has been developed as a thermodynamic theory being
applicable to nonequilibrium phenomena with steep gradients and rapid changes in space-time, which are out
of local equilibrium. Recently, a refined version of ET of rarefied polyatomic gases with 15 fields which
generalizes the Navier-Stokes and Fourier theory has been proposed [3]. The theory describes the relaxation
processes of molecular rotational and vibrational relaxation processes individually.
In this talk, I present the theoretical study of the dispersion relation of a rarefied polyatomic gas basing on the
theory [4]. Its temperature dependence is discussed in the cases where the rotational and vibrational modes
may or may not be excited. The experimental data obtained in the low-frequency region show the validity of
the theory [5]. It is also shown that the curve of the attenuation per wavelength with respect to the frequency
has up to three peaks depending on the temperature and on the relaxation times.
This is the joint work with M. Sugiyama and T. Ruggeri.

Fig.1: Dependence of the attenuation per wave length ߙఒ on the dimensionless frequency ߗ for various
temperatures. The left and right figures show the case that the rotational relaxation time is large and small

respectively. ܿ௩Ƹோ, ܿ௩Ƹ௏are the dimensionless rotational and vibrational specific heats.
[1] I. Müller, T. Ruggeri: Rational Extended Thermodynamics, Springer Tracts in Natural Philosophy 37 (II

edition), Springer-Verlag, New York (1998).
[2] T. Ruggeri, M. Sugiyama: Rational Extended Thermodynamics beyond the Monatomic Gas, Springer,

Cham, Heidelberg, New York, Dordrecht, London (2015).
[3] T. Arima, T. Ruggeri and M. Sugiyama: Entropy 20, 301 (2018).
[4] T. Arima and M. Sugiyama: AIP Conf. Proc. To be published.
[5] T. Arima, T. Ruggeri and M. Sugiyama: Phys. Rev. E, 96, 042143 (2017).



Elvira Barbera, Francesca Brini
Department of Mathematical, Computer, Physical and Earth Sciences,

University of Messina (Italy)
[email protected]

In the last 10 years, a particular attention was devoted to the stationary heat transfer in bounded domains
within the context of extended thermodynamics. It was shown that 13-moments extended thermodynamics is
already able to predict differences from the classical Navier-Stokes thermodynamics and it implies solutions
which are in agreement with the kinetic theory.
The differences are more visible when different geometries are considered and/or a velocity field is present.
The aim of the talk is the presentation of the different results obtained in the context together with some
general considerations and future prospectives.



Benjamin Anwasia, Marzia Bisi, Francesco Salvarani, Ana Jacinta Soares

Department of Mathematics and Computer Science,
University of Parma (Italy)
[email protected]

We present the derivation of a hydrodynamic description of Maxwell-Stefan type for a reactive mixture of
polyatomic gases with a continuous structure of internal energy. The macroscopic equations are derived in the
diffusive limit of a kinetic system of Boltzmann equations for the considered mixture, in the general non-
isothermal setting. More precisely, the Maxwell-Stefan system is deduced as a proper asymptotic limit of the
kinetic system proposed in [1], based on the Borgnakke- Larsen procedure, that describes a mixture of reactive
polyatomic gases by adding to the usual independent variables of the phase-space of the system (time ‫ݐ‬,
position ‫ ݔ‬and velocity ‫ )ݒ‬a continuous positive internal energy variable ‫ ܫ‬which governs, together with the
kinetic energy, the binary encounters both of reactive and of non-reactive type. For simplicity we consider a
mixture of four constituents, subject to a bimolecular and reversible chemical reaction. The asymptotic analysis
of the kinetic system is performed under a reactive-diffusive scaling in which mechanical collisions are
dominant with respect to chemical reactions. The resulting system couples the Maxwell-Stefan equations for
the species diffusive fluxes with the evolution equations for the species number densities and for the
temperature of the mixture. With respect to the standard isothermal non-reactive Maxwell-Stefan system [2],
here the continuity equations for the various species are balance equations including effects of the chemical
reactions on the number densities, and we have also a proper energy balance equation due to transfer of
kinetic energy into internal energy and vice versa. The production terms due to the chemical reaction and the
Maxwell-Stefan diffusion coefficients are explicitly obtained in terms of the collisional kernels and of the
parameters of the kinetic model, including the internal energy of polyatomic particles [3].
[1] L. Desvillettes, R. Monaco, F. Salvarani, A kinetic model allowing to obtain the energy law of polytropic

gases in the presence of chemical reactions, Europ. J. Mech. B/Fluids, 24 (2005), 219-236.
[2] H. Hutridurga, F. Salvarani, On the Maxwell-Stefan diffusion limit for a mixture of monatomic gases,

Math. Meth. Appl. Sci., 40 (2017), 803-813.
[3] B. Anwasia, M. Bisi, F. Salvarani, A.J. Soares, On the Maxwell-Stefan diffusion limit for a reactive

mixture of polyatomic gases in non-isothermal setting, submitted.



Francesca Brini, Tommaso Ruggeri
Department of Mathematics and Alma Mater Research Center on Applied Mathematics AM2,

University of Bologna (Italy)
[email protected]

Rational Extended Thermodynamics (RET) is a well-known phenomenological field theory able to describe
non-equilibrium phenomena and rapid changes in space-time out of local equilibrium. The theory is
constructed starting from the validity requirement of universal principles, such as the objectivity principle and
the entropy principle. This gives the theory a particularly elegant and robust structure both from the
mathematical and the physical points of view. In fact, the RET models are expected to be hyperbolic PDE
systems with a convex extension, so that the well-posedness of the Cauchy problem is guaranteed. The
hyperbolicity property is also very important for a realistic physical description, since it is associated to finite
speeds of disturbances, in contrast to the infinite speed predicted by the parabolic models of Classical
Thermodynamics. Usually, the RET systems are linearized in the neighborhood of an equilibrium state,
thereby providing systems of Grad's type and confining the validity of the convexity requirement only to a
neighborhood of the equilibrium. Consequently, also the hyperbolicity condition remains valid only in some
domain of the state variables (called hyperbolicity region). The analysis about the determination of such
region started more than 25 years ago by Mueller and Ruggeri.
In this talk we present some very promising results in the case of rarefied monatomic or polyatomic gases
and compare them with what is already known in the literature.



Stéphane Brull
CNRS, Bordeaux INP, IMB,
University of Bordeaux (France)
[email protected]

The aim of this method is to develop a deterministic numerical method for kinetic equations that is adaptative
w.r.t the velocity variable. In the classical methods, the velocity grids are chosen identical for each space
point and constant in time. Moreover, the construction of such a global grid is based only on the initial
conditions. However, in the context of rarefied gas flows, such as the airflow around the walls of a shuttle,
important gradients of velocity and temperature can appear. The idea of this work is to define dynamic sets
of discrete velocities independently for every species and every space discretization point. These sets are
then defined according to the local value of the partial moments of each distribution function, by assuming
them to be Maxwellian distributions. To adapt dynamically to the gradients of macroscopic quantities, partial
moments are computed by the use of conservation laws obtained by taking the moments of the discrete
kinetic equations. This formulation allows an implicit treatment of the relaxation operator leading to an
Asymptotic-Preserving scheme for the Euler regime. The method is then implemented and tested on the
BGK model for gas mixtures that has been proposed by Andries, Aoki and Perthame.



Bruno Buonomo

Department of Mathematics and Applications “Renato Caccioppoli”,
University of Naples “Federico II” (Italy)
[email protected]

In this talk we present a recent analysis of optimal time-profiles of public health systems (PHS) Intervention to
favor vaccine propensity [1]. We apply optimal control (OC) to a SIR model with voluntary vaccination and PHS
intervention. We focus on short-term horizons, and on both continuous control strategies resulting from the
forwardbackward sweep deterministic algorithm, and piecewise-constant strategies (which are closer to the
PHS way of working) investigated by the simulated annealing (SA) stochastic algorithm.
For childhood diseases, where disease costs are much larger than vaccination costs, the OC solution sets at
its maximum for most of the policy horizon, meaning that the PHS cannot further improve perceptions about
the net benefit of immunization. Thus, the subsequent dynamics of vaccine uptake stems entirely from the
declining perceived risk of infection (due to declining prevalence) which is communicated by direct contacts
among parents, and unavoidably yields a future decline in vaccine uptake. We find that for relatively low
communication costs, the piecewise control is close to the continuous control. For large communication costs
the SA algorithm converges towards a non- monotone OC that can have oscillations.
[1] B. Buonomo, A. d’Onofrio, P. Manfredi: Optimal time–profiles of Public Health Intervention to shape

voluntary vaccination for childhood diseases. J. Math. Biol., 78, n. 4, 1089–1113 (2019).



Florinda Capone, Roberta De Luca, Salvatore Rionero
Department of Mathematics and Applications “Renato Caccioppoli”,

University of Naples “Federico II” (Italy)
[email protected]

Magneto-hydrodynamic (MHD) convection in horizontal porous layers, filled by electrically conducting fluids,
uniformly heated from below and embedded in an external transverse constant magnetic field, is analysed
[1, 2, 3]. Long-time behaviour of solutions is characterized via the existence of ‫ܮ‬ଶ-absorbing sets. A new
methodology [7, 8] to obtain necessary and sufficient conditions guaranteeing the onset of steady/unsteady
convection, is applied. By mean of the Energy Linearization Principle [5, 6, 7], the absence of subcritical
instabilities without any restrictions on the initial data, is proved. Applications to unsalted and salted porous
fluid layers with the Vadasz inertia term [4, 9, 10], are provided.


[1] F. Capone, S. Rionero, Porous MHD convection: stabilizing effect of magnetic field and bifurcation

analysis. Ric. Mat. 65, (2016), pp. 163186
[2] F. Capone, R. De Luca, Porous MHD convection: effect of Vadasz inertia term. Transp. Porous Media,

(3), (2017), pp. 519-536
[3] F. Capone, R. De Luca, Double diffusive convection in porous media under the action of a magnetic

field. Ric. Mat. DOI:
[4] D. A. Nield, A. Bejan, Convection in Porous Media, 5th ed., Springer, Berlin, (2017)
[5] S. Rionero, Heat and mass transfer by convection in multicomponent Navier-Stokes mixture: absence

of subcritical instabilities and global nonlinear stability via the Auxiliary System Method. Rend. Lincei
Mat. Appl. 25, 368 (2014)
[6] S. Rionero, Dynamic of thermo-MHD flows via a new approach. Atti Accad. Naz. Lincei Cl. Sci. Fis.
Mat. Natur. Rend. Lincei, 28, 21-47, (2017)
[7] S. Rionero, Hopf bifurcations and global nonlinear L2 energy stability in thermal MHD. Atti Accad. Naz.
Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (To appear)
[8] S. Rionero, Hopf bifurcations in dynamical systems. Ric. Mat. (To appear)
[9] P. Vadasz, Coriolis effect on gravity-driven convection in a rotating porous layer heated from below. J.
Fluid Mech. 376, 351375 (1998)
[10] P. Vadasz, Fluid flow and heat transfer in rotating porous media. SpringerBriefs in Applied Sciences
and Technology, Springer, Minneapolis (USA), (2016)



Giancarlo Consolo, Salvatore Federico, Giovanna Valenti

Department of Mathematical, Computer, Physical and Earth Sciences,
University of Messina (Italy)
[email protected]

The possibility to control the magnetization state of nanostructures via electric fields has enabled new
research frontiers that involve multiferroic materials. The coupling between magnetic and mechanical
energies provides these devices with great potential in a wide variety of applications, even though the weak
ferromagnetism at room temperature of natural multiferroics limits their applications. To overcome such a
problem, a valid alternative consists in depositing a thin magnetostrictive layer onto a thick piezoelectric
actuator. In such bilayer structures, the planar strains undergone by the piezoelectric material under the
application of an electric voltage are transferred to the magnetostrictive layer. The resulting piezo-induced
strains may be used to control the propagation of magnetic domain-walls into the magnetostrictive material.
Here, this phenomenon is theoretically investigated in the framework of the extended Landau- Lifshitz-Gilbert
equation [1]. In particular, the present study focuses on elucidating how the crystal symmetry of the
magnetostrictive material may affect the key features exhibited by the propagating walls in both steady and
precessional dynamical regime. To this aim, the most common symmetries of isotropic, cubic and hexagonal
systems are taken into account.
A brief review of the literature is first presented in order to address a comparison with some classical
published results [2]. Special focuses are given to the determination of those physical quantities involved in
the characterization of domain-wall dynamics, such as the second-order stress-free magnetostrictive strain
tensor and the magnetoelastic anisotropy field, starting from the knowledge of more primitive objects, i.e. the
fourth-order magnetostriction tensor [3] and the magnetoelastic energy density, respectively.
Then, results of the analytical solution of the extended Landau-Lifshitz-Gilbert equation calculations are
presented. These reveal that the crystal symmetry affects the travelling-wave profile, the domain wall
mobility, the propagation threshold and the breakdown of the steady solution. Moreover, our analysis
suggests a possible strategy to determine the fourth-order magnetostrictive coefficients. Finally, the results
obtained here are in good qualitative agreement with recent experimental observations and might be also
used to improve the performance of these devices.

[1] G. Consolo and G. Valenti, Journal of Applied Physics 121, 043903 (2017).
[2] W.P. Mason, Physical Review 96, 302 (1954).
[3] S. Federico, G. Consolo and G. Valenti, Mathematics and Mechanics of Solids (2018), DOI:




Berardino D’Acunto

Department of Mathematics and Applications “Renato Caccioppoli”,
University of Naples “Federico II” (Italy)
[email protected]

A continuum approach to mathematical modelling of multispecies biofilm formation and growth is presented.
The general situation of a biofilm constituted by ݊ bacterial species and ݉ substrates, nutrients, is considered.
The biological process is governed by ݊ nonlinear hyperbolic partial differential equations, ݉ semilinear
parabolic partial differential equations for substrate diffusion and an ordinary differential equation for the biofilm
thickness. All equations are mutually connected and lead to free boundary value problems that are essentially
hyperbolic. Theorems that prove uniqueness, existence, positiveness of solutions are discussed. Some new
processes are also considered, such as the invasion of new bacterial species and colonization into an already
constituted biofilm. As engineering and industrial application, a model of biofilm-reactor for the wastewater
treatment plants is presented.



Constantine M. Dafermos
Division of Applied Mathematics,

Brown University (USA)
[email protected]

In the setting of a simple hyperbolic system, I will discuss the process by which the vanishing of the
relaxation time yields as zero relaxation limit the unique admissible solution of the associated "equilibrium"
hyperbolic conservation law.



Monica De Angelis, Fabio De Angelis
Department of Mathematics and Applications “Renato Caccioppoli”,

University of Naples “Federico II” (Italy)
[email protected]

The FitzHugh-Rinzel (FHR) system is derived from the FitzHugh-Nagumo model [1–3] to incorporate bursting
phenomenon of nerve cells. Bursting oscillation is an important phenomenon and it is becoming increasingly
important as it is being detected in many different scientific fields. Indeed, phenomena of bursting have been
observed as electrical behaviours in many nerve and endocrine cells such as hippocampal and thalamic
neurons, mammalian midbrain, and pancreatic in ߚ−cells. (see, f.i. [4] and references therein). In the
cardiovascular system, bursting oscillations are generated by the electrical activity of cardiac cells that excite
the heart membrane to produce the contraction of ventricles and auricles [5]. In addition, bursting phenomena
can be observed in several fields of electromechanical engineering such as devices [6] and computational
simulations of nonlinear structural problems [7].
In this study the following (FHR) system:

߲‫ݑ‬ ൌ ‫ܦ‬ ߲ଶ‫ݕ‬ െ ‫ݓ‬ ൅ ‫ݕ‬ ൅ ݂ሺ‫ݑ‬ሻǢ ߲߲‫ݐݓ‬ ൌ ߝሺെߚ‫ݓ‬ ൅ ܿ ൅ ‫ݑ‬ሻǢ ߲‫ݕ‬ ൌ ߜሺെ‫ݑ‬ ൅ ݇ െ ݀‫ݕ‬ሻ
߲‫ݐ‬ ߲‫ݔ‬ଶ ߲‫ݐ‬

is reduced to a nonlinear integro differential equation and the fundamental solution ‫ܪ‬ሺ‫ݔ‬ǡ ‫ݐ‬ሻ is explicitly
determined. The initial value problem in the whole space is analyzed and, when the source term is linear, by
means of ‫ܪ‬ሺ‫ݔ‬ǡ ‫ݐ‬ሻ the explicit solution is obtained. Otherwise, when the source term is a non linear function, an
integral equation is deduced. Moreover, particular solutions of the FitzHugh-Rinzel system have been explicitly

[1] Izhikevich E.M.: Dynamical Systems in Neuroscience: The Geometry of Excitability and Bursting. The

MIT press. England (2007).
[2] Rinzel J. A Formal Classification of Bursting Mechanisms in Excitable Systems, in Math. Topics in

Population Biology, Lecture Notes in Biomathematics, Springer, NY, 71, 1987.
[3] De Angelis, M. Renno, P Existence, uniqueness and a priori estimates for a non linear integro-differential

equation Ricerche di Mat. 57 (2008).
[4] R. Bertram, M. J. Butte, T. Kiemel and A. Sherman. Topological and phenomelogical classification of

bursting oscillations, Bulletin of Mathematical Biology, Vol. 57, No. 3, pp. 413 .39, 1995.
[5] A. Quarteroni, A. Manzoni and C. Vergara The cardiovascular system: Mathematical modelling,

numerical algorithms and clinical applications Acta Numerica (2017), pp. 365-590.
[6] H. Simo, P. Woafo, Bursting oscillations in electromechanical systems, Mechanics Research

Communications 38 (2011) 537 541.
[7] F. De Angelis, D. Cancellara, L. Grassia, A. D’Amore, The influence of loading rates on hardening effects

in elasto/viscoplastic strain-hardening materials Mechanics of Time-Dependent Materials, 22 (4) (2018)



Vittorio De Falco

Research Centre for Computational Physics and Data Processing, Faculty of Philosophy &

Silesian University in Opava (Czech Republic)
[email protected]

In several radiation processes occurring in high-energy astrophysics between an emitting massive source
(represented by structures around a black hole or a neutron star) and a relatively small-sized body, the
electromagnetic radiation field beside exerting an external radial force, plays also a fundamental role in
removing angular momentum and energy from the affected body through a radiation drag force in a relatively
short time. This is known in the literature as Poynting-Robertson (PR) effect, which is a pure general
relativistic effect, configuring as a viscous force, that induces the matter to spiral in or out towards the
compact object depending on the radiation field strength. Having such a model is extremely important to
describe the matter behavior in strong gravitational fields. Such configurations represent unique and natural
laboratories, which allow us both to test Einstein’s theory in strong field regimes and to infer several critical
information on black holes’ structure and neutron stars’ equations of state.
In my talk, I introduce the fundamental concepts underpinning the general relativistic model of the PR effect
in 2 [2, 3] and 3 [5] dimensions. The governing equations of motion can be obtained as a set of coupled non-
linear first order ODEs through the relativity of observer splitting formalism [1], powerful mathematical
technique in General Relativity (GR) for considerably reducing the complexity of the equations under study.
Due to its non-linear structure, numerical treatments are needed to have insight into the geometrical
structure and to understand the main features of this phenomenon. Selected test particle orbits are
displayed, and their properties are de- scribed. This dynamical system admits the existence of a critical
hypersurface, region where gravitational attraction, radiation pressure, and PR drag force are in equilibrium. I
show how to prove its asymptotical stability through classical techniques in linear stability theory or
alternatively by employing a Lyapunov function (having a more deep physical meaning). To better
understand the radiation processes in GR, an analytical treatment of such effect is performed within the
Lagrangian formalism. I explain how to prove that such a dissipative system in GR admits a Lagrangian
formulation [4], which is a very challenging task in GR, never accomplished before in the literature. Then, I
analytically determine the radiation potential by using a new and innovative method termed energy formalism
[6], which I stress its broad applicability in different physical and mathematical contexts.

[1] Bini, D. et al. (1997). IJMP D, 6:1 – 38.
[2] Bini, D. et al. (2009). CQtt, 26:055009.
[3] Bini, D. et al. (2011). CQtt, 28:035008.
[4] De Falco, V. et al. (2018). PRD, 97:084048.
[5] De Falco, V. et al. (2019a). Physical Review D, 99:023014.
[6] De Falco, V. et al. (2019b). Physical Review D.



Florinda Capone, Roberta De Luca, Maria Vitiello

Department of Mathematics and Applications “Renato Caccioppoli”,
University of Naples “Federico II” (Italy)
[email protected]

The onset of double-diffusive convection in horizontal porous layers for the thermo-diffusive Soret
phenomenon is widely studied in literature due to the numerous applications in the real world phenomena [2,
3, 5]. In [1] the more general case of a Darcy model including inertia term [6] is considered. Necessary and
sufficient conditions guaranteeing the onset of steady or unsteady convection in a closed algebraic form are
obtained. Via the Energy Linearization Principle [4], the coincidence between linear and nonlinear (global)
stability thresholds of the thermo-solute conduction solution, is proved.
[1] F. Capone, R. De Luca, M. Vitiello, Double-diffusive Soret convection phenomenon in porous media:

effect of Vadasz inertia term. Ric. Mat. DOI:
[2] N. Deepika, Linear and nonlinear stability of double-diffusive convection with the Soret effect. Trans.

Porous Med 121, 93-108. (2018).
[3] D.A. Nield, A. Bejan. Convection in Porous Media, 5th Ed. Springer, Berlin (2017).
[4] S. Rionero, Dynamic of thermo-MHD flows via a new approach. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat.

Natur. Rend. Lincei 28, 21-47 (2017).
[5] S. Rionero, Soret effects on the onset of convection in rotating porous layers via the “auxiliary system

method”. Ric. Mat. 62(2), 183 (2013).
[6] P. Vadasz, Coriolis effect on gravity-driven convection in a rotating porous layer heated from below. Int.

Fluid Mech. 376. 351-375 (1998).



Francesco Demontis
Department of Mathematics and Informatics,

University of Cagliari (Italy)
[email protected]

After a quick review of the direct and inverse scattering theory of the focusing Zakharov-Shabat system with
symmetric nonvanishing boundary conditions, we derive the reflectionless solutions of the ʹ ൅ ʹ matrix NLS
equation with vanishing boundary conditions and four different symmetries by using the Marchenko theory.
Since the Marchenko integral kernel has separated variables, the matrix triplet method - consisting of
representing the Marchenko integral kernel in a suitable form - allows us to find the exact expressions of the
reflectionless solutions in terms of a triplet of matrices. Moreover, since these exact expressions contain
matrix exponentials and matrix inverses, computer algebra can be used to “unpack” and graph them. Finally,
it is remarkable that these solutions are also veried by direct substitution in the ʹ ൅ ʹ NLS equation.
This is a joint work with C. van der Mee (University of Cagliari) and Alyssa Ortiz (University of
Colorado at Colorado Springs).



Laurent Desvillettes

Universit´e Paris Diderot (France)
[email protected]

We study a class of cross diffusion systems of the form
߲௧‫ ݑ‬ൌ ο௫ሺߛሺ‫ݒ‬ሻ‫ݑ‬ሻǡ

߲௧‫ ݒ‬ൌ ߜο௫‫ ݒ‬൅ ‫ ݑ‬െ ‫ݒ‬ǡ
where ߛ is a decreasing function of ‫ݒ‬. Those systems naturally arise in chemotaxis under specific assumptions
on the way cells move in presence of the chemoattractant. We show existence of weak or strong solutions
(depending on the dimension), and study the large time behavior of the system.
This is a joint work with Y.J. Kim, A. Trescases and C. Yoon.



Paolo Falsaperla, Andrea Giacobbe, Giuseppe Mulone

Department of Mathematics and Computer Sciences,
University of Catania (Italy)
[email protected]

The instability of steady laminar flow of an electrically conducting fluid between two infinite parallel plates under
a transverse magnetic field has been analyzed by Kakutani [1], Takashima [2] for plane Couette flow. Alexakis
et al. [3] studied shear flows with an applied cross-stream magnetic field using dissipative incompressible
magnetohydrodynamics. This study incorporates exact solutions, the energy stability method, and exact
bounds on the total energy dissipation rate. Recently, Falsaperla et al. [4] proved that the plane Couette and
Poiseuille flows are nonlinearly stable with respect to streamwise perturbations for any Reynolds number. They
also proved nonlinear stability results for plane Couette and Poiseuille flows with respect to tilted perturbation
(2D perturbations with a wave vector not directed along the direction of the basic motion). The aim of this work
is to generalize the results of [4] to the hydromagnetic plane Couette flow. We also compare our results with
Alexakis et al. [3], Takashima [2] and experiments.
[1] T. Kakutani, J. Phys. Soc. Japan 19, 1041 (1964).
[2] M. Takashima, Fluid Dyn. Res. 22, 105 (1998).
[3] A. Alexakis, F. Pétrélis, P. J. Morrison, and Charles R. Doering, Phys. of Plasmas 10 (11), 4324 (2003).
[4] P. Falsaperla, A. Giacobbe and G. Mulone, Nonlinear stability results for plane Couette and Poiseuille

flows, submitted. (2019).



Gaetano Fiore
Department of Mathematics and Applications “Renato Caccioppoli”,

University of Naples “Federico II” (Italy)
[email protected]

Applying a recently developed plane hydrodynamical model to the impact of a very short and intense laser
pulse onto a cold diluted plasma, we explore its consequences for the motion of the plasma electrons shortly
after the beginning of the laser-plasma interaction: where and how long the hydrodynamical description holds,
the formation of a plasma wave, the localization of wave-breaking as a function of the initial plasma density
and of the laser pulse, and its use for self-injection of electrons in the laser wake-field acceleration mechanism.
In our plane model the system of the (Lorentz-Maxwell and continuity) PDEs is reduced into a 1 parameter
family of decoupled systems of Hamilton equations in dim 1, and we use Floquet theory to analyze the
dynamics of an associated periodic ODE system.



Berardino D’Acunto, Luigi Frunzo, Vincenzo Luongo, Maria Rosaria Mattei
Department of Mathematics and Applications “Renato Caccioppoli”,
University of Naples “Federico II” (Italy)
[email protected]

The presentation will concern a mathematical model for dispersal phenomenon in multispecies biofilm based
on a continuum approach and mass conservation principles. The formation of dispersed cells is modeled by
considering a mass balance for the bulk liquid and the biofilm. Diffusion of these cells within the biofilm and in
the bulk liquid is described using a diffusion-reaction equation. Notably, biofilm growth is modeled by a
hyperbolic partial differential equation while the diffusion process of dispersed cells by a parabolic partial
differential equation. The two are mutually connected but governed by different equations that are coupled by
two growth rate terms.
The complete model takes the following form:

μ ݂௜ ൅ μ ሺ‫݂ݑ‬௜ ሻ ൌ ܴெǡ௜ ሺ‫ݖ‬ǡ ‫ݐ‬ǡ ࣒ǡ ࢌǡ ࡿሻǡ Ͳ ൑ ‫ ݖ‬൑ ‫ܮ‬ሺ‫ݐ‬ሻǡ ‫ ݐ‬൐ Ͳǡ
μ‫ݐ‬ μ‫ݖ‬

݂௜ሺ‫ݖ‬ǡ Ͳሻ ൌ ߮௜ሺ‫ݖ‬ሻǡ Ͳ ൑ ‫ ݖ‬൑ ‫ܮ‬଴ǡ ݅ ൌ ͳǡ Ǥ Ǥ Ǥ ǡ ݊ǡ

μ‫ݑ‬ ሺ‫ݖ‬ǡ ‫ݐ‬ሻ ൌ ௡ ܴெǡ௜ ሺ‫ݖ‬ǡ ‫ݐ‬ǡ ࣒ǡ ࢌǡ ࡿሻǡ Ͳ ൏ ‫ ݖ‬൑ ‫ܮ‬ሺ‫ݐ‬ሻǡ – ൒ Ͳǡ


‫ݑ‬ሺͲǡ ‫ݐ‬ሻ ൌ Ͳǡ – ൒ Ͳǡ

‫ܮ‬ሶ ሺ‫ݐ‬ሻ ൌ ‫ݑ‬ሺ‫ܮ‬ሺ‫ݐ‬ሻǡ ‫ݐ‬ሻ ൅ ߪ௔ሺ–ሻ െ ߪௗ൫‫ܮ‬ሺ‫ݐ‬ሻ൯ǡ ‫ܮ‬ሺͲሻ ൌ ‫ܮ‬଴ǡ ‫ ݐ‬൐ Ͳǡ

μ߰௜ െ ߲ ሺ‫ܦ‬ெǡ௜ μ߰௜ ሻ ൌ ܴటǡ௜ ሺœǡ –ǡ ࣒ǡ ࢄǡ ࡿሻǡ ݅ ൌ ͳǡ Ǥ Ǥ Ǥ ǡ ܰǡ Ͳ ൏ ‫ ݖ‬൏ ‫ܮ‬ሺ‫ݐ‬ሻǡ ‫ ݐ‬൐ Ͳǡ
μ‫ݐ‬ ߲‫ݖ‬ ߲‫ݖ‬

߰௜ሺ‫ݖ‬ǡ Ͳሻ ൌ Ͳǡ Ͳ ൑ œ ൑ ଴ǡ μ߰௜ ሺͲǡ ‫ݐ‬ሻ ൌ Ͳǡ ߰௜ሺ‫ܮ‬ሺ‫ݐ‬ሻǡ ‫ݐ‬ሻሻ ൌ ߰௜‫כ‬ሺ‫ݐ‬ሻǡ ݅ ൌ ͳǡ Ǥ Ǥ Ǥ ǡ ܰǡ –൐Ͳ
μ‫ݖ‬ – ൐ ͲǤ

μܵ௝ െ ߲ ሺ‫ܦ‬ௌǡ௝ μܵ௝ ሻ ൌ ‫ݎ‬ௌǡ௝ ሺœǡ –ǡ ࢄǡ ࡿሻǡ ݆ ൌ ͳǡ Ǥ Ǥ Ǥ ǡ ݉ǡ Ͳ ൏ ‫ ݖ‬൏ ‫ܮ‬ሺ‫ݐ‬ሻǡ ‫ ݐ‬൐ Ͳǡ
μ‫ݐ‬ ߲‫ݖ‬ ߲‫ݖ‬

ܵ௝ሺ‫ݖ‬ǡ Ͳሻ ൌ ௝଴ሺœሻǡ Ͳ ൑ œ ൑ ଴ǡ μܵ௝ ሺͲǡ ‫ݐ‬ሻ ൌ Ͳǡ ܵ௝ ሺ‫ܮ‬ሺ‫ݐ‬ሻǡ ‫ݐ‬ሻሻ ൌ ܵ௝‫כ‬ሺ‫ݐ‬ሻǡ ݆ ൌ ͳǡ Ǥ Ǥ Ǥ ǡ ݉ǡ

The Mathematical Modelling of three real special cases will be presented. The first is related to experimental
observations on starvation induced dispersal [1]. The second considers diffusion of a non-lethal antibiofilm
agent which induces dispersal of free cells. The third example considers dispersal induced by a self-produced
biocide agent.

[1] D. Schleheck, N. Barraud, J. Klebensberger, J.S. Webb, D. McDougald, S.A. Rice, S. Kjelleberg. (2009)

Pseudomonas aeruginosa PAO1 preferentially grows as aggregates in liquid batch cultures and

disperses upon starvation PloS one, 4, 5.



Gaetana Gambino, Maria Carmela Lombardo, Gianfranco Rubino, Marco Sammartino
Department of Mathematics and Computer Science,
University of Palermo (Italy)
[email protected]

The FitzHugh-Nagumo model, initially derived as a mathematical simplification of the Hodgkin-Huxley model
to describe the flow of an electric current through the surface membrane of a nerve fiber [1, 2], supports a
rich dynamics: tuning the system parameters it exhibits monostability, excitability or bistability [3].
In this talk the effect on Turing pattern formation of the coupling between the FitzHugh-Nagumo kinetics with
linear cross diffusion will be addressed. The cross diffusion is proved to be crucial for pattern formation when
the FitzHugh-Nagumo system is excitable and it is also responsible of a previously unnoticed Turing
mechanism: out-of-phase patterns arise when the inhibitor rapidly diffuse away from the activator but its
random diffusion is almost slow.
The pattern selection problem in the monostable case is solved performing a close to equilibrium asymptotic
weakly nonlinear analysis, which show the existence of square and super-squares when the bifurcation
takes place through a multiplicity-two eigenvalue without resonance [4].
In the bistable case large amplitude patterns emerge due to the interaction of the Turing instabilities on the
two homogeneous steady states branches of an imperfect pitchfork bifurcation. In order to capture these
subcritical structures, the weakly nonlinear analysis is revised in the neighborhood of the cusp point, next to
the nascent bistability, where the zero mode of the homogeneous perturbation becomes active and interacts
with the spatial critical modes [5]. The resulting bifurcation diagrams reveal large domains of coexisting
stable different structures and localized patterns are numerically obtained in these regions.

[1] Hodgkin A.L., Huxley A.F.: A quantitative description of membrane current and its application to

conduction and excitation in nerve. J. Physiol. 117 n.4, 500–544 (1952).
[2] FitzHugh R.: Thresholds and plateaus in the Hodgkin-Huxley nerve equations. J. Gen. Physiol. 43,

867-896 (1960).
[3] Hagberg A., Meron E.: Pattern formation in non-gradient reaction-diffusion systems: the effects of front

bifurcations. Nonlinearity 7 n.3, 805–835 (1994).
[4] Gambino G., Lombardo M. C., Rubino G., Sammartino M.: Pattern selection in the 2D FitzHugh–

Nagumo model. Ricerche di Matematica, First Online (2018).
[5] Borckmans P., Dewel G.,De Wit A., Dulos E., Boissonade J., Gauffre F., De Kepper, P.: Diffusive

Instabilities and Chemical Reactions. Internat. J. Bifur. Chaos 12 n.11, pp. 2307–2332 (2002).



Francesco Gargano, Giacomo Ponetti, Marco Sammartino, Vincenzo Sciacca

Department of Mathematics and Computer Science,
University of Palermo (Italy)

[email protected]

The Kolmogorov flow is a two-dimensional incompressible viscous flow driven by a streamwise monochromatic
force. It was introduced by Kolmogorov as a toy-model capable of easing the mathematical difficulties of the
full Navier-Stokes equations but still possessing the turbulent regimes typical of the Navier-Stokes solutions.
In this talk we shall investigate the various bifurcations leading from laminar solutions toward weakly chaotic
states, extending the results presented in [1] where the density of the flow is not stratified. New chaotic states
are detected by computing the Lyapunov exponents and analyzed in terms of enstrophy and palinstrophy
growth phases [2].
On the other hand, in the density stratified Kolmogorov flow, the bifurcations leading to chaotic states have not
been studied. Adopting the Boussinesq approximation according to which the base density profile has a linear
relationship with the temperature decreasing from the bottom to the top of the fluid, we shall investigate how
this stabilizing effect influences the bifurcations that occur at low Reynolds numbers in the range of small
Richardson numbers. Besides the obvious observation that higher Reynolds numbers are required to trigger
the instabilities, we shall see that, by increasing the temperature gradient, i.e. the Richardson number, new
structures form in the flow, inducing a richer variety of states leading eventually to the chaotic attractors [2].
[1] D. Armbruster, B. Nicolaenko, N. Smaoui, and P. Chossat, Symmetries and dynamics for 2-D Navier-

Stokes flow, Physica D 95, 81–93 (1996).
[2] F. Gargano, G. Ponetti, M. Sammartino, and V. Sciacca, Route to chaos in the weakly stratified

Kolmogorov flow, Phys. Fluids 31, 024106 (2019).



Florinda Capone, Maurizio Gentile
Department of Mathematics and Applications “Renato Caccioppoli”,

University of Naples “Federico II” (Italy)
[email protected]

Thermal convection of a fluid filling an anisotropic porous medium, uniformly rotating about a vertical axis,
with local thermal non equilibrium, is studied. The linear and nonlinear stability analysis are performed. In
particular, the coincidence between linear instability and nonlinear (global) stability thresholds is proved.



Paolo Falsaperla, Andrea Giacobbe, Giuseppe Mulone
Department of Mathematics and Computer Sciences,
University of Catania (Italy)
[email protected]

We investigate the stability of the basic stationary solution of a model for thermal convection in an inclined
porous layer when the fluid motion obeys to the Darcy-Brinkman law. Inertial effects are also taken into
consideration, and different physical boundary conditions are imposed. The model is an extension of the
work by Rees and Bassom, where the Darcy’s law is adopted. In this model the basic motion is a
combination of hyperbolic and polynomial functions.
We will present a numerical investigation of the linear instability of such basic motion for three-dimensional
perturbations; we will give estimates of nonlinear stability thresholds solving a maximum problem for an
energy Lyapunov functional. For longitudinal perturbations we will prove the coincidence of linear and
nonlinear critical Rayleigh numbers.
These types of fluid flows have applications to geophysics, engineering and many other areas (Straughan,
Nield and Bejan and references therein).
[1] Rees DAS, Bassom AP. 2000 The onset of Darcy-Benard convection in an inclined layer heated from

below. Acta Mech. 144 (1-2), 103?118.
[2] Nield DA, Bejan A. 2017 Convection in Porous Media. Springer, New York, 5th Edition.
[3] Straughan B. 2004 The Energy Method, Stability, and Nonlinear Convection, Springer-Verlag: Ser. In

Appl. Math. Sci., 91, New-York, 2nd Ed.
[4] Straughan B. 2008 Stability, and wave motion in porous media, volume 165 of Appl. Math. Sci.

Springer, New York.
[5] Falsaperla P., Mulone G., 2018 Thermal convection in an inclined po- rous layer with Brinkman law.

Ric. Mat. p. 1-17, ISSN: 0035-5038, doi: 10.1007/s11587-018- 0371-2.
[6] Falsaperla P., Giacobbe A., Mulone G, 2019 Inclined convection in a porous Brinkman layer: linear

instability and nonlinear stability, submitted.



Valeria Giunta
Department of Mathematics and Computer Science,

University of Palermo (Italy)
[email protected]

Multiple Sclerosis (MS) is an inflammatory disorder that affects the central nervous system causing severe
and progressive physical and neurological impairment. MS is characterized by myelin damage and loss,
resulting in the formation of dense, scar-like tissue called plaques.
In [4] and [1] a mathematical model was developed which is able to reproduce many of the typical
pathological hallmarks of the disease.
The aim of the present talk is twofold. First we shall study the aggregation phenomena described by the
reaction-diffusion-chemotaxis model introduced in [1]. In particular, we shall investigate the conditions which
yield the appearance of stationary non constant radially symmetric solutions and, using numerical values of
the parameters taken from the experimental literature, we shall show that the model supports the formation
of stationary patterns that closely reproduce the concentric lesions observed in clinical practice, see [2].
Second we shall investigate the qualitative properties (like existence and uniqueness in the appropriate
function space) of the solutions of the model. We shall in fact show how the inclusion of the volume filling
sensitivity term is able to prevent finite-time blow-up of the solutions, see [5].
Joint work with L. Desvillettes (Université Paris Diderot, France), M.C. Lombardo (University of
Palermo, Italy) and M. Sammartino (University of Palermo, Italy).

[1] M.C. Lombardo, R. Barresi, E. Bilotta, F. Gargano, P. Pantano, and M. Sammartino. Demyelination

patterns in a mathematical model of multiple sclerosis. Journal of Mathematical Biology, 75(2):373-
417, 2017.
[2] E. Bilotta, F. Gargano, V. Giunta, M. C. Lombardo., P. Pantano, and M. Sammartino. Eckhaus and
zigzag instability in a chemotaxis model of multiple sclerosis. Atti della Accademia Peloritana dei
Pericolanti-Classe di Scienze Fisiche, Matematiche e Naturali 96.S3 (2018): 9.
[3] T. Hillen and K.J. Painter. A user’s guide to PDE models for chemotaxis. Journal of Mathematical
Biology, 58(1-2):183?217, 2009.
[4] R.H. Khonsari and V. Calvez. The origins of concentric demyelination: Self- organization in the human
brain. PLoS ONE, 2(1), 2007.
[5] L. Desvillettes, V. Giunta. Well-posedness for a Multiple Sclerosis model, in preparation, 2019.



Henri Gouin, Sergey Gavrilyuk

Aix-Marseille University (France)

[email protected]

Membranes are an important subject of study in physical chemistry and biology. They can be considered as
material surfaces with a surface energy depending on the curvature tensor. Usually, mathematical models
developed in the literature consider the dependence of surface energy only on mean curvature with an added
linear term for Gauss curvature [1, 2]. Therefore, for closed surfaces the Gauss curvature term can be
eliminated because of the Gauss-Bonnet theorem. In [3], the dependence on the mean and Gaussian
curvatures was considered in statics and under a restrictive assumption of the membrane inextensibility.
Thanks to the principle of virtual working, the equations of motion and boundary conditions governing the fluid
membranes subject to general dynamical bending are derived without the membrane inextensibility
assumption. We obtain the dynamic “shape equation” (equation for the membrane surface) and the dynamic
conditions on the contact line generalizing the classical Young-Dupré condition.
[1] W. Helfrich: Elastic properties of lipid bilayers: theory and possible experiments, Z. Naturforsch. C 28,

693–703 (1973).
[2] B. Alberts, A. Johnson, J. Lewis, M. Raff, K. Roberts, P. Walter: Molecular biology of the cell, Garland

Science, New York (2002).
[3] R. Russo, E.G. Virga: Adhesive borders of liquid membranes. Proceedings of the Royal Society of

London A 455, 4145-4168 (1999).
[4] H. Gouin: Vesicle Model with Bending Energy Revisited, Acta Applicandae Mathematicae 132, 347-358

(2014) & arXiv:1510.04824.
[5] S. Gavrilyuk, H. Gouin: Dynamics and boundary conditions for membranes whose surface energy

depends on the mean and Gaussian curvatures, Mathematics and Mechanics of Complex Systems, In
press (2019) & arXiv:1812.06646.



Maria Groppi
Department of Mathematical, Physical and Computer Sciences,

University of Parma (Italy)
[email protected]

Kinetic BGK models are often used in various applications in rarefied gas dynamics and plasma physics,
because of the complexity of nonlinear Boltzmann-type kinetic equations describing the dynamics of
multicomponent gases. In this talk, some consistent relaxation time-approximation models of BGK-type for
inert gas mixtures are presented and their main properties are discussed [1, 2]. Consistency means three
basic properties: correct reproduction of conservation laws, H-theorem and uniqueness of equilibrium
The main peculiarities of the presented BGK models will be highlighted with reference to their continuum
limits obtained by Chapman-Enskog expansions [3]. In particular, it will be shown that a recent BGK model
[2], reproducing the structure of the Boltzmann collision operator for mixtures and well suited to deal with
various intermolecular collisional potentials, can lead in the hydrodynamic limit, in a proper collision
dominated regime, to multitemperature and multivelocity Euler and Navier Stokes closures.
Joint work with M. Bisi, G. Martalò, and G. Spiga, Department of Mathematical, Physical and
Computer Sciences, University of Parma.
[1] M. Groppi, G. Russo, G. Stracquadanio, Semi-Lagrangian approximation of BGK models for inert and

reactive gas mixtures, in “From Particle Systems to Partial Differential Equations V”, Springer
Proceedings in Mathematics and Statistics 258, Patricia Gonçalves and Ana Jacinta Soares (Eds.)
(2018), p. 53-80.
[2] A.V. Bobylev, M. Bisi, M. Groppi, G. Spiga, I.F. Potapenko, A general consistent BGK model for gas
mixtures, Kinet. Relat. Models 11 (2018), 1377–1393.
[3] M. Bisi, A.V. Bobylev, M. Groppi, G. Spiga, Hydrodynamic Equations from a BGK Model for Inert Gas
Mixtures, AIP Conference Proceedings, RGD 31 2018, in press.



Seung Yeal Ha
Department of Mathematical Sciences,

Seoul National University (Korea)
[email protected]

Self-organization of complex systems has received lots of attention in scientific disciplines such as applied
mathematics, biology, control theory of multi-agent system, statistical physics due to many recent
applications in cooperative robot system, unmanned aerial vehicles such as drones and sensor networks etc.
In literature, first-order models have been used in the collective modeling of complex system from the
beginning. In this talk, we will discuss how to lift first-order models to second-order ones by preserving
emergent dynamics and reduction to the first-order model in some limiting situation.
This talk is based on joint works with Dohyun Kim (NIMS).



Jeongho Kim
Department of Mathematical Sciences,

Seoul National University (Korea)
[email protected]

We present a mathematical properties of the relativistic correction for the CS flocking model introduced in the
previous talk by Prof. Tommaso Ruggeri. More precisely, we provide a sufficient framework leading to the
exponential flocking of the relativistic CS model in terms of communication weights. We also show that the
relativistic CS model reduces to the classical CS model, as the speed of light c tends to infinity in any finite-
time interval. For the analytical simplicity, we also present the approximated relativistic CS model, in which
ܱሺܿିଶ ሻ term in the equation was ignored. Then, with this simplified model, we provide the kinetic and
hydrodynamic description of it, as well as their emergent behaviors.
This work was collaborated with Prof. Tommaso Ruggeri and Prof. Seung-Yeal Ha.



Tai-Ping Liu

Institute of Mathematics, Academia Sinica (Taiwan) and
Department of Mathematics, Stanford University (USA)

[email protected]
The traditional interpretation of Hadamard's Well-posedness notion for differential equations is too strong for
incompressible Euler equations and even for compressible Euler equations. Method of convex integration has
produced multiple solutions with a given initial data. The problem arises for the weak solutions. On the other
hand, there is the well-posedness theory for the weak solutions for system of hyperbolic conservation laws. In
this talk, historical perspective will be given and recent work of Shih-Hsien Yu and the author on the well-
posedness of weak solutions for compressible Navier-Stokes equation will be reported. In particular, we
propose that a new notion of well-posedness is necessary. This new notion seems to be physically and
analytically natural.



Maria Carmela Lombardo, Valeria Giunta, Marco Sammartino

Department of Mathematics and Computer Science,
University of Palermo (Italy)

[email protected]

The aim of this talk is to introduce and study a reaction-diffusion-chemotaxis model that describes the initial
stages of a wide class of inflammatory diseases.
Inflammation is the response to outside insults, aimed at eliminating the threat and promote tissue repair and
healing. It is a highly complex process, characterized by the action of both pro- and anti-inflammatory agents
that work synergistically to ensure a quick restoration of tissue health. Inflammation is also believed to play a
central role in the pathophysiology of many common disorders, including some degenerative patologies,
such as Alzheimer’s, atherosclerosis and Multiple Sclerosis.
In the last years several mathematical modeling approaches have been adopted to provide insights on the
major pathological processes involved in inflammation [1, 2]. Despite the increasing interest in this area,
there are only few models that incorporate spatial aspects in the description of inflammation driven diseases
[3, 4, 5].
We shall design a model describing the spatio-temporal dynamics of a population of immune cells and of two
different types of signaling molecules: a pro-inflammatory chemokine, which is the chemoattractant for the
immune cells, and an anti-inflammatory cytokine, which acts, on a longer time scale, as an inhibitor of the
inflammatory state [6].
We are interested in the model capability of reproducing aggregation phenomena leading to the formation of
localized patches of inflammatory activity. To this end we investigate the conditions on the system
parameters that determine the excitation of Turing and wave instabilities. The investigation is conducted by
considering biologically realistic values of the introduced parameters, all of which are taken from the existing
We shall show that, varying the control parameters, the model is able to reproduce qualitatively different
pathological scenarios: a diffused inflammatory state of the type observed in many cutaneous rashes, the
formation of stationary patches of inflammation and the ring-shaped skin rashes observed in Erythema
Annulare Centrifugum (EAC), a very aggressive form of cutaneous eruption [7].
For large values of both the anti-inflammatory time scale and the chemotaxis coefficient, the analysis yields
the presence of large regions in the parameters space where a wave instability occurs, corresponding to the
formation of oscillating-in-time spatial patterns, that qualitatively reproduce the periodic appearance of
localized skin eruptions characteristic of the Recurrent Erythema Multiforme (REM) [8]. Therefore, the
present model proposes a possible mechanism for explaining the insurgence of recurrent inflammations,
whose etiology is still unknown.
The proposed system also displays a cascade of successive bifurcations leading to chaotic behavior, already
observed in existing chemotaxis models. The mechanism of the observed route-to-chaos and its relationship
with self-organized criticality of macrophages will be discussed.
Finally the issue of localized pattern will be addressed: we shall show the presence, in a well-defined region
of the parameter space called the pinning region, of a multiplicity of bifurcating branches of localized states,
whose bifurcation diagram is organized in a characteristic snakes-and-ladders structure, called homoclinic
snaking [9]. The bifurcation structure and the stability properties of the localised solutions will be
investigated, both theoretically and numerically, in the strongly nonlinear regime as the control parameter is
varied away from the primary bifurcation value.


[1] Painter, K.J. Mathematical models for chemotaxis and their applications in self-organisation

phenomena (2018) Journal of Theoretical Biology. Article in Press.
[2] Ramirez-Zuniga, I., Rubin, J.E., Swigon, D., Clermont, G. Mathematical modeling of energy

consumption in the acute inflammatory response (2019) Journal of Theoretical Biology, 460, pp. 101-
[3] Penner, K., Ermentrout, B., Swigon, D. Pattern formation in a model of acute inflammation (2012)
SIAM Journal on Applied Dynamical Systems, 11 (2), pp. 629-660.
[4] Chalmers, A.D., Cohen, A., Bursill, C.A., Myerscough, M.R. Bifurcation and dynamics in a
mathematical model of early atherosclerosis: How acute inflammation drives lesion development
(2015) Journal of Mathematical Biology, 71 (6-7), pp. 1451-1480.
[5] Lombardo, M.C., Barresi, R., Bilotta, E., Gargano, F., Pantano, P., Sammartino, M. Demyelination
patterns in a mathematical model of multiple sclerosis (2017) Journal of Mathematical Biology, 75 (2),
pp. 373-417.
[6] Giunta, V., Lombardo, M.C., Sammartino, M. Pattern formation and transition to chaos in a
mathematical model of acute inflammation (2019) submitted.
[7] Bilotta, E., Gargano, F., Giunta, V., Lombardo, M.C., Pantano, P., Sammartino, M. Axisymmetric
solutions for a chemotaxis model of Multiple Sclerosis (2018) Ricerche di Matematica, pp. 1-14. Article
in Press.
[8] Lerch, M., Mainetti, C., Terziroli Beretta-Piccoli, B., Harr, T. Current Perspectives on Erythema
Multiforme (2018) Clinical Reviews in Allergy and Immunology, 54 (1), pp. 177-184.
[9] Knobloch, E. Spatial localization in dissipative systems (2015) Annual Review of Condensed Matter
Physics, 6 (1), pp. 325-359.



Berardino D’Acunto, Luigi Frunzo, Vincenzo Luongo, Maria Rosaria Mattei
Department of Mathematics and Applications “Renato Caccioppoli”,
University of Naples “Federico II” (Italy)
[email protected]

A mathematical model for heavy metals interaction in a nitrifying multispecies biofilm is proposed. The model
is based on a continuum approach and mass conservation principles. A diffusion-reaction equation describes
the dynamics of a toxic heavy metal within the multispecies biofilm system. Two systems of hyperbolic partial
differential equations define the binding sites evolution during biofilm growth. The latter is governed by a
system of hyperbolic equations for microbial species growth and a nonlinear ordinary differential equation
describing the biofilm thickness evolution as a free boundary problem. The substrate diffusion-reaction within
the biofilm is described with an additional system of parabolic partial differential equations. The model is
applied to a case study reproducing a biofilm devoted to municipal wastewater treatment. Numerical
simulations confirm the model consistency and highlight the adaptive behaviour of such complex microbial



Armando Consiglio, Francesco Mainardi

Department of Physics and Astronomy,
University of Bologna (Italy)

[email protected]

In physics, process involving the phenomena of diffusion and wave propagation have great relevance; these
physical processes are governed, from a mathematical point of view, by partial differential equations of order
1 and 2 in time. It is known that, whereas the diffusion equation describes a process where the disturbance
spreads infinitely fast, the wave-front velocity of the disturbance is finite for the wave equation. By introducing
a fractional derivative of order α in time with ͳ ൑ ߙ ൑ ʹ, we are lead to processes that, in mathematical
physics, we may refer to as fractional diffusive waves. The use of the Laplace transform in the analysis of the
Cauchy and Signalling problems leads to a special function of the Wright type, nowadays known as M-Wright
function. In this work we want to show that the time-fractional diffusion-wave equation interpolates between
the two different responses, studying and simulating both the situations in which the data function (initial signal)
is a Dirac delta generalized function, that leads to the fundamental solution, and the one in which the data
function (initial signal) is provided by a box-function. In the latter case the solutions are obtained by a
convolution of the Green function with the initial data function.
The work of FM has been carried out in the framework of the activities of the National Group of Mathematical
Physics (INdAM-GNFM).
[1] Yu. Luchko and F. Mainardi, Cauchy and signaling problems for the time-fractional diffusion-wave

equation, ASME Journal of Vibration and Acoustics 136 No 5 (2014), 050904/1–7. DOI:
10.1115/1.4026892; E-print arXiv:1609.05443.
[2] F. Mainardi, The fundamental solutions for the fractional diffusion-wave equation, Applied Mathematics
Letters 9 No 6 (1996), 23–28.



Natale Manganaro

University of Messina (Italy)
[email protected]

Simple wave solutions are of great interest for nonlinear wave problems. Such a class of solutions are
admitted by first order quasilinear hyperbolic systems in the homogeneous case and they are useful for
solving different problems of interest in the applications as, for instance, Riemann problems. Unfortunately
simple waves are not usually admitted by hyperbolic systems when dissipative effects are taken into account
(non-homogeneous case). Within the theoretical framework of the method of differential constraints, here we
describe a possible strategy aimed at generalizing simple wave solutions for nonhomogeneous hyperbolic
systems. An example of interest in fluid dynamics is presented.



Paolo Maremonti

Department of Mathematics and Physics,
University of Campania “L. Vanvitelli” (Italy)

[email protected]

In [1] we investigate on the Navier-Stokes initial boundary value problem in 3D-exterior domains with non
decaying initial data. We are interested to prove the existence of weak solutions defined for all ‫ ݐ‬൐ Ͳ for large

non decaying initial data. We extend the technique already developed in the paper [2] to the 3D-exterior


This special problem is part of the peculiar literature related to the solutions global in time with non finite energy,

that in the last decade has attracted the interest of some authors. We believe that this literature can be referred

to two different branches. One branch looks for the existence of solutions to the problem in the class of the

physically reasonable steady solutions of the Navier-Stokes equations. The relevant and physically meaning

question is the investigation on a steady fluid motion physically reasonable, governed by the Navier-Stokes

equations, that can be seen as limit of an un- steady motion or converse if the steady motion can have the

transition to an unsteady motion. Another branch, nevertheless physically interesting, looks for non decaying
solutions in connection with the turbulence problems. In this case a priori for the kinetic field ‫ ݒ‬or its translated
‫ ݒ‬െ ‫ݒ‬ஶ, with assigned vector ‫ݒ‬ஶ, no sort of limit property at infinity is possible to assume.
In connection with the second question, we are able to prove the following result (the symbol ሺ͵ǡ denotes the
initial data and the symbol ‫ܬ‬଴௣ሺȳሻ ؔ completion of ࣝ଴ሺπሻ with respect to ‫ ׏ צ‬ή‫צ‬௣ for some ‫א ݌‬ λሻሻǣ

Theorem - For all ‫ݒ‬଴ ‫ܮ א‬ஶሺߗሻ ‫ܬ ת‬଴௣ሺߗሻǡ ‫ ݌‬൐ ͵, there exists a suitable weak solution ‫ݒ‬ to the initial boundary
value problem in an exterior domain. Moreover, the initial data is assumed continuously in the norm of ‫ܮ‬ஶሺߗሻ.

[1] P. Maremonti and S. Shimizu, Global existence of weak solutions to 3-D Navier–Stokes IBVP with non-

decaying initial data in exterior domains. Submitted for the publication.

[2] P. Maremonti and S. Shimizu, Global existence of solutions to 2-D Navier–Stokes flow with non-
decaying initial data in half-plane, J. Differential equations, 265 (2018) 5352-5383,



Marzia Bisi, Maria Groppi, Giorgio Martalò
Department of Mathematical, Physical and Computer Sciences,

University of Parma (Italy)
[email protected]

Kinetic theory is the classical framework to describe the dynamics of rarefied gases. However, some
particular problems in this context can be initially investigated at the hydrodynamic level by means of
macroscopic equations instead of kinetic ones.
Hydrodynamic equations can be conceived as a model system providing qualitative indications about kinetic
solutions, even when the two approaches give solutions that do not coincide accurately.
A special class of 1-dimensional stationary problems is constituted by the half space problem of evaporation
and condensation, that has been widely investigated for a single component gas [1].
By using typical qualitative methods of dynamical systems theory, we will discuss from a mathematical point
of view the main features of the evaporation-condensation problem for a binary mixture of rarefied gases
modeled by a set of Navier-Stokes equations [2], obtained as hydrodynamic limit of a recent BGK description
[3] by classical Chapman-Enskog theory.
Some numerical result about evaporation-condensation solutions for a mixture of noble gases will be
presented and discussed.
[1] A. V. Bobylev, S. Ostmo and T. Ytrehus, Qualitative analysis of the Navier-Stokes equations for

evaporation-condensation problems, Phys. Fluids 8(7), 1764-1773 (1996).
[2] M. Bisi, A. Bobylev, M. Groppi and G. Spiga, Hydrodynamic Equations from a BGK Model for Inert

Gas Mixtures, in AIP Conf. Proc., in press.
[3] A. V. Bobylev, M. Bisi, M. Groppi, G. Spiga and I. F. Potapenko, A general consistent BGK model for

gas mixtures, Kinet. Relat. Mod. 11(6) (2018).



Berardino D’Acunto, Luigi Frunzo, Vincenzo Luongo, Maria Rosaria Mattei
Department of Mathematics and Applications “Renato Caccioppoli”,
University of Naples “Federico II” (Italy)
[email protected]

The establishment of mixed species biofilms results from the interplay of different factors, such as mass
transfer, detachment forces, communication (typically via quorum sensing), and metabolic cooperation or
competition. Recent advances in microbial ecology have identified motility as one of the main mediators of
the development and shape of multispecies communities. Indeed, motile cells with high kinetic energy and
acting as invaders can lead to the dissolution of heterologous biofilms and re-population of the matrix or can
result in the development of several beneficial phenotypes. To fill in the gap in modeling the establishment of
such mixed species communities mediated by the invasion process, a one-dimensional continuous model is
developed by considering two state variables representing the planktonic and sessile phenotypes and
reproducing the transition from one state to the other. Different planktonic cell motion behaviors can be
described, as well as by including regulatory regimes triggered by the external chemical dynamics. The
proposed model is solved numerically to simulate biofilm evolution during biologically relevant conditions and
provides interesting insights towards the qualitative and quantitative understanding of biofilm dynamics and



Andrea Mentrelli, Tommaso Ruggeri

Department of Mathematics and Alma Mater Research Center on Applied Mathematics AM2,
University of Bologna (Italy)
[email protected]

Linear closure of the moment equations has been for a long time the standard closure in Extended
Thermodynamics. The resulting limited hyperbolicity domain, which restricted the applicability of the theory to
a relatively small neighborhood of the equilibrium, and its validity only for monatomic gases, were weaknesses
of the theory that hampered its adoption in practical applications. After the theory was pushed beyond its long-
standing boundaries with the extension to polyatomic gases, very recently Brini and Ruggeri have shown that
the maximum entropy principle with a second order closure allows for an extension also of the hyperbolicity
region, proving that the theory has still much to offer. We discuss how the maximum entropy principle with
second and higher order closures allow to predict shock structure solutions closer to those predicted by the
kinetic theory and in much closer agreement with the experiments, with respect to the classical linear closure,
further increasing the appeal of the theory.



Giuseppe Mulone
Department of Mathematics and Computer Sciences,

University of Catania (Italy)
[email protected]

An overview of linear instability and nonlinear stability results for laminar flows in fluid-dynamics is given. Plane
Couette and Poiseuille flows are nonlinearly stable with respect to streamwise perturbations for any Reynolds
number Re (see [3]). In this case the coefficient of time-decay of the energy is ߨଶȀሺʹܴ݁ሻ, and it is a bound from
above of the time-decay of streamwise perturbations of linearized equations.
Plane Couette and Poiseuille flows are linearly and nonlinearly energy stable if the Reynolds number Re is

less than: ܴത ൌ ܴ݁ை௥௥ሺʹߨȀሺߣ •‹ ߠሻሻȀ •‹ ߠ

when a perturbation is a tilted perturbation in the direction ‫ݔ‬ᇱ which forms an angle ߠ ‫ א‬ሺͲǡ ߨȀʹሿ with the
direction ࢏ of the motion and does not depend on ‫ݔ‬ᇱ. ܴ݁ை௥௥ is the ܱ‫ ݎݎ‬critical Reynolds number for spanwise
perturbations which is evaluated at the wave number ʹߨȀሺߣ •‹ ߠሻ, ߣ being any positive wavelength. By taking
the minimum of ܴത with respect to ߣ, we obtain the critical energy Reynolds number: for plane Couette flow:
ܴ݁ை௥௥ ൌ ͶͶǤ͵ •‹ ߠ and for plane Poiseuille flow: ܴ݁ை௥௥ ൌ ͺ͹Ǥ͹ •‹ ߠ (in particular, for ߠ ൌ ߨȀʹ we have the
classical values ܴ݁ை௥௥ ൌ ͶͶǤ͵ for Couette ܴ݁ை௥௥ ൌ ͺ͹Ǥ͹ for Poiseuille flow). Here the non-dimensional interval
between the planes bounding the channel is ሾെͳǡͳሿ.
In particular, these results improve those obtained by Joseph [2], who found for streamwise perturbations a
critical nonlinear value of ʹͲǤ͸ͷ in the Couette case, and those obtained by Joseph and Carmi who found the
value ͶͻǤͷͷ for plane Poiseuille flow for streamwise perturbations. If we fix some wavelengths from the
experimental data of Prigent et al. [4], and the numerical simulations of Barkley and Tuckerman [6], Tsukahara
et al. [5], the critical Reynolds numbers we obtain are in a very good agreement both with the experiments and
the numerical simulation. These results partially solve the Couette-Sommerferld paradox.

[1] W. M'F. Orr, Proc. Roy. Irish Acad. A 27 9-68 and 69-138 (1907).
[2] D. D. Joseph, J. Fluid Mech. 33 part 3, 617-621 (1966).
[3] K. Moffatt, in Whither turbulence, J. Lumley (ed), Springer, 250-257 (1990).
[4] A. Prigent, G. Grégoire, H. Chaté and O. Dauchot, Physica D 174 100-113 (2003).
[5] T. Tsukahara, Y. Seki, H. Kawamura and D. Tochio, In Proc. 4th Intl Symp. On Turbulence and Shear

Flow Phenomena, pp. 935-940 (2005).
[6] D. Barkley and L. S. Tuckerman, J. Fluid Mech. 576 109-137 (2007).
[7] P. Falsaperla, A. Giacobbe and G. Mulone, Nonlinear stability results for plane Couette and Poiseuille

flows, submitted (2019).



Giovanni Nastasi

Department of Mathematics and Computer Sciences,
University of Catania (Italy)
[email protected]

Charge transport in suspended monolayer graphene is simulated by a numerical deterministic approach,
based on a discontinuous Galerkin (DG) method, for solving the semiclassical Boltzmann equation for
electrons. Both the conduction and valence bands are included and the inter-band scatterings are taken into
account. The use of a Direct Simulation Monte Carlo (DSMC) approach, which properly describes the inter-
band scatterings, is computationally very expensive because the valence band is very populated and a huge
number of particles is needed. Also the choice of simulating holes instead of electrons does not overcome
the problem because there is a certain degree of ambiguity in the generation and recombination terms of
electron hole pairs. Often, direct solutions of the Boltzmann equations with a DSMC neglect the inter-band
scatterings on the basis of physical arguments. The DG approach does not suffer from the previous
drawbacks and requires a reasonable computing effort. It is found out that the inclusion of the inter-band
scatterings produces huge variations in the average values, as the current, with zero Fermi energy while, as
expected, the effect of the inter- band scattering becomes negligible by increasing the absolute value of the
Fermi energy. If the presence of an oxide substrate is also included then it is necessary to add the
scatterings of the charge carriers with the impurities and the phonons of the substrate, besides the
interaction mechanisms already present in the graphene layer. It results that the presence of a substrate
leads to a degradation of the electron and hole mobility.

[1] M. Coco, A. Majorana, G. Nastasi, V. Romano, High-field mobility in graphene on substrate with a

proper inclusion of the Pauli exclusion principle, Atti Accad. Pelorit. Pericol. Cl. Sci. Fis. Mat. Nat. (in
[2] M. Coco, A. Majorana, V. Romano, Cross validation of discontinuous Galerkin method and Monte
Carlo simulations of charge transport in graphene on substrate, Ricerche mat., 66, 201–220 (2017).
[3] A. Majorana, G. Nastasi, V. Romano, Simulation of Bipolar Charge Transport in Graphene by Using a
Discontinuous Galerkin Method, Commun. Comput. Phys., Vol. 26, No. 1, pp. 114-134 (2019).
[4] G. Nastasi, V. Romano, Improved mobility models for charge transport in graphene, Commun. Appl.
Ind. Math. (in press).
[5] V. Romano, A. Majorana, M. Coco, DSMC method consistent with the Pauli exclusion principle and
comparison with deterministic solutions for charge transport in graphene, J. Comput. Phys., 302, 267–
284 (2015).



Daniele Bernardo Panaro, Florinda Capone, Maria Rosaria Mattei,
Vincenzo Luongo, Luigi Frunzo

Department of Mathematics and Applications "Renato Caccioppoli",
University of Naples “Federico II” (Italy)
[email protected]

Most of the existing models on Anaerobic Digestion of waste biomasses are based on nonlinear ordinary
differential equations describing the biological activities of bacteria living in continuous stirred tank reactors.
The perfect mixing ensured by this reactor configuration results in the possibility of neglecting any functional
dependence of biological activities on space. In many real cases, the conversion of solid waste biomasses is
carried out in plug-flow reactors, where the position along the reactor strongly affects biological activities and
reactor performances. A new mathematical model describing the anaerobic bioconversion of solid wastes in a
plug-flow reactor is here presented. The model is based on mass balance considerations for different state
variables and results in nonlinear partial differential equations accounting for the convection-diffusion-reaction
of particulate and dissolved compounds within the bioreactor. Numerical simulations are performed to show
model consistency.



Maria Cristina Carrisi, Sebastiano Pennisi
Department of Mathematics and Informatics,
University of Cagliari (Italy)
[email protected]

We aim to discuss here the following set of balance equations for the description of relativistic polyatomic

߲ఈܸఈ ൌ Ͳǡ ߲ఈܶఈఉ ൌ Ͳǡ ߲ఈ‫ܣ‬ఈఉఊ ൌ ܲఉఊǡ ߲ఈ‫ܪ‬௏ఈ ൌ ܲǤ (1)

In [1], the authors considered only first the two of these equations and the traceless part of (1)3, i.e.,
߲ఈ‫ܣ‬ఈழఉఊவ ൌ ‫ܫ‬ழఉఊவ; the reason behind this choice was that they wanted to find, in the non relativistic limit and
in the monoatomic limit, the results of the 14 moments models of the articles [2]-[6].

In the present article we investigate what happens if we don’t take away the trace of (1)3. In this way one
obtains a 15 moments model and we want to investigate it. But, in the meanwhile, another 15 moments

model [7] has been produced in the classical context. To avoid confusion between these two models we

prefer to consider both of them compacted in only one and in the relativistic context, even if at the cost of
obtaining a 16 moments model. For this reason we include (1)4 in the field equations; these are expressed in
terms of the tensors

ାஶ ାஶ

ܸఈ ൌ ݉ܿ න න න ݂‫݌‬ఈ߮ሺ‫ܫ‬ோሻ߰ሺ‫ܫ‬௏ሻ݀‫ܫ‬ோ݀‫ܫ‬௏݀ܲሬԦǡ
Ըయ ଴ ଴

ܶఈఉ ାஶ ାஶ ݂‫݌‬ఈ‫݌‬ఉ ݉‫ܿܫ‬ଶ൰ ߮ሺ‫ܫ‬ோ ሻ߰ሺ‫ܫ‬௏ ሻ݀‫ܫ‬ோ ݀‫ܫ‬௏ ݀ܲሬԦǡ (2)

ൌ ܿ න නන ൬ͳ ൅

Ըయ ଴଴

‫ܣ‬ఈఉఊ ൌ ܿ ାஶ ାஶ ൬ͳ ൅ ݉ʹܿ‫ܫ‬ଶ൰ ߮ሺ‫ܫ‬ோ ሻ߰ሺ‫ܫ‬௏ ሻ݀‫ܫ‬ோ ݀‫ܫ‬௏ ݀ܲሬԦǡ
න න න ݂‫݌‬ఈ‫݌‬ఉ‫݌‬ఊ

Ըయ ଴ ଴

‫ܪ‬௏ఈ ାஶ ାஶ ʹ‫ܫ‬௏ ߮ሺ‫ܫ‬ோ ሻ߰ሺ‫ܫ‬௏ ݀ܲሬԦǡ
ൌ ݉ න න න ݂‫݌‬ఈ ቆͳ ൅ ݉ܿଶ ቇ ሻ݀‫ܫ‬ோ݀‫ܫ‬௏
Ըయ ଴ ଴

where ݀ܲሬԦ ൌ ௗ௣భௗ௣మௗ௣య ǡ ‫ܫ‬ோ is the rotational energy of a molecule, ‫ܫ‬௏ its vibration energy and ‫ܫ‬ ൌ ‫ܫ‬ோ ൅ ‫ܫ‬௏.

We will calculate the non relativistic limit of the full set of eqs. (1), finding a 16 moments model for classical

extended thermodynamics of polyatomic gases. It encloses two important subsystems: the natural

extension of [1] which is obtained neglecting eq. (1)4 and the model [7] which comes out by neglecting the

trace of eq. (1)3. After that, we will impose the Maximum Entropy Principle for these field equations and
compare the results with those of [1]. The resulting system is hyperbolic for every timelike congruence and

this property assures that the characteristic velocities don’t exceed the speed of light.


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