WASCOM 2019

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

XX International

Conference on

Waves and Stability

in Continuous Media

Book of Abstracts

CONFERENCE INFORMATION

Overview

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

techniques.

LEVICO (TN)

BOLOGNA

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

VILLASIMIUS (CA) VULCANO (ME)

TAORMINA (ME)

MONDELLO (PA)

PALERMO ACIREALE (CT)

CATANIA

SCICLI (RG)

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

1

ORGANIZERS

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

(Italy)

Seung-Yeal Ha Tai-Ping Liu

(Korea) (USA and Taiwan) Masaru Sugiyama

(Japan)

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)

GNFM

Dipartimento di

Matematica e

Informatica

2

WASCOM 2019

JUNE 10-14, 2019

CONFERENCE PROGRAM

3

WASCOM 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

CHAIRMAN: TOMMASO RUGGERI

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

CHAIRMAN: TAI-PING LIU

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)

CHAIRMAN: MASARU SUGIYAMA

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

Stability

17.00 - 17.25 COFFEE BREAK

CHAIRMAN: HENRI GOUIN

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)

4

WASCOM 2019

Tuesday, June 11th

CHAIRMAN: SEUNG-YEAL HA

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

CHAIRMAN: PETER VADASZ

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)

CHAIRMAN: MARCO SAMMARTINO

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

CHAIRMAN: GIUSEPPE MULONE

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

Graphene

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)

5

WASCOM 2019

Wednesday, June 12th

CHAIRMAN: SALVATORE RIONERO

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

CHAIRMAN: GIUSEPPE TOSCANI

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

Diseases

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)

CHAIRMEN: SALVATORE RIONERO - TOMMASO RUGGERI

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)

6

WASCOM 2019

Thursday, June 13th

CHAIRMAN: VITTORIO ROMANO

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

CHAIRMAN: GIUSEPPE SACCOMANDI

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

CHAIRMAN: LAURENT DESVILLETTES

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

Thermodynamics

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

CHAIRMAN: FLORINDA CAPONE

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)

7

WASCOM 2019

JUNE 10-14, 2019

ABSTRACTS

8

DISPERSION RELATION OF A RAREﬁED POLYATOMIC GAS WITH MOLECULAR

RELAXATION PROCESSES BASED ON RATIONAL EXTENDED THERMODYNAMICS WITH

15 FIELDS

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, ܿ௩Ƹare the dimensionless rotational and vibrational specific heats.

References

[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).

9

STATIONARY FLOW AND HEAT TRANSFER IN EXTENDED THERMODYNAMICS

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.

10

MAXWELL-STEFAN EQUATIONS FOR A REACTIVE MIXTURE OF POLYATOMIC GASES

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].

References

[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.

11

ON THE HYPERBOLICITY PROPERTY OF EXTENDED THERMODYNAMICS MODELS FOR

RAREFIED GASES

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.

12

LOCAL DISCRETE VELOCITY GRIDS FOR MULTI-SPECIES RAREFIED FLOW

SIMULATIONS

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.

13

OPTIMAL PUBLIC HEALTH SYSTEMS INTERVENTION TO FAVOR VACCINE PROPENSITY

FOR CHILDHOOD DISEASES

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.

References

[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).

14

RECENT RESULTS ON THE ONSET OF MHD CONVECTION IN POROUS MEDIA

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.

References

[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: https://doi.org/10.1007/s11587-018-0417-5

[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)

15

PROPAGATION OF MAGNETIC DOMAIN WALLS IN MAGNETOSTRICTIVE MATERIALS

WITH DIFFERENT CRYSTAL SYMMETRY

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.

References

[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:

10.1177/1081286518810741.

16

MATHEMATICAL MODELLING OF MULTISPECIES BIOFILMS

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.

17

UNIQUENESS OF ZERO RELAXATION LIMIT

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.

18

ON SOLUTIONS RELATED TO FITZHUGH-RINZEL MODEL

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

determined.

References

[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)

533-551.

19

THE GENERAL RELATIVISTIC POYNTING-ROBERTSON EFFECT: NON-LINEAR

DISSIPATIVE SYSTEM IN GENERAL RELATIVITY

Vittorio De Falco

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

Science,

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.

References

[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.

20

ONSET OF DOUBLE-DIFFUSIVE CONVECTION IN POROUS MEDIA WITH SORET EFFECT

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.

References

[1] F. Capone, R. De Luca, M. Vitiello, Double-diffusive Soret convection phenomenon in porous media:

effect of Vadasz inertia term. Ric. Mat. DOI: https://doi.org/10.1007/s11587-018-0428-2.

[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).

21

REFLECTIONLESS SOLUTIONS FOR SQUARE MATRIX NONLINEAR SCHROEDINGER

EQUATION WITH VANISHING BOUNDARY CONDITIONS

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).

22

ABOUT A CLASS OF CROSS DIFFUSION SYSTEMS ARISING IN CHEMOTAXIS

Laurent Desvillettes

IMJ-PRG,

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.

23

NEW STABILITY RESULTS FOR HYDROMAGNETIC PLANE COUETTE FLOWS

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.

References

[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).

24

ON THE IMPACT OF SHORT LASER PULSES ON COLD DILUTED PLASMAS

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.

25

MATHEMATICAL MODELING OF DISPERSAL PHENOMENON IN BIOFILMS

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 ǡ ሻǡ Ͳ ݖ ܮሺݐሻǡ ݐ Ͳǡ

μݐ μݖ

݂ሺݖǡ Ͳሻ ൌ ߮ሺݖሻǡ Ͳ ݖ ܮǡ ݅ ൌ ͳǡ Ǥ Ǥ Ǥ ǡ ݊ǡ

μݑ ሺݖǡ ݐሻ ൌ ܴெǡ ሺݖǡ ݐǡ ǡ ǡ ሻǡ Ͳ ൏ ݖ ܮሺݐሻǡ Ͳǡ

μݖ

ୀଵ

ݑሺͲǡ ݐሻ ൌ Ͳǡ Ͳǡ

ܮሶ ሺݐሻ ൌ ݑሺܮሺݐሻǡ ݐሻ ߪሺሻ െ ߪௗ൫ܮሺݐሻ൯ǡ ܮሺͲሻ ൌ ܮǡ ݐ Ͳǡ

μ߰ െ ߲ ሺܦெǡ μ߰ ሻ ൌ ܴటǡ ሺǡ ǡ ǡ ǡ ሻǡ ݅ ൌ ͳǡ Ǥ Ǥ Ǥ ǡ ܰǡ Ͳ ൏ ݖ൏ ܮሺݐሻǡ ݐ Ͳǡ

μݐ ߲ݖ ߲ݖ

߰ሺݖǡ Ͳሻ ൌ Ͳǡ Ͳ ǡ μ߰ ሺͲǡ ݐሻ ൌ Ͳǡ ߰ሺܮሺݐሻǡ ݐሻሻ ൌ ߰כሺݐሻǡ ݅ ൌ ͳǡ Ǥ Ǥ Ǥ ǡ ܰǡ Ͳ

μݖ ͲǤ

μܵ െ ߲ ሺܦௌǡ μܵ ሻ ൌ ݎௌǡ ሺǡ ǡ ǡ ሻǡ ݆ ൌ ͳǡ Ǥ Ǥ Ǥ ǡ ݉ǡ Ͳ ൏ ݖ൏ ܮሺݐሻǡ ݐ Ͳǡ

μݐ ߲ݖ ߲ݖ

ܵሺݖǡ Ͳሻ ൌ ሺሻǡ Ͳ ǡ μܵ ሺͲǡ ݐሻ ൌ Ͳǡ ܵ ሺܮሺݐሻǡ ݐሻሻ ൌ ܵכሺݐሻǡ ݆ ൌ ͳǡ Ǥ Ǥ Ǥ ǡ ݉ǡ

μݖ

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.

References

[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.

26

SPATIAL PATTERNS AND BISTABILITY IN A CROSS-DIFFUSIVE FITZHUGH-NAGUMO

SYSTEM

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.

References

[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).

27

TRANSITION TO TURBULENCE IN THE WEAKLY STRATIFIED KOLMOGOROV FLOW

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].

References

[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).

28

THERMAL CONVECTION IN A ROTATING HORIZONTALLY ISOTROPIC POROUS MEDIUM

WITH LTNE

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.

29

INCLINED CONVECTION IN A POROUS BRINKMAN LAYER: LINEAR INSTABILITY AND

NONLINEAR STABILITY

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).

References

[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.

30

AGGREGATION PHENOMENA AND WELL-POSEDNESS FOR A MULTIPLE SCLEROSIS

MODEL

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).

References

[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.

31

MEMBRANES AND VESICLES

Henri Gouin, Sergey Gavrilyuk

CNRS,

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.

References

[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.

32

CONSISTENT BGK MODELS FOR GAS MIXTURES AND HYDRODYNAMIC EQUATIONS

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

solution.

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.

References

[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.

33

ON THE SECOND-ORDER EXTENSIONS OF FIRST-ORDER COLLECTIVE MODELS

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).

34

FROM THE RELATIVISTIC MIXTURE OF GASES TO THE RELATIVISTIC CUCKER-SMALE

FLOCKING: MATHEMATICAL ANALYSIS

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.

35

ON WELL-POSEDNESS OF WEAK SOLUTIONS

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.

36

COHERENT STRUCTURES IN A CHEMOTAXIS MODEL OF ACUTE INFLAMMATION

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

literature.

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.

37

References

[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-

114.

[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.

38

BIOSORPTION OF HEAVY METALS IN A NITRIFYING BIOFILM

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

community.

39

ON THE EVOLUTION OF FRACTIONAL DIFFUSIVE WAVES

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.

Acknowledgments

The work of FM has been carried out in the framework of the activities of the National Group of Mathematical

Physics (INdAM-GNFM).

References

[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.

40

GENERALIZED SIMPLE WAVES FOR HYPERBOLIC SYSTEMS

Natale Manganaro

MIFT,

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.

41

WEAK SOLUTIONS TO THE NAVIER-STOKES EQUATIONS WITH NON DECAYING DATA

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

domains.

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.

ݒ

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 ܬሺȳሻ ؔ completion of ࣝሺπሻ with respect to צήצ for some א λሻሻǣ

Theorem - For all ݒ ܮ אஶሺߗሻ ܬ תሺߗሻǡ ͵, 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.

References

[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,

doi.org/10.1016/j.jde.2018.07.004.

42

ANALYSIS OF EVAPORATION-CONDENSATION PROBLEMS FOR A BINARY GAS

MIXTURE

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.

References

[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).

43

MODELING CELL MOTILITY IN BIOFILMS

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

ecology.

44

COMPARISON OF SHOCK STRUCTURE BEHAVIOURS FOR INCREASING ORDER OF

CLOSURES

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.

45

NEW NONLINEAR STABILITY RESULTS FOR PLANE COUETTE AND POISEUILLE FLOWS

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 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: ܴ݁ை ൌ ͺǤ ߠ (in particular, for ߠ ൌ ߨȀʹ we have the

classical values ܴ݁ை ൌ ͶͶǤ͵ for Couette ܴ݁ை ൌ ͺǤ 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 ʹͲǤͷ 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.

References

[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).

46

NUMERICAL SOLUTIONS OF THE SEMICLASSICAL BOLTZMANN EQUATION FOR

BIPOLAR CHARGE TRANSPORT IN GRAPHENE

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.

References

[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

press).

[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).

47

ANAEROBIC DIGESTION IN PLUG-FLOW REACTORS: A MATHEMATICAL MODEL

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.

48

A 16 MOMENTS MODEL IN RELATIVISTIC EXTENDED THERMODYNAMICS OF

RAREFIED POLYATOMIC GAS

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

gases:

߲ఈܸఈ ൌ Ͳǡ ߲ఈܶఈఉ ൌ Ͳǡ ߲ఈܣఈఉఊ ൌ ܲఉఊǡ ߲ఈܪఈ ൌ ܲ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 ݂ఈ߮ሺܫோሻ߰ሺܫሻ݀ܫோ݀ܫ݀ܲሬԦǡ

Ըయ

ܶఈఉ ାஶ ାஶ ݂ఈఉ ݉ܿܫଶ൰ ߮ሺܫோ ሻ߰ሺܫ ሻ݀ܫோ ݀ܫ ݀ܲሬԦǡ (2)

ൌ ܿ න නන ൬ͳ

Ըయ

ܣఈఉఊ ൌ ܿ ାஶ ାஶ ൬ͳ ݉ʹܿܫଶ൰ ߮ሺܫோ ሻ߰ሺܫ ሻ݀ܫோ ݀ܫ ݀ܲሬԦǡ

݉

න න න ݂ఈఉఊ

Ըయ

ܪఈ ାஶ ାஶ ʹܫ ߮ሺܫோ ሻ߰ሺܫ ݀ܲሬԦǡ

ൌ ݉ න න න ݂ఈ ቆͳ ݉ܿଶ ቇ ሻ݀ܫோ݀ܫ

Ըయ

where ݀ܲሬԦ ൌ ௗభௗమௗయ ǡ ܫோ is the rotational energy of a molecule, ܫ its vibration energy and ܫ ൌ ܫோ ܫ.

బ

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.

49