Book_of_Abstracts_Wascom19

References
[1] S. Pennisi, T. Ruggeri, Relativistic Extended thermodynamics of rarefied polyatomic gas, Annals of

Physics, 377 (2017), 414-445, doi: 10.1016/j.aop.2016.12.012.
[2] Arima, T.; Taniguchi, S.; Ruggeri, T.; Sugiyama, M. Extended Thermodynamics of dense gases.

Continuum Mech. Thermodyn. 2012, 24, 271–292.
[3] M.C. Carrisi, S. Pennisi, T. Ruggeri, Monatomic Limit of Relativistic Extended Thermodynamics of

Polyatomic Gas, Continuum Mech. Thermodyn., doi: 10.1007/s00161- 018-0694-y, (2018).
[4] T. Ruggeri, M. Sugiyama, Rational Extended Thermodynamics beyond the Monatomic Gas, Springer,

Cham Heidelberg New York Dorderecht London (2015).
[5] Liu, I.-S.; Muller, I.; Ruggeri, T. Relativistic thermodynamics of gases. Ann. Phys. (N.Y.) 1986, 169,

191–219.
[6] Muller, I.; Ruggeri, T. Rational Extended Thermodynamics, 2nd ed.; Springer Tracts in Natural

Philosophy. Springer, New York, NY, USA, 1998.
[7] Arima, T.; Taniguchi, S.; Ruggeri, T.; Sugiyama, M. Extended thermodynamics of rarefied polyatomic

gases: 15-field theory incorporating relaxation processes of molecular rotation and vibration. Entropy
2018, 20, 301-321. DOI: 10.3390/e20040301.

50

HOPF BIFURCATIONS IN THERMAL MHD AND SPECTRUM INSTABILITY DRIVEN BY
PERTURBATIONS TO PRINCIPAL ENTRIES

Salvatore Rionero
University of Naples “Federico II” (Italy)
Fellowship Accademia Nazionale dei Lincei, Roma (Italy)

[email protected]

The transition from a steady state to an unsteady oscillatory state (Hopf bifurcation) is a scenario:
i) “less continuous” than the transition from a steady state to a (close) steady state (steady
bifurcation);
ii) more detectable and impressive from the physical point of view.

Further, an Hopf bifurcation is a limit cycle candidate, of known frequency, to the nonlinear system at stark.

Let

ߣ௡ െ ‫ܫ‬ଵ௡ߣ௡ିଵ ൅ ‫ܫ‬ଶ௡ߣ௡ିଶ ൅ ‫ ڮ‬൅ ሺെͳሻ௡‫ܫ‬௡ ൌ Ͳ

be the spectrum equation. The new approach for the onset of bifurcation is based on a suitable use of each
instability condition

‫ܫ‬௠ ൌ Ͳǡ ݉ ‫ א‬ሼͳǡʹǡ ǥ ǡ ݊ሽǡ

expecially for characterizing the occurring of unsteady bifurcations (Hopf, steady-Hopf, unsteady aperiodic,
…). In thermal MHD, at growth of the Chandrasekhar number ܳଶ െ measuring the growth of the magnetic
field in which the electrically conducting fluid is embedded െ an Hopf bifurcation can occur only when the
ܳnuଶ mhabsertoܲ௥reisaclehsfsorthtahne number. Calling Hopf bifurcation number the threshold
Prandtl the Prandtl magnetic bifurcation, we, via the new approach, show that ܳ௖, in
ܳ௖ that occurring of an Hopf

the free-free case, is given by ͳ ൅ ܲ௥
ܲ௠ െ ܲ௥
ܳ௖ ൌ ߨଶ

and analogous values are furnished for different cases. In particular spectrum instabilities driven by
perturbations to the principal entries and applications to convection in multicomponent fluid layers are
analyzed.

51

MATHEMATICAL MODELING OF CHARGE TRANSPORT IN GRAPHENE

Vittorio Romano

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

The last years have witnessed a great interest for 2D-materials due to their promising applications. The most
investigated one is graphene which is considered as a potential new material to exploit in nano-electronic
and optoelectronic devices.
Charge transport in graphene can be described with several degrees of physical complexity. At quantum
level an accurate model is represented by the Wigner equation but in several cases its semiclassical limit,
the Boltzmann equation, constitutes a fully acceptable model. However, the numerical difficulties
encountered in the direct solution of both the Wigner and the semiclassical Boltzmann equation has
prompted the development of hydrodynamical, energy transport and drift diffusion models, in view of the
design of a future generation of electron devices where graphene replaces standard semiconductors like
silicon and gallium arsenide. Moreover, thermal effects in low dimensional structures play a relevant role
and, therefore, also phonon transport must be included.
Interesting new mathematical issues related to the peculiar features of graphene arise. The main aspects will
be discussed and recent results [1-10] illustrated in the perspective of future developments, in particular the
optimization of graphene field effect transistors.

References
[1] L. Luca, V. Romano, Quantum corrected hydrodynamic models for charge transport in graphene,

Annals of Physics 406, 30-53 (2019).
[2] A. Majorana, G. Nastasi, V. Romano, Simulation of bipolar charge transport in graphene by using a

discontinuous Galerkin method, Comm in Comp. Physics 26, 114-134 (2019).
[3] L. Luca, V. Romano, Comparing linear and nonlinear hydrodynamical models for charge transport in

graphene based on the Maximum Entropy Principle, Int. J. of Non-Linear Mech. (2018).
[4] M. Coco, V. Romano, Simulation of Electron–Phonon Coupling and Heating Dynamics in Suspended

Monolayer Graphene Including All the Phonon Branches, J. Heat Transfer 140, 092404 (2018).
[5] M. Coco, A. Majorana, V. Romano, Cross validation of discontinuous Galerkin method and Monte

Carlo simulations of charge transport in graphene on substrate, Ricerche di Mat., 66, 201— 220,
2017.
[6] G. Mascali, V. Romano, Charge transport in graphene including thermal effects, SIAM J. Applied
Math. Vol 77 (2), 593-613 (2017).
[7] M. Coco, G. Mascali, V. Romano, Monte Carlo Analysis of Thermal Effects in Monolayer Graphene, J.
Of Computational and Theoretical Transport 45(7), 540-553, 2016.
[8] A. Majorana, G. Mascali, V. Romano, Charge transport and mobility in monolayer graphene, J. Math.
Industry 7:4, https://doi.org/10.1186/s13362-016-0027-3 , 2016.
[9] 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.
[10] V. D. Camiola, V. Romano, Hydrodynamical Model for Charge Transport in Graphene, J. Stat. Phys.
157, 1114-1137, 2014.

52

RESONANT TURING PATTERNS IN THE FITZHUGH-NAGUMO MODEL WITH CROSS
DIFFUSION

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

In this talk we shall describe the formation of resonant spatial structures in the FitzHugh-Nagumo (FN)

system [2, 5]: ൜‫ݑ‬௧‫ݒ‬ൌ௧ െ‫ݑ‬ଷ ൅ ߚ‫ݒݑ‬ ൅ ‫ ݑ‬െ ‫ ݒ‬൅ ‫׏‬ଶ‫ ݑ‬൅ ݀௩‫׏‬ଶ‫ݒ‬
ൌ ߝሺߛ‫ݑ‬ െ‫ݒ‬ െ ܽሻ ൅ ݀௨‫׏‬ଶ‫ ݑ‬൅ ݀‫׏‬ଶ‫ݒ‬

where the linear cross diffusion effect is considered [6].
In the parameters region where the above system supports bistability of the homogeneous steady states,
diffusive instabilities lead to the emergence of subcritical large amplitude patterns. In this case the classical
weakly nonlinear analysis, as e.g. given in [3], fails. Therefore, to describe the resulting structures, we take
into account the coupling of the zero mode with the spatial unstable mode.
We will also study the interaction of the patterns emerging from the two different homogeneous equilibria: on
a 2D spatial domain, if the unstable interacting modes satisfy some resonant conditions [1], complex
quasiperiodic structures emerge. The selection of these superlattice patterns [4] is addressed via normal
form reduction. We show that the numerical simulations of the reaction-diffusion system corroborate the
predictions obtained through the normal form.

References
[1] Bachir M., Sonnino G., Tlidi M.: Predicted formation of localized superlattices in spatially distributed

reaction-diffusion solutions. Physical Review E, 86, 045103(R), (2012).
[2] FitzHugh R.: Impulses and physiological states in theoretical models of nerve membrane. Biophysical

Journal, 1(6):445-466, (1961).
[3] Gambino G., Lombardo M.C., Rubino G., Sammartino M., Pattern selection in the 2D FitzHugh-

Nagumo model. Ricerche di Matematica 10.1007/s11587-018-0424-6, (2018).
[4] Judd S.L., Silber M.: Simple and superlattice Turing patterns in reaction-diffusion systems: bifurcation,

bistability, and parameter collapse. Physica D, 136:45-65, (2000).
[5] Metens S., Borckmans P., and Dewel G., Large amplitude patterns in bistable reaction-diffusion

systems. 10.1007/978-94-011-4247-2, (2000).
[6] Zemskov E.P., Epstein I.R., Muntean A.: Oscillatory pulses in FitzHugh-Nagumo type systems with

cross-diffusion. Mathematical Medicine and Biology, 28(2):217-226, (2011).

53

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

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

University of Bologna (Italy)
[email protected]

We present a relativistic model for a mixture of Euler gases with multi temperatures. For the proposed
relativistic model, we explicitly determine production terms resulting from the interchange of energy-
momentum between the constituents via the entropy principle. We use the analogy with the homogeneous
solutions of a mixture of gases and the thermomechanical Cucker-Smale flocking model in a classical setting
(Ha, S.-Y., Ruggeri, T.: Emergent dynamics of a thermodynamically consistent particle model. Arch. Rational
Mech. Anal. 223, 1397–1425 (2017)) to derive a relativistic counterpart of the TCS model. Moreover, we
employ the theory of principal subsystem to derive the relativistic Cucker-Smale model.
In the next talk Jeongho Kim will present the mathematical analysis for the derived relativistic CS
model. This is a jointly work with Seung-Yeal Ha and Jeongho Kim.

54

HELMHOLTZ-TYPE SOLITARY SOLUTIONS IN NON-LINEAR ELASTODYNAMICS

Giuseppe Saccomandi

Department of Engineering,
University of Perugia (Italy)
[email protected]

The nonlinear equations descriptive of transverse wave propagation in an isotropic, incompressible, elastic
solid on an elastic foundation support interesting class of solitary solutions [1, 2, 4]. The aim of the present talk
is to discuss some of such solutions and their stability illustrating some connections with nonlinear optics via
a NLS reduction [3].
References:
[1] Rogers, Colin, Giuseppe Saccomandi, and Luigi Vergori “Helmholtz-Type Solitary Solutions in

Quasilinear Elastodynamics” submitted.
[2] Rogers, Colin, Giuseppe Saccomandi, and Luigi Vergori. "Cnoidal and gausson phenomena in nonlinear

elastodynamics." Acta Mechanica 229.8 (2018): 3489-3500.
[3] Rogers, Colin, Giuseppe Saccomandi, and Luigi Vergori. "Nonlinear elastodynamics of materials with

strong ellipticity condition: Carroll-type solutions." Wave Motion 56 (2015): 147- 164.
[4] Rogers, Colin, Giuseppe Saccomandi, and Luigi Vergori "Carroll-type deformations in nonlinear

elastodynamics." Journal of Physics A: Mathematical and Theoretical 47.20 (2014): 205204.

55

VISCOUS MHD VORTICITY-CURRENT EQUATIONS WITH DATA IN L1(R2)

Marco Sammartino
Department of Engineering,
University of Palermo (Italy)
[email protected]

In this talk we shall consider the 2D viscous MHD equations:

߲࢛ ൅ ࢛ ή સ࢛ ൅ સ‫݌‬ ൌ ‫ݒ‬ο࢛ ൅ ࡮ ή સ࡮ǡ
߲‫ݐ‬

߲࡮ ൅ ࢛ ή સ࡮ ൌ ߤο࡮ ൅ ࡮ ή સ࢛ǡ
߲‫ݐ‬

સ ή ‫ ܝ‬ൌ Ͳǡ સ ή ۰ ൌ Ͳǡ

where ࢛ ൌ ሺ‫ݑ‬ଵǡ ‫ݑ‬ଶሻ is the fluid velocity, ‫ ݌‬is the pressure, ࡮ ൌ ሺܾଵǡ ܾଶሻ is the magnetic field, ‫ ݒ‬and ߤ are
respectively the viscosity and the resistivity.
Introducing the vorticity ߱ and the current density ݆

߱ ൌ સ ൈ ࢛ ൌ ߲ଵ‫ݑ‬ଶ െ ߲ଶ‫ݑ‬ଵǡ
݆ ൌ સ ൈ ࡮ ൌ ߲ଵܾଶ െ ߲ଶܾଵǡ

one can write the MHD equations in the vorticity-current formulation:

߲߱ ൅ ࢛ ή સ߱ ൌ ‫ݒ‬ο߱ ൅ ࡮ ή સ࢐ǡ (1)
߲‫ݐ‬ ή સ‫ݑ‬ଶ െ (2)
߲݆
߲‫ݐ‬ ൅ ࢛ ή સ݆ ൌ ߤο݆ ൅ ࡮ ή સ߱ ൅ ʹ߲ଵ࡮ ʹ߲ଶ࡮ ή સ‫ݑ‬ଵǡ

where ࢛ and ࡮ are written, in terms of ߱ and ݆, through the Biot-Savart law:
࢛ሺ࢞ǡ ‫ݐ‬ሻ ൌ ሺࡷ ‫߱ כ‬ሻሺ࢞ǡ ‫ݐ‬ሻ ൌ න ࡷሺ࢞ െ ࢟ሻ߱ሺ࢟ǡ ‫ݐ‬ሻ݀࢟

Թ మ

࡮ሺ࢞ǡ ‫ݐ‬ሻ ൌ ሺࡷ ‫݆ כ‬ሻሺ࢞ǡ ‫ݐ‬ሻ ൌ න ࡷሺ࢞ െ ࢟ሻ݆ሺ࢟ǡ ‫ݐ‬ሻ݀࢟

Թమ

with ͳ ‫ݔ‬ଶ ȁ࢞‫ݔ‬ଵȁଶ൰Ǥ
ʹߨ ȁ࢞ȁଶ
ࡷሺ࢞ሻ ൌ ൬െ ǡ

The initial condition for (1) and (2) are

߱ሺ࢞ǡ ‫ ݐ‬ൌ Ͳሻ ൌ ߱଴ሺ࢞ሻǡ (3)
݆ሺ࢞ǡ ‫ ݐ‬ൌ Ͳሻ ൌ ݆଴ሺ࢞ሻǤ (4)

We shall prove the following result:
Theorem 1. Suppose ߱଴ǡ ݆଴ ‫ܮ א‬ଵ. Then Eqs. (1)-(2) with initial data given by (3)-(4), admit, globally in time, a
unique solution ߱ and ݆. Moreover ߱ and ݆ are smooth for ‫ ݐ‬൐ Ͳ and, in particular,
߱ǡ ݆ ‫ܮ א‬ஶሺሾͲǡ ܶሿǡ ‫ܮ‬ଵሺԹଶሻ ‫ܮ ת‬ஶሺԹଶሻሻ, for each ܶ ൐ Ͳ.
This is joint work with V. Sciacca and M. Schonbek.

56

EXTENSION OF THE AUXILIARY EQUATION METHOD BY MEANS OF HYPERELLIPTIC
FUNCTIONS

Michele Sciacca

Department of Agricultural, Food and Forest Sciences,
University of Palermo (Italy)
[email protected]

The interest to find exact solutions of (partial) differential equations is an old problem which has involved many
researchers over the years. Among the different methods, I consider the auxiliary equation method, which
uses an auxiliary equation, whose solutions are known, to find some exact solutions of the (partial) differential
equation under investigation.
Some years ago a direct method was proposed, which uses the hyperelliptic functions Ե௜௝௞ [1–5]. Here I
generalize this method and I show that it can be also applied to non-autonomous differential equation (for
instance to non- autonoums Korteweg de Vries).
References
[1] H.F. Baker, Multiply Periodic Functions, Cambridge Univ. Press, 1907.
[2] E.D. Belokolos, V.Z. Enolskii, J. Math. Sci. 106 (6) (2001) 3395.
[3] E.D. Belokolos, V.Z. Enolskii, J. Math. Sci. 108 (2002) 295.
[4] T. Brugarino, M. Sciacca, Phys. Lett. A 372 (2008) 1836.
[5] Y. Feng, Y. Dong, Q. Ding, H. Zhang, Appl. Math. Comp. 215 (2010) 3868.

57

UP-WIND DIFFERENCE APPROXIMATION AND SINGULARITY FORMATION FOR A SLOW
EROSION MODEL

Vincenzo Sciacca

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

We consider a model for a granular flow in the slow erosion limit, introduced in [1, 2], which describes the
erosion of a mountain profile caused by small avalanches. The model takes the form of a nonlocal first-order
conservation law

‫ݑ‬௧ ൌ ሺ݂ሺ‫ݑ‬ሻ‫ܧ‬ሾ‫ݑ‬ሿሻ௫ ൌ Ͳǡ
where ‫ ݑ‬൅ ͳ gives the slope of the standing profile of granular matter; ݂ሺ‫ݑ‬ሻ is the erosion function and has
the meaning of the erosion rate per unit length in space covered by the avalanches; and ‫ܧ‬ሾήሿ is a nonlocal, in
space, integral operator. The function ݂ is defined on ሺെͳǡ Ͳሿ, with a singularity at ‫ ݑ‬ൌ െͳ. Well-posedness
results was established in [2, 3], for initial data ‫ݑ‬଴ ‫ܸܤ א‬ሺԹሻ and െͳ ൅ ߙ ൑ ‫ݑ‬଴ ൑ Ͳ, with ߙ ൐ Ͳ; and in [4], with
‫ݑ‬଴ ‫ܮ א‬ଵሺԹሻǡ െͳ ൑ ‫ݑ‬଴ ൑ Ͳǡ ݂ሺ‫ݑ‬଴ሻ ‫ܮ א‬ଵሺԹሻ ‫ܮ ׫‬ఙሺԹሻ and ͵ ൑ ߪ ൑ λ.
No rigorous numerical results are known for this model, excluding the wave-front tracking method in [5].
We propose an up-wind numerical scheme for this problem and show that the approximate solutions generated
by the scheme converge to the unique entropy solution. Numerical examples are also presented showing the
reliability of the scheme. We study also the finite time singularity formation for the model with the singularity
tracking method [6, 7, 8], and we characterize the singularities as shocks in the solution.
Joint work with Giuseppe M. Coclite (Polytechnic University of Bari - Italy) and Francesco Gargano
(University of Palermo - Italy).

References
[1] K.P. Hadeler and C. Kuttler. Dynamical models for granular matter. Granular Matter, 2(1):9-18, 1999.
[2] W. Shen and T. Y. Zhang. Erosion profile by a global model for granular flow. Arch. Ration. Mech. Anal.

204:837-879, 2012.
[3] D. Amadori and W. Shen. An integro-differential conservation law arising in a model of granular flow. J.

of Hyperb. Diff. Eqs., 9(1):105-131, 2012.
[4] G.M. Coclite and E. Jannelli. Well-posedness for a slow erosion model. J. of Math. Anal. and Appl.,

456(1):337-355, 2017.
[5] D. Amadori and W. Shen. Front tracking approximations for slow erosion. Disc. and Cont. Dyn. Syst.,

32(5):1481-1502, 2012.
[6] G. Della Rocca, M.C. Lombardo, M. Sammartino, and V. Sciacca. Singularity tracking for Camassa-

Holm and Prandtls equations. Appl. Numer. Math., 56(8):1108- 1122, 2006.
[7] F. Gargano, M. Sammartino, V. Sciacca, and K. W. Cassel. Analysis of com- plex singularities in high-

Reynolds-number Navier-Stokes solutions. J. Fluid Mech., 747:381-421, 2014.
[8] F. Gargano, G. Ponetti, M. Sammartino, and V. Sciacca. Complex singularities in KdV solutions.

Ricerche di Matematica, 65(2):479-490, 2016.

58

RATIONAL EXTENDED THERMODYNAMICS OF A RAREFIED POLYATOMIC GAS WITH
RELAXATION PROCESSES OF MOLECULAR ROTATION AND VIBRATION

Masaru Sugiyama
Nagoya Institute of Technology (Japan)

[email protected]

Rational extended thermodynamics (RET) [1, 2] is the theory for describing highly nonequilibrium
phenomena that are out of the validity range of thermo-dynamics of irreversible processes.
The purpose of the present talk is to show the RET theory of a rarefied polyatomic gas in which molecular
rotational and vibrational relaxation processes are treated individually. The theory is justified, at mesoscopic
level, by a generalized Boltzmann equation. Its distribution function depends on two internal variables, by
which we can study the energy exchange among the different molecular modes of a gas, that is,
translational, rotational, and vibrational modes. Then a triple hierarchy of the moment system is necessary,
and the system of balance equations is closed via the maximum entropy principle. The production terms in
the system, which are suggested by a generalized BGK-type collision term in the Boltzmann equation, are
adopted.
Firstly, in order to cast a spotlight on the dynamic pressure, a simplified RET theory with seven independent
fields (ET7): mass density, velocity, translational energy density, molecular rotational energy density, and the
molecular vibrational energy density [3] is explained. Secondly, by taking into account also the viscous stress
and the heat flux, the RET theory with 15 independent fields (ET15) is explained [4]. This is a direct
generalization of the Navier-Stokes and Fourier (NSF) theory of viscous heat- conducting fluids. In fact, the
NSF theory can be derived from the ET15 as a limiting case of small relaxation times via the Maxwellian
iteration. The relaxation times introduced in the theory are shown to be related to the shear and bulk
viscosities and heat conductivity. Some applications of the RET theories explained above to the ultrasonic
waves will be shown in the next talk by T. Arima.
This is the joint work with T. Arima and T. Ruggeri [3,4].

References
[1] I. Müller and T. Ruggeri: Extended Thermodynamics, (1st edition) Springer 1993; Rational Extended

Thermodynamics, (2nd edition) Springer 1998.
[2] T. Ruggeri and M. Sugiyama: Rational Extended Thermodynamics beyond the Monatomic Gas,

Springer 2015.
[3] T. Arima, T. Ruggeri and M. Sugiyama: Phys. Rev. E 96 (2017) 042143.
[4] T. Arima, T. Ruggeri and M. Sugiyama: Entropy 20 (2018) 301.

59

ON THE SIMILARITY SOLUTION OF STRONG SPHERICAL SHOCK WAVES BASED ON
EXTENDED THERMODYNAMICS

Shigeru Taniguchi, Tommaso Ruggeri

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

Rational Extended Thermodynamics (RET) [1, 2] has been developed for analyzing highly non-equilibrium
phenomena out of local equilibrium. It has been shown that the RET theory can explain the features of the
structure of a plane shock wave in a polyatomic gas in which the internal modes, namely, the rotational or
vibrational modes in a molecule relax very slowly and the theoretical predictions by the RET theory agree with
experimental data and the predictions by kinetic theory quantitatively [3, 4, 5].
In this talk, we study similarity solutions of a spherical shock wave in rarefied polyatomic gases on the basis
of the RET theory with only six independent fields; the mass density, velocity, pressure and dynamic pressure.
By adopting the strategy proposed in [6] based on the Lie group theory, we derive a system depending on a
similarity variable and numerically solve this system with the boundary conditions for a strong shock [7], In
particular, the deviation from the well-known Sedov-von Neumann-Taylor solution is addressed quantitatively
and an important role of the dynamic pressure will be discussed [7].
References
[1] I. Muller and T. Ruggeri, Rational Extended Thermodynamics (Springer- Verlag, New York, 1998).
[2] T. Ruggeri and M. Sugiyama, Rational Extended Thermodynamics beyond the Monatomic Gas

(Springer, Heidelberg, 2015).
[3] S. Taniguchi, T. Arima, T. Ruggeri and M. Sugiyama, Phys. Rev. E Vol. 89, 013025 (2014).
[4] S. Kosuge and K. Aoki, Phys. Rev. Fluids Vol. 3, 023401 (2018).
[5] S. Taniguchi, T. Arima, T. Ruggeri and M. Sugiyama, J. Phys. Conf. Ser. Vol. 1035, 012009 (2018).
[6] A. Donato and T. Ruggeri, J. Math. Anal. Appl. Vol. 251, 395 (2000).
[7] R. Nagaoka, S. Taniguchi and T. Ruggeri, submitted to AIP Conf. Proc. (2019).

60

MODELLING OF ECOLOGY IN A PHOTOTROPHIC-HETEROTROPHIC BIOFILM

AlbertoTenore, Berardino D’Acunto, Maria Rosaria Mattei,
Vincenzo Luongo, Luigi Frunzo

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

The work presents a 1D mathematical model for the analysis and prediction of microbial interactions within
mixotrophic biofilms composed of microalgae and heterotrophic bacteria. The model combines equations for
biomasses growth and decay, diffusion-reaction of substrates, and detachment process. In particular, the
colonization of external species invading the biofilm is considered. The biofilm growth is governed by nonlinear
hyperbolic PDEs while substrate and invading species dynamics are dominated by semilinear parabolic PDEs.
It follows a complex system of PDEs on a free boundary domain. The equations are numerically integrated by
using the method of characteristics. The model has been applied to simulate the ecology of a mixotrophic
biofilm formed by phototrophic and heterotrophic species. For this purpose, the main factors influencing
microbial interactions has been included, light as well as nutrients.

61

KINETIC MODELING OF ALCOHOL CONSUMPTION

Giuseppe Toscani

Department of Mathematics, University of Pavia (Italy) and
IMATI - National Research Council, Pavia (Italy)
[email protected]

In most countries, alcohol consumption distributions have been shown to possess universal features. Their
unimodal right-skewed shape is usually modeled in terms of the Lognormal distribution, which is easy to fit,
test, and modify. However, empirical distributions often deviate considerably from the Lognormal model, and
both Gamma and Weibull distributions appear to better describe the survey data. In this talk we explain the
appearance of these distributions by means of classical methods of kinetic theory of multi-agent systems. The
microscopic variation of alcohol consumption of agents around a universal social accepted value of
consumption, is built up introducing as main criterion for consumption a suitable value function in the spirit of
the prospect theory of Kahneman and Twersky. The mathematical properties of the value function then
determine the unique macroscopic equilibrium which results to be a generalized Gamma distribution. The
modeling of the microscopic kinetic interaction allows to clarify the meaning of the various parameters
characterizing the generalized Gamma equilibrium.

62

THE EXTENDED THERMODYNAMICS FOR A.C AND D.C. DYNAMIC HIGH-FIELD
TRANSPORT IN GRAPHENE

Massimo Trovato

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

Using the maximum entropy principle (MEP) [1], we present a general theory to describe ac and dc high-field
transport in monolayer graphene [2] within a dynamical context. In particular:

i) The connections between the conductivity effective mass and the introduction of a Lorentz factor
for the system, and, more generally, the analogies between the monolayer graphene and other
physical systems in which we have a saturation velocity for the charge carriers are explicitly
explained.

ii) By keeping unchanged the modulus of the group velocity, we show that the external field and the
scattering processes can only align or randomize its direction with respect to the applied field.
Here, we prove that the alternation and the competition between these processes, together with
the effects of linear band structure, can lead to the onset of negative differential mobility (NDM)
for both the average velocity and for other deviatoric moments of higher order.

iii) By using the small-signal analysis, the nature and the characteristics of the collisional processes
can be easily investigated. We show that the electron transport is characterized by the streaming
motion regime due to the combined action of the electric field and scattering phenomena. The
streaming motion is also present in correspondence of a very few collisional events and it extends
to larger values of the external field than for the usual semiconductors.

In general, by using the present approach, the effects imputable to a linear band structure, the role of
conductivity effective mass of carriers, and their connection with the coupling between the driving field and the
dissipation phenomena are analyzed both qualitatively and quantitatively for different electron densities.
We conclude that, under conditions very far from thermal equilibrium, the HD results are found to compare
well with those obtained by analogous MC simulations. Consequently, the overall agreement is used to validate
the theoretical approach and to provide a systematic physical insight into the microscopic dynamics. Therefore,
the present HD-MEP method can be fruitfully applied to describe transport properties in graphene with the
relevant following advantages: (a) to provide a closed analytical approach and a reduced computational effort
with respect to other competitive numerical methods at a kinetic level; (b) to investigate and classify in a
systematic way the behavior of the macroscopic moments in ac and dc dynamic conditions; (c) to distinguish
the different regimes of transport by identifying, from an analysis of collisional frequencies, the dominant
scattering mechanisms for a given range of electric field.

References
[1] I. Müller, T. Ruggeri, Rational Extended Thermodynamics: Springer Tracts in Natural Philosophy, Vol.

37, (Springer-Verlag New York) (1998).
[2] M. Trovato, P. Falsaperla, L. Reggiani, Maximum-entropy principle for a.c. and d.c. dynamic high-field

transport in monolayer graphene (2019), to appear on Journal of Applied Physics.

63

UNCERTAINTY AND SENSITIVITY ANALYSIS FOR BACTERIAL INVASION IN MULTI-
SPECIES BIOFILMS

Andrea Trucchia, Luigi Frunzo, Maria Rosaria Mattei,
Vincenzo Luongo, Mélanie C. Rochoux

BCAM - Basque Center for Applied Mathematics (Spain)
[email protected]

In this work, we present a probabilistic analysis of a detailed one-dimensional continuum biofilm model which
explicitly accounts for planktonic bacterial invasion in a multi-species biofilm. The objective of the presented
research is i) to quantify and understand how the uncertainty in the new parameters of the invasion sub-model
influences the biofilm model predictions and ii) to spot which parameters are the most important factors with
respect to the biofilm model response. A surrogate of the biofilm model is trained using an experimental design
with limited size. A comparison of different types of surrogates (generalized Polynomial Chaos expansion -
gPC, Gaussian process model - GP) is performed; results show that the best performance (measured in terms
of the Q_2 predictive coefficient) is retrieved using a Least-Angle Regression (LAR) gPC-type expansion,
where a sparse polynomial basis is constructed to reduce the problem size and where the basis coordinates
are obtained using a regularized least-square minimization. The resulting LAR gPC-expansion is found to
capture the raise in complexity of the biofilm structure due to niche formation. Sobol’ sensitivity indices show
the prevalence in the invasion sub-model of the maximum colonization rate of autotrophic bacteria on biofilm
composition.

64

INSTABILITY, WEAK TURBULENCE AND CHAOS IN POROUS MEDIA

Peter Vadasz

Department of Mechanical Engineering,
Northern Arizona University (USA)
[email protected]

A review of the research on the instability of steady state convection in a porous layer heated from below is
presented. The latter leads to chaos (weak turbulence) and the possibility of controlling this transition from
steady convection to chaos is considered. The governing equations consisting of the continuity, the extended
Darcy, and the energy equations subject to the assumption of local thermal equilibrium and the Boussinesq
approximation are converted into a set of three nonlinear ordinary differential equations by assuming two-
dimensional convection and expansion of the dependent variables into a truncated spectrum of modes.
Solutions to the resulting set of equations via analytical (weak nonlinear), computational (Adomian
decomposition) as well as numerical (Runge-Kutta-Verner) methods are presented and compared to each
other. The analytical solution for the transition point to chaos is identical to the computational and numerical
solutions in the neighborhood of a convective fixed point and deviates from the accurate computational and
numerical solutions as the initial conditions deviate from the neighborhood of a convective fixed point. The
control of this transition is also discussed.

65

LINEARLY DEGENERATE SYSTEMS OF PDES AND INTERACTING WAVES

Raffaele Vitolo

Department of Mathematics and Physics,
University of Salento, Lecce (Italy)
[email protected]

In this talk we will review recent results on the theory of linearly degenerate (or completely exceptional)
quasilinear systems of first-order PDEs in two independent variables. We will show that a wide family of such
systems is determined by the requirement that there exists a Hamiltonian formulation of a distinguished type.
In the 3-component case, such systems are all integrable and are related to the Zakharov-Manakov system of
3 interacting waves. This nontrivial map yields a correspondence between solutions that will be discussed.

66

GLOBAL ANALYSIS OF MATHEMATICAL MODELS FOR NONLOCAL EPIDEMIC DISEASES

Wendi Wang

School of Mathematics and Statistics,
Southwest University (China)
[email protected]

The mathematical models of dengue fever and lyme disease are presented, which are described by nonlocal
reaction-diffusion equations. The basic reproduction numbers of disease transmission are derived and are
shown to be the threshold values of disease invasion. The influences of population mobility and spatial
heterogeneity on disease outbreaks are also analyzed by numerical simulation.

67

THE RIEMANN PROBLEM OF RELATIVISTIC EULER EQUATIONS

Qinghua Xiao

Wuhan Institute of Physics and Mathematics,
Chinese Academy of Sciences (China)
[email protected]

We study the Riemann problem of relativistic Euler equations with Synge energy for rarefied monatomic gas
and polyatomic gas. Constitutive equations of these relativistic Euler equations are related to the modified
Bessel functions of the second kind. We provide detailed investigation of basic hyperbolic qualities and
properties of elementary waves for the relativistic Euler equations, especially for the properties of shock waves
for the relativistic Euler equations describing the rarefied monatomic gas and some polyatomic gas.
Mathematical theory of the Riemann problem for these relativistic Euler equations, which is analogous to the
corresponding theory of the classical Euler equations, is rigorously provided.
This is a joint work with Prof. Tommaso Ruggeri and Prof. Huijiang Zhao.

68

LIST OF SPEAKERS IN ALPHABETICAL ORDER (A - L)

Arima, Takashi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . pag. 9
Barbera, Elvira . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . pag. 10
Bisi, Marzia. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . pag. 11
Brini, Francesca . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . pag. 12
Brull, Stéphane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . pag. 13
Buonomo, Bruno . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . pag. 14
Capone, Florinda . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . pag. 15
Consolo, Giancarlo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . pag. 16
D’Acunto, Berardino . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .pag. 17
Dafermos, Costantine M. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . pag. 18
De Angelis, Monica . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . pag. 19
De Falco, Vittorio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . pag. 20
De Luca, Roberta . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . pag. 21
Demontis, Francesco . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . pag. 22
Desvillettes, Laurent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . pag. 23
Falsaperla, Paolo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . pag. 24
Fiore, Gaetano . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . pag. 25
Frunzo, Luigi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . pag. 26
Gambino, Gaetana . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . pag. 27
Gargano, Francesco . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . pag. 28
Gentile, Maurizio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . pag. 29
Giacobbe, Andrea . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . pag. 30
Giunta, Valeria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . pag. 31
Gouin, Henri . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .pag. 32
Groppi, Maria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . pag. 33
Ha, Seung-Yeal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . pag. 34
Kim, Jeongho. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . pag. 35
Liu, Tai-Ping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . pag. 36
Lombardo, Maria Carmela . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . pag. 37
Luongo, Vincenzo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . pag. 39

LIST OF SPEAKERS IN ALPHABETICAL ORDER (M - Z)

Mainardi, Francesco . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . pag. 40
Manganaro, Natale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . pag. 41
Maremonti, Paolo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . pag. 42
Martalò, Giorgio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . pag. 43
Mattei, Maria Rosaria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . pag. 44
Mentrelli, Andrea . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . pag. 45
Mulone, Giuseppe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . pag. 46
Nastasi, Giovanni . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . pag. 47
Panaro, Daniele B. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . pag. 48
Pennisi, Sebastiano . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . pag. 49
Rionero, Salvatore . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . pag. 51
Romano, Vittorio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . pag. 52
Rubino, Gianfranco . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . pag. 53
Ruggeri, Tommaso . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . pag. 54
Saccomandi, Giuseppe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . pag. 55
Sammartino, Marco . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . pag. 56
Sciacca, Michele . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . pag. 57
Sciacca, Vincenzo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . pag. 58
Sugiyama, Masaru . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . pag. 59
Taniguchi, Shigeru . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . pag. 60
Tenore, Alberto . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . pag. 61
Toscani, Giuseppe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . pag. 62
Trovato, Massimo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . pag. 63
Trucchia, Andrea . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . pag. 64
Vadasz, Peter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .pag. 65
Vitolo, Raffaele . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .pag. 66
Wang, Wendi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . pag. 67
Xiao, Qinghua . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . pag. 68

WASCOM 2019