76300 - Mathematical Methods for Continuum Mechanics

Learning outcomes

At the end of the course the student: - has deep notions of Continuum Mechanics in their main mathematical aspects; - is able to analyze autonomously the most recent developments of the mentioned matters and their most important problems, related to applications in Physics, Biology and Astrophysics.

Course contents

Introduction to compartmental models, formulated in a continuum mechanics background, SI, SIR/SIV type models in detail, for the description of real life phenomena, from ecological applications to either the spread of virus/e-cigarettes epidemics or the diffusion of bacteria.

The role of a spatial diffusion: from Dynamical Systems to Diffusion-Reaction Models governed by parabolic/hyperbolic PDES..

Equilibrium solutions and related stability analysis.

Cross-diffusion terms, within the Fick's Law, and logistic type (Verhulst) reactions towards the onset of Turing instabilities.

Examples: from the Lotka-Volterra prey-predator system, with fishing and logistics type effects, even in the presence of self/cross spatial diffusion, to very recent epidemiological models generalizing the pionieristic SIR type Kermack-McKendrick model, accounting for demography, vaccination and information, VS either the recent chemotactic model for virus infection SIV , V=Virus, by Bellomo-Tao, based on the Nowak-May dynamical system or the SIL type model by Vanag-Epstein for crimo-taxis.

The special role of forward and backward bifurcations, towards the definition of the typical Basic Reproduction Numbers, in short BRN, of the model under study.

The reaction-diffusion (parabolic/hyperbolic) models of Abramson and Kenkre for the spread of the Hantavirus and the aggregation (parabolic/hyperbolic) models of Keller-Segel for chemotactic mechanisms, also in the presence of a stabilizing logistic type term on the bacterial cell.

Introduction to first- and second-order PDES in two real independent variables: the Cauchy problem, the role of characteristic curves within classification and propagation of first/second order Jump type singularities, towards the hyperbolic waves of hyperbolic models.

The formal generalization to parabolic, hyperbolic and elliptic models in the 3D version: the special role of characteristic surfaces.

Wave propagation properties of Hyperbolic PDEs systems: as a special example, the 1D and 3D versions of the Euler model for a barotropic perfect gas.

Hyperbolic  sound waves VS dispersive sound waves, within the Fourier Normal Modes tool.

Classical and non-classical second order PDEs: analytical tools for the study of qualitative mathematical properties.

The Energy Method in L^2: Boundary Conditions towards Uniqueness Theorems and non linear Stability results.

A comparative study among the classical diffusion, wave and damped wave linear models with the semilinear Navier-Stokes model for viscous liquids.

Quasi-linear first-order PDEs in a conservative form: The Rankine-Hugoniot equations, weak solutions and shocks.

Introduction to Continuum Mechanics.

Preliminaries of tensorial calculus and analysis.

Localizations, configurations, deformations, motions and kinematical properties, in both formalisms eulerian and lagrangian: the Transport Theorem in its various versions.

General balance laws, in integral and local forms, even in the presence of a singular surface relative to the scalar/vectorial unknown field.

The role of constitutive relations for vectorial/tensorial influxes.

Derivation of the Rankine-Hugoniot jump-type equation.

The conservation principles of the Continuum Mechanics and their local divergence and convective forms: the Cauchy and Kinetic Energy Theorems.

Classical and non classical constitutive theories for fluids and solids: the Euler model for barotropic perfect fluids, the Navier-Stokes model for linearly viscous/dissipative fluids, the Maxwell model for viscoelastic fluids and the Navier model for linear and homogeneous elastic solids.

The two Thermodynamics Principles, in both integral and local forms.

Rigid heat conduction: the energy equation, the empirical stationary Fourier law vs alternative constitutive theories with a single relaxation time, like the hyperbolic Maxwell-Cattaneo correction. 

Construction, in terms of constitutive relations and balance equations, of  quasi-linear parabolic and hyperbolic mathematical models for traffic flows, in population dynamics and in bio-medical settings, also accounting for a relaxation/delay time.

A special focus on reaction-diffusion models by Abramson and Kenkre for the spread of the Hantavirus, the reaction-diffusion-drift models by Keller-Segel and the hydrodynamic model by Chavanis-Sire for chemotactic aggregations..

Mathematical models to analyze the effect of e-cigarettes on smoking cessation: qualitative and sensivitivity analysis.

A brief review of  continuum mathematical models for the onset and diffusion of the Alzheimer's disease.

Analogies between the chemotactic collapse in cellular aggregation processes and the Jeans instability in an astrophysical setting, within star and galaxy formation processes.

Readings/Bibliography

F.John: Partial Differential Equations, Springer, 1991.

M.Renardy, R.C.Rogers: Introduction to PDEs, Springer, 2006.

I- Shih Liu: Continuum mechanics, Springer 2002.

T.Ruggeri: Introduzione alla termomeccanica dei sistemi continui ed ai sistemi iperbolici, UNITEXT, Springer 2025.

B.Straughan: The energy method, stability and nonlinear convection, Springer New York, 2004.

B.Straughan: Heat Waves Applied Mathematical Sciences, 177, Springer New York, 2011.

J.D.Murray:Mathematical Biology. I: An Introduction, vol.17 of Interdisciplinary Appl.Math. Springer, New York, 2003a.

J,D.Murray:Mathematical Biology. II: Spatial models and biomedical applications, vol.17 of Interdisciplinary Appl. Math. Springer, New York, 2003b

Lecture notes of the teacher and research articles published on Virtuale.

Teaching methods

The course, consists of class-room lectures where the theoretical aspects of each topics are dealt with, stressing the importance of the knowledge of dynamical systems and nonlinear PDE's to build novel mathematical models, and introducing the analytical techniques to approach their study, with the aim to mathematically describe the experimental properties of real life phenomena in the bio-medical, physical and astrophysical fields.

Attendance is stongly recommended, but it is not mandatory.

Assessment methods

The assessment method consists in an oral exam, where the first question is a discussion on a topic/model, related to the  subjects covered within the course, chosen by the student, followed by questions from the teacher that, starting from this topic, could range across the whole program, with the aim of ascertaining the knowledge of the various formalisms developed within the course itself, along with their goals.

Office hours

See the website of Franca Franchi