# 76300 - Mathematical Methods for Continuum Mechanics

• Campus: Bologna
• Corso: Second cycle degree programme (LM) in Mathematics (cod. 5827)
• from Sep 18, 2023 to Dec 20, 2023

## 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, the SI, SIR/SIV models in detail, for the description of natural phenomena, from ecological applications to epidemic/e-cigarettes spread.

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

Equilibrium solutions and stability analysis.

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

Examples: from the Lotka-Volterra prey-predator system, even diffusive, to epidemic models generalizing the classic Kermack-McKendrick SIR model VS  either the chemotactic model for virus infection SIV by Bellomo-Tao or the SIL model by Vanag-Epstein for crimo-taxis.

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

Wave propagation properties of Hyperbolic PDEs.

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

The Energy Method in L^2: uniqueness and non linear stability.

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

Hyperbolic quasi-linear first-order systems 1D and 3D with applications in social and life sciences.

Discontinuity/hyperbolic waves, within the characteristic  curves/surfaces topic VS dispersive waves, within the Fourier normal modes tool.

The travelling waves tool with examples.

Introduction to Continuum Mechanics.

Preliminaries of tensorial calculus and analysis.

Localizations, configurations, deformations, motions and kinematical properties, in both formalisms eulerian and lagrangian.

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.

Analysis of linear and non-linear stability, uniqueness and wave propagation properties.

The two Thermodynamics Principles, in integral and local forms.

The Energy equation, the empirical Fourier's law and alternative heat theories with a single/dual phase lag time.

The Entropy inequality, the Helmholtz free energy and the Clausius-Duhem inequality: the classical Gibbs forms and the constitutive restrictions.

Thermodynamic compatibility of a mathematical model.

The Navier-Stokes-Fourier thermo-viscous model and the Euler model for a polytropic gas.

Rigid heat conduction, classical and with a thermal relaxation time: a comparative analysis among linear and non-linear parabolic heat equations and hyperbolic ones.

Construction, in terms of constitutive relations and balance equations, of  quasi-linear parabolic and hyperbolic mathematical models for traffic flows, in dynamics of population and in bio-medical contexts, accounting for a relaxation/delay time; the non-linear porous medium equation and its properties of finite wave propagation.

The reaction-diffusion models of Abramson and Kenkre for the spread of the Hantavirus and the aggregation models of Keller-Segel and Chavanis-Sire for chemotactic mechanisms.

Wave propagation properties of new hyperbolic diffusion-reaction models for chemotaxis, generalizing the Dolak-Hillen strategy.

Linear stability analysis VS chemotactic collapse formation: stabilization effects.

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

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

Mathematical models for a self-gravitating interstellar gas cloud, in an expanding universe: gravito-sonic waves VS gravitational collapse formation.

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

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 continui, Monduzzi editore 2007.

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 nonlinear PDE's to build mathematical models, and introducing the analytical techniques to approach their study, with the aim to describe the experimental properties in physical,  astrophysical and biomedical situations.

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, with the aim of ascertaining the knowledge of the different formalisms developed within the course itself. content of the course.

## Office hours

See the website of Franca Franchi

### SDGs

This teaching activity contributes to the achievement of the Sustainable Development Goals of the UN 2030 Agenda.