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

Preliminaries of tensorial calculus and analysis.

Introduction to partial differential equations: the Cauchy problem, characteristic curves and surfaces, classification and propagation of singularities.

Conservative form: weak solutions and shocks.

Hyperbolic quasi-linear first-order systems 1D and 3D.

Discontinuity/hyperbolic waves VS dispersive waves.

The travelling waves tool with examples.

Introduction to Continuum Mechanics.

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.

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.

Convective instabilities: the Boussinesq approximation and the classical Bénard problem.

Nonlinear hyperbolic models in a conservative form: weak solutions and shock waves. The Euler model in an adiabatic regime.

Nonlinear parabolic and hyperbolic mathematical models for traffic flow, 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 reacton-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.

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.

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 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.L. Vazquez: An introduction to the mathematical theory of the porous medium equation Oxford Univ.Press, 2007.

Lecture notes of the teacher and research articles distributed within lessons.

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.

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