The activity in mathematical analysis is mainly focussed on ordinary and partial differential equations, on dynamical systems, on the calculus of variations, and on control theory. Connections of these topics with differential geometry are also developed.The activity in mathematical modelling is oriented to subjects for which the main technical tools come from mathematical analysis. The present themes are multiscale analysis, mechanics of materials, micromagnetics, modelling of biological systems, and problems related to control theory.The applications of mathematics developed in this course are related to the numerical analysis of partial differential equations and of control problems. This activity is organized in collaboration with MathLab for the study of problems coming from the real world, from industrial applications, and from complex systems.

## Turbulent compressible fluid dynamics

Turbulence plays a fundamental role in many different applications varying from aeronautical to environmental simulations. This course aims at giving an overview of the phenomenology and mathematical modelling of compressible turbulent flows. We will start from a first part focused on gasdynamics (thermodynamics, compressible navier-stokes equations, speed of sound, shock waves). The second part of the course will instead be focused on turbulence phenomenology and modelling.

## Calculus of variations

Programme: Minumum problems for integral functionals in one independent variable: necessary and sufficient optimality conditions, solution to classical problems, problems invariant under reparametrization and geodesics. Minumum problems for multiple integrals: direct methods, lower semicontinuity and relaxation results for multiple integrals, quasiconvexity and polyconvexity.

Number of lectures: 30 two hour lectures

Period: November 15 - March 29

## Semigroup theory and applications

**Program**

Bochner integral; Pettis and Bochner theorems; vector valued distributions and Sobolev functions.

Elements on unbounded operators: closed, dissipative and maximal dissipative operators.

Semigroups and their generators.

Cauchy problem for abstract equations, Duhamel formula.

Hille-Yosida, Lumer-Phillips and Stone theorems, construction of (semi)groups associated to Heat, Wave, Klein Gordon and Schrödinger equations.

## Advanced FEM techniques

## Topics in computational fluid dynamics

### Topics/Syllabus

- Introduction to CFD, examples.
- Constitutive laws
- Incompressible flows.
- Numerical methods for potential and thermal flows
- Boundary layer theory
- Thermodynamics effects, energy equation, enthalpy and entropy
- Vorticity equations
- Introduction to turbulence
- Numerical methods for viscous flows: steady Stokes equations
- Stabilisation (SUPG) and inf-sup condition

## Computational mechanics by reduced order methods

**Mathematics Area, PhD in Mathematical Analysis, Modelling and Applications (AMMA)**

**Master in High Performance Computing (MHPC)**

**Lectures Prof Gianluigi Rozza, Tutorials coordinated by Dr Michele Girfoglio, Dr Federico Pichi and Dr Ivan Prusak.**