Ing Ind - Inf (Mag.)(ord. 270) - MI (487) MATHEMATICAL ENGINEERING - INGEGNERIA MATEMATICA
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095963 - ADVANCED PARTIAL DIFFERENTIAL EQUATIONS
Goals
To present mathematical models and advanced techniques in the theory of partial differential equations, of frequent use in the applied sciences.
Expected learning outcomes
At the end of the course the student is expected:
1. To know the basic theory of Hilbertian Sobolev spaces: density and approximation by smooth functions, traces, compactness, embeddings, Bochner integral. The theory of the Faedo-Galerkin method and its applications. Applications of fixed point theorems. The basic theory of the fundamental differential equations in mathematical physics.
2. To be able to derive the variational formulation of the most common boundary and initial boundary value problems for stationary and evolution PDEs and analyze their well-posedness. Knowledge of some nonlinear equations such as the stationary Navier-Stokes system or semilinear equations. To be able to analyze a physical model and to set up the related partial differential equation.
Topics
1. Sobolev spaces. Definitions of the Hilbert-type spaces and their duals. Traces. Poincaré inequalities. Embedding theorems. Time dependent Sobolev spaces, the Bochner integral.
2. Elliptic equations. Variational formulation of the most common boundary value problems for equations in divergence form. Lax-Milgram Theorem and analysis of the well posedness. Maximum principles, positivity preserving. Eigenvalues and eigenfunctions. The Stokes system, equilibrium of a plate. Nonlinear equations: steady Navier-Stokes equations, semilinear elliptic equations, critical point theory.
3. Evolution equations. First order quasilinear equations: characteristic lines and shocks. Hilbert triples. Abstract second order evolution problems: existence, uniqueness, stability for abstract equations. Parabolic equations: weak formulation of the most common initial-boundary value problems. Weak maximum principles. Wave equation: weak formulation of the most common initial-boundary value problems. The Faedo-Galerkin method and analysis of the well posedness. Energy methods and applications to semilinear parabolic and hyperbolic problems.
4. Conservation laws. Solitary waves for evolution equations, basic examples. Conservation of mass, momentum, energy. The Korteweg-de Vries equations.
Pre-requisites
Basic knowledge of the most common equations of mathematical Physics: Heat and Laplace equation (separation of variables, maximum principles, fundamental solution). Wave equation (d'Alembert, Kirchhoff formulas). Basic calculus with Distributions.
Lebesgue integration and spaces of p-summable functions. Banach and Hilbert spaces. Riesz Representation and Lax-Milgram Theorems. Fourier series. Weak convergence. Banach-Alaoglu Theorem.
Assessment
Written examination: the student is required to solve boundary value problems, initial-boundary value problems, to set up variational formulations, to solve problems through enrgy methods. Highly insufficient grades (<13/30) in the written examination are not allowed to take part at the oral discussion.
Oral discussion: it is on the whole program, the student must be able to explain the main concepts and prove the main results with property of language.
Bibliography
A. Ferrero, F. Gazzola, M. Zanotti, Elements of advanced mathematical analysis for physics and engineering}, , Editore: Ed. Esculapio, Anno edizione: 2013
F. Gazzola, F. Tomarelli, M. Zanotti, Analytic Functions, Integral Transforms, Differential Equations, Editore: Ed. Esculapio, Anno edizione: 2020
F. Gazzola, H.C. Grunau, G. Sweers, Polyharmonic boundary value problems, Editore: LNM 1991, Springer, Anno edizione: 2010 http://www1.mate.polimi.it/~gazzola/book_GGS.pdf Note:
downladable for free
Notes from prof. Gazzola Note:
Complements to some parts of the program
Software used
No software required
Learning format(s)
Type of didactic form
Ore di attività svolte in aula
(hh:mm)
Ore di studio autonome
(hh:mm)
Lesson
48:00
72:00
Training
32:00
48:00
Computer Laboratory
0:00
0:00
Experimental Laboratory
0:00
0:00
Project Laboratory
0:00
0:00
Total
80:00
120:00
Information in English to support internationalization
English