Evolutionary Partial Differential Equations
A.Y. 2026/2027
Learning objectives
The course aims to provide the basic functional-analytic and variational tools for the study of the main classes of evolution equations, with particular emphasis on parabolic and hyperbolic problems, as well as on the Navier-Stokes equations. The course also presents models arising from applications within a rigorous mathematical framework.
Expected learning outcomes
At the end of the course, students will be able to analyze evolution problems in appropriate functional spaces, discuss existence, uniqueness, continuous dependence, and regularity of solutions by means of classical techniques such as the Faedo-Galerkin method and time discretization.
Lesson period: First semester
Assessment methods: Esame
Assessment result: voto verbalizzato in trentesimi
Single course
This course cannot be attended as a single course. Please check our list of single courses to find the ones available for enrolment.
Course syllabus and organization
Single session
Responsible
Lesson period
First semester
Course syllabus
Preliminary program
1. Review of prerequisite topics
1.1. L^p spaces, Hilbert spaces, and Sobolev spaces. Weak and weak-* convergence.
1.2. Function spaces with values in Banach spaces.
1.3. Linear operators.
2. Parabolic Equations
2.1. Variational formulation of linear initial-boundary value problems.
2.2. Existence of solutions: the Faedo-Galerkin method.
2.3. Uniqueness, continuous dependence on the data, and regularity of solutions.
2.4. Nonlinear reaction-diffusion equations.
3. Hyperbolic Equations
3.1. Variational formulation of linear initial-boundary value problems.
3.2. Existence of solutions via time discretization methods.
3.3. Uniqueness, continuous dependence on the data, and regularity of solutions.
3.4. An example of a nonlinear hyperbolic equation: the sine-Gordon equation.
4. The Navier-Stokes Equations
4.1. Weak formulation of the Navier-Stokes equations.
4.2. Existence of weak solutions in three dimensions and uniqueness of weak solutions in two dimensions.
4.3. Existence of strong solutions in two dimensions.
5. Subject to the available time: An Introduction to the theory of attractors
5.1. Definition of a dynamical system. Dissipative dynamical systems.
5.2. Definitions of trajectory, complete and bounded trajectories, and equilibrium points.
5.3. The ω-limit set and its characterization. Absorbing sets.
5.4. Global attractors: uniqueness, properties, and the existence theorem for global attractors.
5.5. Applications to reaction-diffusion equations, the sine-Gordon equation, and the Navier-Stokes equations.
1. Review of prerequisite topics
1.1. L^p spaces, Hilbert spaces, and Sobolev spaces. Weak and weak-* convergence.
1.2. Function spaces with values in Banach spaces.
1.3. Linear operators.
2. Parabolic Equations
2.1. Variational formulation of linear initial-boundary value problems.
2.2. Existence of solutions: the Faedo-Galerkin method.
2.3. Uniqueness, continuous dependence on the data, and regularity of solutions.
2.4. Nonlinear reaction-diffusion equations.
3. Hyperbolic Equations
3.1. Variational formulation of linear initial-boundary value problems.
3.2. Existence of solutions via time discretization methods.
3.3. Uniqueness, continuous dependence on the data, and regularity of solutions.
3.4. An example of a nonlinear hyperbolic equation: the sine-Gordon equation.
4. The Navier-Stokes Equations
4.1. Weak formulation of the Navier-Stokes equations.
4.2. Existence of weak solutions in three dimensions and uniqueness of weak solutions in two dimensions.
4.3. Existence of strong solutions in two dimensions.
5. Subject to the available time: An Introduction to the theory of attractors
5.1. Definition of a dynamical system. Dissipative dynamical systems.
5.2. Definitions of trajectory, complete and bounded trajectories, and equilibrium points.
5.3. The ω-limit set and its characterization. Absorbing sets.
5.4. Global attractors: uniqueness, properties, and the existence theorem for global attractors.
5.5. Applications to reaction-diffusion equations, the sine-Gordon equation, and the Navier-Stokes equations.
Prerequisites for admission
Basic concepts of functional analysis: Banach spaces, Hilbert spaces, and L^p spaces. Dual spaces, weak convergence, compact operators, and their spectrum. Basic knowledge of elliptic partial differential equations (the necessary results will be briefly reviewed during the course).
Suggested references for the prerequisites:
1) S. Salsa, Partial Differential Equations, Springer
2) H. Brezis, Functional Analysis, Liguori Editore
Suggested references for the prerequisites:
1) S. Salsa, Partial Differential Equations, Springer
2) H. Brezis, Functional Analysis, Liguori Editore
Teaching methods
Lectures in presence and course notes on selected topics.
Teaching Resources
Suggested bibliography
· H. Brezis, Analisi funzionale, Liguori Editore.
· J. C. Robinson, Infinite-Dimensional Dynamical Systems, Cambridge University Press.
· S. Salsa, Equazioni a derivate parziali, Springer.
· L. C. Evans, Partial Differential Equations, American Mathematical Society.
· Lecture notes.
Addtional reference for selected topics
· R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Applied Mathematical Sciences 68, Springer-Verlag.
· H. Brezis, Analisi funzionale, Liguori Editore.
· J. C. Robinson, Infinite-Dimensional Dynamical Systems, Cambridge University Press.
· S. Salsa, Equazioni a derivate parziali, Springer.
· L. C. Evans, Partial Differential Equations, American Mathematical Society.
· Lecture notes.
Addtional reference for selected topics
· R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Applied Mathematical Sciences 68, Springer-Verlag.
Assessment methods and Criteria
The assessment consists of an oral examination on the topics and proofs covered during the lectures and listed in the final course syllabus.
MATH-03/A - Mathematical Analysis - University credits: 6
Lessons: 42 hours
Professors:
Aspri Andrea, Cavaterra Cecilia
Professor(s)
Reception:
appointment via email
Dipartimento di Matematica, Via Saldini 50 - ufficio n. 2060