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Equazioni di evoluzione quasilineari
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Quasi-linear Evolution Equations
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Academic year 2019/2020
- Teacher
- Albert James Milani
- Type
- Basic
- Course disciplinary sector (SSD)
- MAT/05 - analisi matematica
- Delivery
- Formal authority
- Language
- English
- Attendance
- Obligatory
- Type of examination
- Oral
- Prerequisites
- Functional Analysis
- Propedeutic for
- Evolution Equations and Dynamical Systems
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Sommario del corso
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Course objectives
Offer an elementary, unified approach to the theory of strong solvability of the simplest type of initial-boundary value problem for quasi-linear evolution equations of hyperbolic and parabolic type (quasi-linear wave and heat equations) in a bounded domain, with homogeneous Dirichlet boundary conditions.
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Course delivery
Lectures
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Learning assessment methods
Interaction with audience
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Program
1 Introduction. 1.1 Second Order PDEs of Evolution. 1.2 Overview. 1.3 Examples. 1.4 Technical Results. 1.4.1 Notations. 1.4.2 Bases of L^2(Ω) and H^1_0(Ω). 1.4.3 Sobolev Imbeddings and Related Inequalities. 1.4.4 Sobolev Spaces Involving Time. 1.4.5 Regularization. 1.4.6 Elliptic Regularity.
2 Hyperbolic Equations. 2.1 Linear Equations. 2.1.1 Weak Solutions. 2.1.2 A Regularity Result. 2.2 Regular Solutions. 2.3 Quasi-Linear Equations. 2.3.1 Local Regular Solutions. 2.3.2 The Compatibility Conditions.
3 Parabolic Equations. 3.1 Linear Equations. 3.1.1 Weak Solutions. 3.1.2 A Regularity Result. 3.2 Regular Solutions. 3.3 Quasi-Linear Equations. 3.3.1 Local Regular Solutions. 3.3.2 The Compatibility Conditions. 3.4 Equations in Divergence Form. 3.4.1 Weak Solutions. 3.4.2 A Regularity Result.
4 On Global Existence. 4.1 Quasi-Linear Wave Equations. 4.2 Quasi-Linear Heat Equations. 4.3 Parabolic Equations in Divergence Form 4.4 Gradient Estimates.
Suggested readings and bibliography
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Will be provided in class
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Note
Will be available (in English)
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