Fourier Analysis and Nonlinear PDE (Winter 2023-2024)

Phan Thành Nam, Lecture Notes (typed by Arianna Rast), Homework, Moodle (ID 30600, pass F&PDE2324)

General Information

Goal: In this course we will use classical tools in harmonic analysis to study nonlinear differential equations. In the first part, we will discuss some popular evolution equations such as heat equations, wave equations, transport equations, and Schrödinger equations where basic Fourier analysis tools are helpful. In the second part, we will focus on concrete models coming from fluid mechanics (e.g. Navier–Stokes and Euler equations) for which the Littlewood-Paley decomposition plays a prominent role.

Audience: Bachelor and Master students of Mathematics, TMP Master. Other participants will get "Schein" if pass the course.

The course is suitable for master students and motivated bachelor students. Prerequisites: Lebesgue integration and L^2 theory of the Fourier transform. Knowing some basic results from harmonic analysis (e.g. Maximal inequalities, Littlewood–Paley theory, ...) are helpful, but not mandatory since they will be recalled properly.

Time and place:

• Lectures: Thursday 10:15-12:00 (B 046), Friday 14:15-16:00 (B 251).
• Exercises: Friday 16:15-18:00 (B 251).
• Tutorial: Thursday 08:30-10:00 (B 039).

References:

• Hajer Bahouri, Jean-Yves Chemin and Raphaël Danchin: Fourier Analysis and Nonlinear Partial Differential Equations. Springer 2011.

Grade: Final grade is determined by your total performance of the exams:

• You can get up to 100 points in the final exam.
• You can get up to 25 points in the midterm exam.

You need 50 points to pass the course and 85 points to get the final grade 1.0.

Exams: The midterm exam and final exam take place on Thursday December 8, 2023 and Friday February 9, 2024, respectively. You can use your notes (e.g. lecture notes, homework sheets) in the exams.

Contents of the lectures

19.10. Introduction.

20.10. Chapter 1: Basic analysis. L^2 theory of Fourier transform. Schwartz space. Tempered distributions. Fourier transform of tempered distributions.

26.10. Tempered distributions S'(R^d) vs. General distributions D'(R^d). Homogeneous Sobolev Spaces. Completeness of the homogeneous Sobolev norm when 2s < d.

27.10. Relation between Sobolev spaces and distributional derivatives. Duality of Sobolev spaces. A quadratic form representation of fractional Sobolev space.

3.11. Sobolev's continuous embedding theorem. Non-homogeneous Sobolev spaces. Weak convergence and compactness. Sobolev's compact embedding theorem.

9.11. Moser-Trudinger inequality and BMO inequality in the critical case s=d/2. Hardy inequality. Perron-Frobenius principle. Trace operator.

10.11. Chapter 2: Some basic evolution equations. Heat equation. Fundamental solution. Smoothing effect. Asymptotic behavior at time zero and infinity. Uniqueness in tempered distributions. Tychonoff's non-uniquness.

16.11. Schrödinger equation. Conservation laws and dispersive estimates. Nonlinear ODE on Banach space. Osgood's extension of Grönwall's lemma. Cauchy-Lipschitz theorem for local solution.

17.11. Blow-up criterion. Applications to nonlinear Schrödinger equation. Existence of NLS in H^s(R^d) when s>d/2.

23.11. Critical power of NLS regularity predicted by scaling. Strichartz estimates.

24.11. Local existence of cubic NLS in H^1(R^3). Global well-posedness and scattering theory.

30.11. Chapter 3: Littlewood-Paley theory. Bernstein type lemmas. Smoothing effect for heat equation.

1.12. Dyadic partition of unity. Homogeneous Besov spaces. Continuous embedding theorem for Besov spaces.

8.12. Midterm exam.

14.12. Example of the function |x|^{-a}. Convergence of series in homogeneous Besov spaces. Completeness and incompleteness of homogeneous Besov spaces. Fatou property.

15.12. Characterizations of homogeneous Besov spaces. Embedding theorems with Lebesgue spaces.

21.12. Nonhomogeneous Besov spaces. Completeness and Fatou property. Sobolev embedding theorem for Besov spaces.

18.1. 2024. Chapter 4: Incompressible Navier-Stokes equation. Weak formulation and Leray condition. Energy spaces invariant under scaling. A general fixed point theorem. Local well-posedness with H^{d/2-1} initial data.

19.1. Global existence in 2D with L^2 initial data. Global existence in 3D with small \dot{H}^{1/2} data.

25.1. Properties of global solutions in 3D. The set of globally solvable data is open in \dot{H}^{1/2} (R^3).

26.1. L^p approach. The local existence for L^3(R^3) data. The global existence for small L^3 data.

1.2, 2.2, 8.2. The endpoint space for Picard’s iteration method.

9.2. Final exam