Fourier Analysis applied to Partial Differential Equations

Subscription
Subscription to the mailing list is also possible by sending an email with subject "subscribe" and empty email body to lecture-salguero-s26-join@mis.mpg.de

---

Next lectures
28.04.2026, 10:00 (E2 10 (Leon-Lichtenstein))
05.05.2026, 10:00 (E2 10 (Leon-Lichtenstein))
12.05.2026, 10:00 (E2 10 (Leon-Lichtenstein))
19.05.2026, 10:00 (E2 10 (Leon-Lichtenstein))
26.05.2026, 10:00 (E2 10 (Leon-Lichtenstein))
02.06.2026, 10:00 (E2 10 (Leon-Lichtenstein))
09.06.2026, 10:00 (E2 10 (Leon-Lichtenstein))
16.06.2026, 10:00 (E2 10 (Leon-Lichtenstein))
23.06.2026, 10:00 (E2 10 (Leon-Lichtenstein))
30.06.2026, 10:00 (E2 10 (Leon-Lichtenstein))
07.07.2026, 10:00 (E2 10 (Leon-Lichtenstein))

---

Fourier analysis has been a key tool in the study of partial differential equations since its first use in the heat equation. In the 1930s, Littlewood and Paley developed a powerful theory to decompose functions into spectrally localized pieces, which proved to be very effective in handling differential operators. But it was not until the early 1980s, with the pioneering work of J. M. Bony, that the Littlewood-Paley decomposition gave rise to a systematic framework for tackling nonlinear differential equations: paradifferential calculus. Since then, this machinery has proven highly successful in the analysis of nonlinear PDEs. In these lectures, we will introduce the Littlewood-Paley decomposition, Besov spaces, and the paraproduct decomposition (the essential ingredient in paradifferential calculus). Furthermore, we will discuss some applications of this theory to classical models in fluid dynamics.

Keywords
Fourier analysis, Littlewood-Paley theory, Besov spaces, paradifferential calculus, PDE

Prerequisites
Basic background in PDE and functional analysis is needed