Introduction to Nonlinear Evolutionary PDEs
This class will be an in-person class at Shuangqing C-654.
The focus of this course will be on evolution equations of both Hamiltonian (conservative, such as wave, Klein-Gordon, or Schroedinger) and dissipative kinds (heat equation)on $\mathbb{R}^{d},d\geq 2$, but the main focus will essentially be $d=3$. Our goal is to provide an introduction to the \textbf{concentration compactness} technique (or profile decomposition to capture the defect of compactness), which is a core method in modern large data theory. To develop intuition, we first cover the simpler elliptic analogue, which addresses the loss of compactness due to the action of symmetry groups. We begin with a self-contained discussion of wave and Klein-Gordon equations and carefully cover some basic aspects leading to problems of Fourier analysis, such as Sobolev embeddings and Strichartz estimates. A prior course in PDE is not required, but basic knowledge of ODEs and functional analysis is helpful. Advanced undergraduate students and graduate students looking to get into the PDE research should find this introduction class helpful. Postdocs are encouraged, too, but may find this class elementary.
The focus of this course will be on evolution equations of both Hamiltonian (conservative, such as wave, Klein-Gordon, or Schroedinger) and dissipative kinds (heat equation)on $\mathbb{R}^{d},d\geq 2$, but the main focus will essentially be $d=3$. Our goal is to provide an introduction to the \textbf{concentration compactness} technique (or profile decomposition to capture the defect of compactness), which is a core method in modern large data theory. To develop intuition, we first cover the simpler elliptic analogue, which addresses the loss of compactness due to the action of symmetry groups. We begin with a self-contained discussion of wave and Klein-Gordon equations and carefully cover some basic aspects leading to problems of Fourier analysis, such as Sobolev embeddings and Strichartz estimates. A prior course in PDE is not required, but basic knowledge of ODEs and functional analysis is helpful. Advanced undergraduate students and graduate students looking to get into the PDE research should find this introduction class helpful. Postdocs are encouraged, too, but may find this class elementary.
Lecturer
Puskar Mondal
Date
16th September ~ 16th December, 2025
Location
Weekday | Time | Venue | Online | ID | Password |
---|---|---|---|---|---|
Tuesday,Friday | 13:30 - 15:05 | Shuangqing-C654 | ZOOM 14 | 712 322 9571 | BIMSA |
Prerequisite
Basic ODE and functional analysis
Syllabus
\section{Tentative Syllabus}
ODE and Dynamical systems, Lyapunov function, Hartman-Grobman theorem, Stable-Unstable manifolds. Nonlinear wave and Klein-Gordon equations: Cauchy problem, energy method, long-time behavior, basic Fourier analysis such as Littlewood-Paley theory, Strichartz estimates, Introduction to Concentration-Compactness technique for dispersive PDEs. If time permits, we will see how this concentration-compactness method can be applied to the equations of advection-diffusion type, e.g., the incompressible 3-d Navier-Stokes equation, and prove an interesting result: mild solutions which remain bounded in the space $\dot{H}^{\frac{1}{2}}$ (critical space) do not become singular in finite time.
ODE and Dynamical systems, Lyapunov function, Hartman-Grobman theorem, Stable-Unstable manifolds. Nonlinear wave and Klein-Gordon equations: Cauchy problem, energy method, long-time behavior, basic Fourier analysis such as Littlewood-Paley theory, Strichartz estimates, Introduction to Concentration-Compactness technique for dispersive PDEs. If time permits, we will see how this concentration-compactness method can be applied to the equations of advection-diffusion type, e.g., the incompressible 3-d Navier-Stokes equation, and prove an interesting result: mild solutions which remain bounded in the space $\dot{H}^{\frac{1}{2}}$ (critical space) do not become singular in finite time.
Reference
Notes are to be provided, as we will not follow any particular textbook (otherwise, what's the point in teaching?). But the following book and a couple of articles should be good friends.
C. d. Sogge, Lectures on Non-linear Wave equations
CONCENTRATION COMPACTNESS FOR CRITICAL WAVE MAPS, Joachim Kreiger, Wilhelm Schlag, European Mathematical Society,
Carlos E. Kenig, Frank Merle, Global well-posedness, scattering and blow-up for the energy-critical focusing non-linear wave equation, Acta
Math. 201 (2) (2008) 147–212.
C. d. Sogge, Lectures on Non-linear Wave equations
CONCENTRATION COMPACTNESS FOR CRITICAL WAVE MAPS, Joachim Kreiger, Wilhelm Schlag, European Mathematical Society,
Carlos E. Kenig, Frank Merle, Global well-posedness, scattering and blow-up for the energy-critical focusing non-linear wave equation, Acta
Math. 201 (2) (2008) 147–212.
Audience
Advanced Undergraduate
, Graduate
, Postdoc
Video Public
Yes
Notes Public
Yes
Language
English
Lecturer Intro
Puskar Mondal is currently a postdoctoral fellow at CMSA (dept. of Mathematics) and a lecturer at the Math department at Harvard University. His mentor is Prof. Shing-Tung Yau. Before this, he was a Ph.D. student at Yale university where he worked on Mathematical General Relativity under the supervision of Prof. Vincent Moncrief.