2026 Spring PDE2 (Modern PDE) Course Webpage

English Version|中文版

Basic Info

Instructor: Junyan ZHANG, yx3x@ustc.edu.cn

Teaching assistants: Yuchen YIN, yuchenyin@mail.ustc.edu.cn; Yilu LIU, lystcl@163.com.

Time & Location: 5402@East campus, Week 1~16, 2(3,4,5), 4(6,7).

Prerequisites
    Classical PDEs (PDE1), Real analysis, Functional analysis. Specifically, students taking this course are expected to possess the following prerequisites:
  • Real analysis: Lebesgue theory of measure and integral (Convergence theorems, Lebesgue differentiation theorems), L^p spaces.
  • Linear functional analysis: Riesz representation theorem for Hilbert spaces, weak/weak-* convergence, spectrum of compact operators.
  • It would be better if you've learned Ch. 6, 8, 9 of Folland's real analysis book (L^p interpolation, Fourier transform, Distributions).
Course contents

Textbook: My PDE2-Lecture Notes (in Chinese). Last updated: 04/03/2026. Lastest errata: Errata to 20260316 version.

    Course contents: 60% (Chapter 1~3 of lecture notes) are selected from Chapter 5~8 in Evans' PDE book. The remaining 40% (Chapter 5~7 of lecture notes) covers Fourier charaterization of Sobolev spaces, Scrodinger and wave equations.
  • 1. Sobolev space W^{k,p} ([1, Ch 5])
  • 2. Elliptic equations: Weak solution and max principle for linear elliptic equations ([1, Ch 6]), Eigenvalue problems, constrained calculus of variations and Mountain-Pass ([1, Ch 6.5, 8.4, 8.5]), *De Giorgi-Nash-Moser iteration ([6, Ch 4.1]), Pohozaev identity ([1, Ch 9.4]).
  • 3. Parabolic equations: Weak solution and max principle, vanishing viscosity method ([1, Ch 7.1, 7.3]).
  • 4. Fourier charaterization of Sobolev spaces:([3, Appendix A], [4, Ch 1]).
  • 5. Schrodinger equations: decay and Strichartz estimates, small-data GWP and scattering for mass-critical NLS, Virial identity. ([5, Ch 1]).
  • 6. Wave equations: LWP of linear and quasilinear waves, blow-up criteria, GWP of cubic NLW in 3D. ([2, Ch 4~7]).
  • 7. *Noether's theorem ([1, Ch 8.6]).

Part of manual solutions (in Chinese) to Evans PDE book (Ch 5-8, 12) by myself (written in 2017~2018) can be foundhere.

Exams and Grades

Your total score = max{30%HW + 20%Midterm +50%Final, 30%HW + 70%Final. 100%Final}.

Midterm exam: 9:50~12:00, 04/21/2026@5402 east campus. Midterm covers Ch 1.1~2.6 of our lecture notes.

Final exam will take place in week 16 or the beginning of week 17 (no later than 06/23/2026).

HW and Tutorials

HW: NO COPYING, otherwise you'll receive 0 score.

    Rules for HW:
  • HWs are published on this webpage and QQ group.
  • Late HW will be deducted 25%-40% scores.
  • Hand-written HW, scanned PDF and TeX-complied/Markdown HW are accepted.

Expected Schedules



Week Date Course Contents Homework Due
  Week 1     3.3     Weak derivatives and Sobolev spaces
  Smooth approximations of Sobolev functions  		   Smooth approximation in \R^d     HW1, TeX code.
  Due on 03/24/2026.
  HW1-Solution  
  3.5     Smooth approximations of Sobolev functions
  Basic calculus of Sobolev functions
  Trace Theorem  		   [1.2] 3, 4, 6; [1,3] 3 (bonus)  
  Week 2     3.10     Zero-Trace Theorem, Extension (proof skipped)
  Sobolev embeddings (GNS inequality) and compact embeddings  		   [1.4] Q4, Q5, Q1(bonus)  
  3.12     Poincare inequalities, Morrey's embedding     [1.4] 3, 7  
  Week 3     3.17     Lipschitz continuity, H^{-1} space     [1.5] 1  
  3.19     Existence theorems of linear elliptic equations     [2.2] 2, 3, Q1; [2.3] 1     HW2, TeX code.
  Due on 04/14/2026.    
  Week 4     3.24     Eigenvalue theory of symmetric elliptic operators
  Lyusternik constraint calculus of variation (not required)  		   [2.4] 3(bonus)  
  3.26     Mountain-Pass and its applications
  (Existence of ground state of mass-critical NLS, not required)  		   None  
  Week 5     3.31     Elliptic regularity theorem, Tutorial 1     None  
  4.2     Weak maximum principles, Bernstein's technique, Harnack inequality     [2.6]1, 2, Q2  
  Week 6     4.7     Hopf lemma, strong maximum principle
  *De Giorgi-Nash-Moser iteration (not required)  		   None  
  4.9     *De Giorgi-Nash-Moser iteration, Pohozaev identity     None     HW3, TeX code.
  Due on 05/07/2026.  
  Week 7     4.14     Pohozaev identity, *Noether Theorem     [2.8] 1, 2  
  4.16     Tutorial 2, Space-time Sobolev spaces.     None  
  Week 8     4.21     Midterm exam     None  
  4.23     Existence of linear parabolic equation (Galerkin's method)
  Parabolic maximum principle  		   [3.2] 1, 2  
  Week 9     4.28     Parabolic maximum principle, Vanishing viscosity method     [3.4] Q1, [3.5] Q1  
  4.30     L^p interpoaltion, Fourier transform         HW4
  Due on 05/26/2026.  
  Week 10     5.5     Holiday of Labor Day     None  
  5.7     Tempered distributions      
  Week 11     5.12     Fourier charaterization of H^s(\R^d), Tutorial 3      
  5.14     Refined Sobolev inequalities and trace theorems      
  Week 12     5.19     Decay and Strichartz estimates of Schrodinger equations
  Mass-critical NLS  		    
  5.21     Mass-critical NLS, Virial identities         HW5
  Due on 06/09/2026.  
  Week 13     5.26     Existence, regularity and finite propagation speed of linear wave equations      
  5.28     LWP of quasilinear wave equations      
  Week 14     6.2     Examples of blowup and GWP of quasilinear waves, blow-up criterions
  Tutorial 4  		    
  6.4     GWP of cubic NLW in 3D      
  Week 15     6.9     Introduction to calculus of variation     None  
  6.11     Introduction to calculus of variation     None  
  Week 16     6.16     Final review, Tutorial 5     None  
  6.18     Q&A     None