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: 05/01/2026. Lastest errata: Errata to 20260403 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 my lecture notes.
Midterm exam score: average 75.15, median 76, highest 107, highest bonus score 10. Please click here to download the solution (in Chinese).

Final exam: 9:50~12:10, 06/16/2026@5402 east campus. Final exam covers Ch 2.8~end of this course.

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.
  HW2-Solution  
  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 (courtesy to Yilu LIU), Space-time Sobolev spaces.     None  
  Week 8     4.21     Midterm exam (09:50~12:00)     None  
  4.23     Existence of linear parabolic equation (Galerkin's method)     [3.2] 1, 2  
  Week 9     4.28     Parabolic regularity (proof skipped)
  Parabolic maximum principle, Vanishing viscosity method  		   [3.4] Q1  
  4.30     Vanishing viscosity method: an introduction to boundary layer     [3.5] Q1  
  Week 10     5.5     Holiday of Labor Day     None  
  5.7     Fourier transform, Tempered distributions         HW4
  Due on 05/26/2026.  
  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      
  Week 13     5.26     Existence, regularity and finite propagation speed of linear wave equations         HW5
  Due on 06/09/2026.  
  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     Final review.   Deadline for late HW: 06/11/2026.     None  
  Week 16     6.16     Final Exam (Must attend!)     None  
  6.18     No Class     None