MAP6437 (1) Advanced Partial Differential Equations

• Instructor: Dr. Xiaoming Wang
  
• Office: 312 LOV
  
• Phone: 644-6419
  
• Hour: Tue.Thur. 9:30-10:45AM
  
• Place: Lov 104
  
• Office Hours: Tue. 2:00-2:50PM and Thur. 10:50-11:40AM and/or by appointment
• Textbook: Partial Differential Equations. by L.C. Evans published by American Mathematical Society . See also Supplementary Textbook
• Prereq: Advanced calculus, Elementary PDEs, or authorization by the instructor
• In case you have any questions, please do not hesitate to contact me via phone at 644-6419 via e-mail at wxm@math.fsu.edu or drop by my office at 312 J. Love

Course Topics (I&II)

• General introduction to PDE, examples and issues (I)
• Review of ODE theory, existence , uniqueness and dependence on data (I)
• Nonlinear first order PDE (including conservation laws and Hamilton-Jacobi equations) and the method of characteristics (I)
• Fourier transform method (I)
• Sobolev spaces: weak derivatives, extensions and traces, Sobolev inequalities, imbeddings (I)
• Second order elliptic equations: weak solutions, Lax-Milgram theorem, energy estimates, Fredholm alternative, regularities, maximum principle, eigenvalue and eigenfunctions (I)
• Second order parabolic equations: existence, uniqueness, regularity, maximum principle (II)
• Second order hyperbolic equations: existence, uniqueness, speed of propagation (II)
• Introduction to semi-group theory (II)
• Introduction to the calculus of variation (II)
• Introduction to non-variational techniques (II)
• Introduction to Hamilton-Jacobi Equations (II)
• Introduction to Navier-Stokes systems (II)
• Introduction to hyperbolic systems (II)
• Introduction to kinetic equations (II)
• Selected topics if time permits.

Supplementary Textbooks

• Partial Differential Equations by Fritz John, published by Springer-Verlag New York.
• Partial Differential Equations: Methods and Applications by Robert C. McOwen, published by Prentice Hall.
• Second Order Parabolic Partial Differential Equations by Gary Lieberman, published by World Scientific, 1996.
• Methods of Mathematical Physics by Richard Courant and David Hilbert, published by Wiley-Interscience, 1962
• Quelques Methodes de Resolution des Problemes aux Limits non Lineares , by Jacques Louis Lions, published by Dunod, 1969.
• Navier-Stokes equations and Nonlinear Functional Analysis by Roger Temam, Published by SIAM.
• Partial Differential Equations by Jeffrey Rauch, Published by Springer-Verlag New York.
• Systems of Conservation Laws I & II by Denis Serre, published by Cambridge University Press
• Nonlinear Dynamics and Statistical Theories for Basic Geophysical Flows by Andrew Majda and Xiaoming Wang, published by Cambridge University Press, 2006
• Elliptic problems in nonsmooth domains by P. Grisvard, published by Pitman, 1985

Required Work and Grading Policy

• 20% Classroom participation.
• 80% Problem Sets (about 4).

Homework Assignments and Announcements

• Advanced PDE1 HW1. Due: Thur. Sept. 24, 2009
  • Prove the following Gronwall type inequality: suppose
    dy/dt ≤ f(t)y(t)+g(t), for t ≥ t₀, then
    y(t) ≤ y(t₀)exp(∫_t0^t f(s)ds) +∫_t0^t exp(∫_s^t f(τ)dτ)g(s)ds.
  • Prove that the solution of a given ode dx/dt = f(t,x), x(t₀)=x₀ depends on the initial data in a continuous fashion. Here we asume that f is continuous in (t, x) and Lipschitz continuous in x.
  • Solve the following first order linear PDE: u_t+x²u_x = u, u(0,x)=u₀(x). Describe the region where the solution exists.
  • Let Φ be the fundamental solution to the Laplace operator on Rⁿ and let f be a twice continuously differentiable function with bounded support. Show that u(x)= ∫_Rⁿf(y)Φ(x-y)dy solves the Poisson equation -Δ u = f.
• Advanced PDE1 HW2. Due: Thur. Oct. 22, 2009
  • Consider the oblique derivative problem on the upper half plane R²₊:
    -Δu+βu=f in R²₊, u_y+αu_x=g on y=0
    where β>0, α are constants. Show that there exists at most one twice continuously differentiable solution that vanishes at infinity. (Hint: try the abc method, i.e., multiply the equation by a linear combination of u and its first derivatives and integrate over the domain.)
  • Problems 4, 5, 10, 11, 12, 13 on pages 86-88 from the textbook.
  • (Optional, F2009. This is related to the so-called coupled continuum pipe flow model for flow is karst aquifer.) Consider the following type of equation on the unit disk -Δu(x) + u(0)δ(0) = f(x); where δ(0) is the Dirac delta function centered at 0. Assume f is radially symmetric. Discuss if there is a solution (separate the one dimensional and two dimensional case).
• Advanced PDE1 HW3. Due: Wed. final week (extra credit for solving the 1, 3,4 optional problem)
  • (optional) Show that the wave equation on a bounded smooth domain U with homogeneous boundary condition is structurally stable in the sense that the solution u of the following IBVP depends continuously on the wave speed c.
    u_tt-c²Δu = f, u = 0 on ∂U, u(x,0)=g(x), u_t(x,0)=h(x).
    You may assume that f, g, h are smooth and satisfies appropriate compatibility conditions.
  • Problems 17, 18 on pages 88-89.
  • (optional) Let u be the unique entropy solution to the scalar hyperbolic conservation law
    u_t +f(u)_x=0, u(x,0)=g(x).
    Define v(x,t)=u(x,t+t₀), t₀>0. Show that v is also an entropy solution with initial data v(x,0)=u(x, t₀).
  • (optional) Show that time is reversable for smooth solutions to hypebolic conservations laws. Are conservation laws always time reversable? Why?
  • (optional) Show that smooth solutions to conservation law enjoy infinitely many conserved quantities. Are conservations laws conservative all the time? Why?
  • Find the entropy solution to the Burgers type equation u_t + u³u_x=0 subject to the initial condition u(x, 0)=0, for x<0 and u(x, 0)=1 for x>0. (Hint: try self-similar solution for the rarefaction wave part of the solution)
  • Problems 3, 13 (assume piecewise smooth and continuous solution), 14 on pages 163-165.
• APDE1 HW4. Due: TBA
  • pp. 290-292, #6, 7, 8, 13, 17, 18
  • (scaling and trace, optional) Show via scaling/dimensional/unit argument that the trace of   H^s(Rⁿ)  function on   x_n=0   should belong to   H^s-1/2(R^n-1) :.
  • (even extension, optional) Show that if a∈ H¹(B+) where B+ denotes the upper unit half ball, and if ã denotes the even extension of a to the whole unit ball B, then ã∈ H¹(B).
• APDE2 HW3. Due: Tue. April 28, 2009
  • pp. 345-347, #1, 2,3,
• APDE2 HW4 Due: TBA
  • pp. 425-426, #1,2,6,7,
  • (stabilizing effect of diffusion): 1. show that the origin is unstable for the ODE u_t+u⁵-u³=0; 2. show that the origin is asymptotically stable for the PDE u_t-ε u_xx+u⁵-u³=0 with the homogeneous Dirichlet boundary condition u(0,t)=u(1,t)=0 for all positive t, where ε is an arbitrary positive number. (Hint: use energy method to show that solutions with small enough initial data all converge to zero). What can you say about the equation u_t-ε u_xx+u³-u=0 ?
• (projection and trace) Suppose we have a nice smooth domain Ω in Rⁿ and a smooth surface Γ in Ω and let f and g be two smooth function on Ω that agrees with each other on Γ. Let P_m be an orthogonal projection in H¹(Ω) onto the space spanned by the first m functions of a smooth ONB. Is it necessary that P_mf agrees with P_mg on Γ?
• (average and trace) For u ∈ H¹(Rⁿ+), we have lim_ε→0∫_0≤xn≤ε u(x) dx / ε = u(x', 0) with the right hand side understood in the sense of trace.
• Show that the following trace type inequality is true. for u ∈ H¹(Rⁿ+), we have |u|_L²(R^n-1) ≤ c |u|^1/2_L²(Rⁿ+) |u|^1/2_H¹(Rⁿ+) .
• (periodic extension) Suppose that u∈ H²(0,1), and u(0)=u(1)=0. Let u_ext be the odd extension of u to (-1, 1). Show that u_ext∈ H²_per(-1, 1). Similarly, suppose that u∈ H²(0,1), and u'(0)=u'(1)=0. Let u_ext be the even extension of u to (-1, 1). Show that u_ext∈ H²_per(-1, 1).
• (Hardy's inequality) Show that for f ∈ C¹[0,1] and f(0)=0, we have ∫_I f(x)²/x²  dx ≤ 4 ∫_I f'(x)²  dx where I=[0,1].
• (Heaviside function and differentialbility) Show that the Heaviside function H(x)=1 for x≥0, H(x)=0 for x<0 on [-1,1] is an element in H^s_per([-1, 1]) for s &isin (0, 0.5).
  

  
  

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