Math 251B (Topics in Dispersive PDE).
Lectures: MWF 11:00-11:50am in Geology 6704.
Instructor: Monica Visan, MS 6167. Email address: firstname.lastname@example.org
Office Hours: by appointment.
- Linear wave equation: fundamental solution, monotonicity formulas, dispersive and Strichartz
- Semilinear wave equations: sharp local wellposedness, wellposedness in the energy space for
energy-subcritical and energy-critical nonlinearities, global wellposedness for the defocusing
energy-subcritical and energy-critical NLW.
- Illposedness results for focusing and defocusing NLW.
- Quasilinear wave equations: the null condition.
-  M. Christ, J. Colliander, T. Tao, Asymptotics, frequency modulation, and low
regularity ill-posedness for canonical defocusing equations. Amer. J. Math. 125 (2003), no. 6,
-  L. Hormander, Lectures on nonlinear hyperbolic differential equations. Mathematiques &
(Berlin) [Mathematics & Applications], 26. Springer-Verlag, Berlin, 1997. MR1466700
-  C. Sogge, Lectures on non-linear wave equations. Second edition. International Press, Boston, MA,
-  T. Tao, Nonlinear dispersive equations. Local and global analysis. CBMS Regional Conference
Mathematics, 106. Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society,
Providence, RI, 2006.
Grading: Students will give one-hour presentations. Here is a list of possible topics:
- (1) Global wellposedness for the energy-critical NLW: blowup implies potential energy concentration.
Propositions 5.3-5.6 in .
- (2) Global wellposedness for the energy-critical NLW: non-concentration of potential energy.
Proposition 5.9 in .
- (3) An energy estimate for a quasilinear equation: Proposition 2.1 and Theorem 3.1 in . See also
Proposition 6.3.2 and the proof of (6.3.18') in .
- (4) Existence and uniqueness for linear equations: page 105 in . See also Theorem 3.2 in .
- (5) Local existence for quasilinear equations: Theorems 4.1 and 4.3 in . Set up the iteration
scheme for the existence result in Theorem 4.1, prove Lemma 4.2, and conclude with (4.10).
- (6) Local existence for quasilinear equations: Theorems 4.1 and 4.3 in . Using the results
presented in (5), prove the convergence of the iteration scheme, the uniqueness of solutions and the
blowup criterion. If time allows, prove Theorem 4.3.
- (7) Klainerman-Sobolev inequality: Proposition 6.5.1 in .
- (8) Global existence for quasilinear equations in dimensions d>3: Theorem 6.5.2 in . If time
permits, discuss also Theorem 6.5.3 in .