Math 251B (Topics in Dispersive PDE).

Instructor: Monica Visan, MS 6167. Email address: visan@math.ucla.edu

Office Hours: by appointment.

Topics:

• Linear wave equation: fundamental solution, monotonicity formulas, dispersive and Strichartz estimates.
• 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.

• [1] 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, 1235-1293. MR2018661
• [2] L. Hormander, Lectures on nonlinear hyperbolic differential equations. Mathematiques & Applications (Berlin) [Mathematics & Applications], 26. Springer-Verlag, Berlin, 1997. MR1466700
• [3] C. Sogge, Lectures on non-linear wave equations. Second edition. International Press, Boston, MA, 2008.
• [4] T. Tao, Nonlinear dispersive equations. Local and global analysis. CBMS Regional Conference Series in 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 [4].
• (2) Global wellposedness for the energy-critical NLW: non-concentration of potential energy. Proposition 5.9 in [4].
• (3) An energy estimate for a quasilinear equation: Proposition 2.1 and Theorem 3.1 in [3]. See also Proposition 6.3.2 and the proof of (6.3.18') in [2].
• (4) Existence and uniqueness for linear equations: page 105 in [2]. See also Theorem 3.2 in [3].
• (5) Local existence for quasilinear equations: Theorems 4.1 and 4.3 in [3]. 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 [3]. 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 [2].
• (8) Global existence for quasilinear equations in dimensions d>3: Theorem 6.5.2 in [2]. If time permits, discuss also Theorem 6.5.3 in [2].