| 1 |
Jan. 7 |
brief organizational meeting |
| 2 |
Jan. 9 |
Part I. Parabolic Equations. The heat equation: fundamental solution, mean-value property, maximum principle, uniqueness (Evans 2.3). |
| 3 |
Jan. 11 |
Heat equation: maximum principle, (non-)uniqueness on R^n, backward uniqueness (Evans 2.3, John). |
| 4 |
Jan. 14 |
Linear parabolic equations:
existence and uniqueness of weak solutions
(Evans 7.1). |
| 5 |
Jan. 16 |
Linear parabolic:
regularity of weak solutions (Evans 7.1). |
| 6 |
Jan. 18 |
Linear parabolic:
Nash-Moser theory, L^\infty bound (Taylor III 15.9). |
| 7 |
Jan. 21 |
Linear parabolic:
maximum principles, Harnack inequality (Evans 7.1). |
| 8 |
Jan. 23 |
Linear parabolic: proof of Harnack inequality in a special case
(Evans 7.1). |
| 9 |
Jan. 25 |
Linear parabolic: Nash-Moser theory,
weak Harnack inequality and Holder continuity (Taylor III 15.9).
|
| 10 |
Jan. 28 |
Linear parabolic: quick overview of semigroup
theory (Evans 7.1). |
| 11 |
Jan. 30 |
Semilinear parabolic equations: abstract local existence theorem
(Taylor III 15.1). |
| 12 |
Feb. 4 |
Semilinear parabolic: local existence for reaction-diffusion
systems in various spaces (McOwen 11.2). |
| 12 |
Feb. 6 |
Semilinear parabolic: global existence examples: harmonic map
heat-flow, reaction diffusion (Taylor III 15.1-15.2). |
| 13 |
Feb. 8 |
Semilinear parabolic: blow-up and asymptotic behaviour for some reaction-diffusion equations (McOwen 11.3). |
| 14 |
Feb. 11 |
Gradient flows: of convex functionals in Hilbert space (Evans 9.6). |
| 15 |
Feb. 13 |
Gradient flows: application to some quasilinear parabolic equations (Evans 9.6). |
| 16 | Feb. 15 |
Gradient flows: general metrics. |
| 17 | Feb. 25 |
Gradient flows: in the Wasserstein metric, and the porous medium equation (Villani 8.3). |
| 18 |
Feb. 27 |
Part II. Hyperbolic and Dispersive Equations. Wave equation: fundamental solutions and basic properties (Evans 2.4). |
| 19 |
Feb. 29 | Free Schroedinger equation: fundamental solutions and basic properties (Cazenave 2). |
| 20 |
Mar. 3 | Linear hyperbolic equations: weak solutions and regularity (Evans 7.2). |
| 21 |
Mar. 5 |
Linear hyperbolic: geometric optics, Hamilton-Jacobi equations, characteristics (Evans 7.2). |
| 22 |
Mar. 7 |
Linear hyperbolic: finite speed of propagation (Evans 7.2). |
| 23 |
Mar. 10 |
Semilinear wave equations: local existence, global existence, blow-up (McOwen 12.3; Strauss 4). |
| 24 |
Mar. 12 |
Nonlinear Schroedinger equations:
Strichartz estimates (Cazenave 2.3). |
| 25 |
Mar. 14 |
Nonlinear Schroedinger:
local existence (Cazenave 4). |
| 26 |
Mar. 17 |
Nonlinear Schroedinger:
global existence vs. blow-up (Cazenave 6). |
| 27 |
Mar. 19 |
Nonlinear Schroedinger: standing
waves and their stability (Cazenave 8). |
| 28 |
Mar. 26 |
Conservation laws: 1D scalar: characteristics,
weak solutions, R-H condition, shocks (Evans 3.4) |
| 29 |
Mar. 28 |
Conservation laws: 1D scalar: rarefaction waves,
non-uniqueness, entropy condition (Evans 3.4) |
| 30 |
Mar. 31 |
Conservation laws: 1D scalar: uniqueness of entropy solutions |
| 31 |
Apr. 2 |
Conservation laws: 1D scalar: Riemann problem, Lax-Oleinik formula, asymptotics (Evans 3.4) |
| 32 |
Apr. 4 |
Conservation laws: systems in 1D: hyperbolicity, simple waves |
| 33 |
Apr. 7 |
Conservation laws: systems in 1D: k-rarefaction waves, k-shocks |
| 34 |
Apr. 9 |
Conservation laws: systems in 1D: local solution of Riemann's problem |