Blow-up dynamics for the self-dual Chern-Simons-Schrödinger equation

Abstract

We study the blow-up dynamics for the self-dual Chern-Simons-Schrödinger equation (CSS) under equivariance symmetry. (CSS) is $L^2$-critical, has the pseudoconformal symmetry, and admits a static solution $Q$ for each equivariance index $m \geq 0$. An application of the pseudoconformal transformation to $Q$ yields an explicit finite-time blow-up solution $S(t)$ which contracts at the pseudoconformal rate $|t|$, with zero asymptotic profile. In the case of higher equivariance indices $m \geq 1$, we first construct pseudoconformal blow-up solutions $u$ with nonzero asymptotic profiles (thus $u \neq S(t)$ necessarily), and moreover exhibit an instability mechanism, the rotational instability, of such solutions. As complementary to this result, we show that pseudoconformal blow-up solutions can arise from a codimension one manifold of initial data. These results are based on the joint works with Soonsik Kwon [40, 41] and occupy Chapters 2 and 3. In Chapter 4, we consider the most physically relevant, but the most delicate radial case $m=0$. In this regime, $S(t)$ is no longer a finite energy blow-up solution. Interestingly enough, there are smooth finite energy blow-up solutions whose blow-up rates differ from the pseudoconformal rate by a power of logarithm. This result is based on the joint work [42] with S. Kwon and Sung-Jin Oh. We obtain all these results via modulation analysis.