Associate Professor @ Korea Advanced Institute of
Science and Technology, Daejeon, Korea
Professor Kwon received his Ph.D. at UCLA in
2008. Then he worked at Princeton University as an
instructor in 20082010. Since 2010, he is appointed as
Associate Professor at the Korea Advanced Institute of
Science and Technology, Daejeon, Korea.

Abstract
In this talk, I present a blow up construction  jointly with Kihyun Kim 
on the selfdual ChernSimonsSchrödinger equation (CSS), also
known as a gauged nonlinear Schrödinger equation (NLS). CSS is
$L^{2}$critical, admits solitons, and has the psuedoconformal symmetry.
These features are similar to the $L^{2}$critical NLS. In this work,
we consider pseudoconformal blowup solutions under $m$equivariance,
$m\geq1$. Our result is threefold. Firstly, we construct a pseudoconformal
blowup solution $u$ with given asymptotic profile $z^{\ast}$:
\[
\Big[u(t,r)\frac{1}{t}Q\Big(\frac{r}{t}\Big)e^{i\frac{r^{2}}{4t}}\Big]
e^{im\theta}\to z^{\ast}\qquad\text{in }H^{1}
\]
as $t\to0^{}$, where $Q(r)e^{im\theta}$ is a static solution. Secondly,
we show that such blowup solutions are unique in a suitable class.
Lastly, yet most importantly, we exhibit an instability mechanism
of $u$. We construct a continuous family of solutions $u^{(\eta)}$,
$0\leq\eta\ll1$, such that $u^{(0)}=u$ and for $\eta>0$, $u^{(\eta)}$
is a global scattering solution. Moreover, we exhibit a rotational
instability as $\eta\to0^{+}$: $u^{(\eta)}$ takes an abrupt spatial
rotation by the angle
\[
\Big(\frac{m+1}{m}\Big)\pi
\]
on the time interval $t\lesssim\eta$.
We are inspired by works in the $L^{2}$critical NLS. In the seminal
work of Bourgain and Wang (1997), they constructed such pseudoconformal
blowup solutions. Merle, Raphaël, and Szeftel (2013) showed an
instability of BourgainWang solutions. Although CSS shares many features
with NLS, there are essential differences and obstacles over NLS.
Firstly, the soliton profile to CSS shows a slow polynomial decay
$r^{(m+2)}$. This causes many technical issues for small $m$. Secondly,
due to the nonlocal nonlinearities, there are strong longrange interactions
even between functions in far different scales. This leads to a nontrivial
correction of our blowup ansatz. Lastly, the instability
mechanism of CSS is completely different from that of NLS. Here, the phase rotation
is the main source of the instability. On the other hand, the selfdual
structure of CSS is our sponsor to overcome these obstacles. We exploited
the selfduality in many places such as the linearization, spectral
properties, and construction of modified profiles.
In the talks, I will present background of the problem, main theorems,
and outline of the proof with emphasis on heuristics of main features,
such as the long range interaction between blow up profile and asymptotic
profile $z$, and rotational instability mechanism.
