By "Boussinesq type systems" we mean systems that are bidirectional versions of well known scalar nonlinear dispersive equations such as the Korteweg-de Vries or Benjamin-Ono equations. The first example of such systems was derived by Boussinesq (1877). They offer a variety of new and challenging problems. We will survey the known results and many open questions.

The Cauchy problem for compressible Euler system from inviscid limit of Navier-Stokes remains completely open, as a challenging issue in fluid dynamics. In this talk, I will give a first resolution for this problem in the 1D isentropic case. We will show the global well-posedness of entropy solutions with small BV initial data in the class of inviscid limits from the associate Navier-Stokes. More precisely, any small BV entropy solutions are inviscid limits from Navier-Stokes. Those are unique and stable among inviscid limits from Navier-Stokes. The proof is based on the three main methodologies: the modified front tracking algorithm; the a-contraction with shifts; the method of compensated compactness. This is a joint work with Geng Chen (U. Kansas) and Alexis Vasseur (UT-Austin).

The self-dual Chern-Simons-Schrödinger (CSS) equation is viewed as a gauged version of the 2D cubic nonlinear Schrödinger (NLS) equation. It admits static solutions and explicit blow-up solutions via the pseudoconformal transformation. I will discuss recent progress on its blow-up dynamics, from constructions, instability mechanism, and rigidity of blow-up rate. I will highlight remarkable features of (CSS) dynamics, which differ from related models like 2D cubic NLS, Schrödinger maps, wave maps, and others. This talk is based on joint works with Kihyun Kim and Sung-Jin Oh.

We consider the incompressible Euler equations and related PDEs in scaling critical Sobolev spaces, which are also critical for local well-posedness. We show various ill/well-posedness results for the initial value problem at critical regularity. Then, we discuss some applications of understanding critical dynamics, including singularity formation and enhanced dissipation for the dissipative counterparts. 

We consider axisymmetric incompressible inviscid flows without swirl. When the axial vorticity is non-positive in the upper half space and odd in the last coordinate, we call the flow anti-parallel and we may expect a head-on collision of anti-parallel vortex rings. By establishing monotonicity and infinite growth of the vorticity impulse on the upper half-space,  we obtain infinite growth of vorticity maximum at infinite time for certain classical vorticity. On the other hand, a finite but faster growth for some smooth vorticity is obtained thanks to the stability of Hill's vortex. This talk is based on joint work with In-Jee Jeong(SNU). 

According to the Fujita-Kato principle, the Helmholtz decomposition is one of the key ingredients needed to study the solvability of the incompressible Navier-Stokes equations in a functional analytic approach. The Helmholtz decomposition in the Lp-setting was well-studied for 1 < p < ∞. However, it is not suitable to investigate this issue for the L∞ vector fields. In this talk, we will present the Helmholtz decomposition of BMO vector fields in a domain, as a substitute theory for the L∞-setting. This talk is based on a series of joint works with Professor Yoshikazu Giga (The University of Tokyo).