TBA

We discuss local regularity of solutions for Stokes system near boundary in three and higher dimensions. We construct a weak solution whose normal derivatives are unbounded near boundary for the Stokes system in a half space. Similar construction can be made for the Navier-Stokes equations via the method of perturbation as well.

We consider the equations of a non-Newtonian incompressible fluid in  a general  {time-space} cylinder $Q_{T}= \Omega \times (0,T)  \subset \R^{n} \times \R, n \ge 2$. We assume that the rheology of the fluid is changing with respect to time and space and satisfies, for each $(x,t) \in Q_{T}$, the associated  power law $ |\bD|^{p(x,t) } \bD $ {of the symmetric gradient $\bD$ of $u$}.

Under the assumption that $\frac{2n}{n+2} < p_{0} \le p(x,t) \le p_{1} < +\infty$ and the set of discontinuity of $p$ is closed and of measure zero, { where $p_0,p_1$ are constants, } we show the existence of a weak solution to the corresponding equations of PDEs for any given  initial velocity in $L^{2}_{\sigma } (\Omega ) $.

We propose the infinity wave equation which can be derived from the exponential wave equation through the limit $p \to \infty$.

The solution of  infinity Laplacian equation can be considered as a static solution of the  infinity wave equation.

We present basic observations and find some special solutions.  

We consider the quasi-geostrophic equation with its principal part $(-\Delta)^{\alpha}$ for $\alpha >0$ in ${\mathbb R}^n$ with $n \ge 2$. We show that for every initial data $\theta_0 \in \dot B^{1-2\alpha + \frac{n}{r}}_{r, q}$ with $1< r < \infty$ and $1 \le q \le \infty$, there exists a unique solution $\theta$ in the class of maximal Lorentz-Besov regularity theorem $\partial_t\theta, (-\Delta)^\alpha \theta \in L^{\gamma, q}(0, T; \dot B^s_{p, 1})$ for $2\alpha/\gamma + n/p -s =4\alpha -1$ with $n/p \le n/r < 2\alpha/\gamma + n/p$ and $s>\max\{-1, 1-4\alpha + n/r\}$, where $0 < T \le \infty$. If $\theta_0$ is sufficiently small,then we may take $T=\infty$. Notice that both classes of initial data and solutions are scaling invariant. This is the joint work with Peer C. Kunstmann(Karlsruhe) and Senjo Shimizu(Kyoto).

The logarithmic vortex is a solution to the incompressible inviscid fluid equations which lie on the borderline of well-posedness theory. For this reason, dynamics near the logarithmic vortex may show various phenomena which are impossible for regular solutions. We derive the evolution equation for the bounded part of the vorticity advected by the logarithmic vortex and discuss its long-time dynamics. 

In this talk, we introduce two models proposed by Lust and Moffatt. We first discuss a model proposed by L\"ust who appears the first one to correct some generalized Ohm's law to guarantee the energy conservation. We then provide some open problems to Moffatt's magnetic relaxation equations.

We study  a Stokes-Magneto system  in $\mathbb{R}^d$ ($d\geq 2)$ with fractional diffusions   $\Lambda^{2\alpha}\mathbf{u}$ and $\Lambda^{2\beta} \mathbf{b}$ for the velocity $\mathbf{u}$ and the magnetic field $\mathbf{b}$, respectively. Here $\alpha,\beta$ are positive constants and $\Lambda^s = (-\Delta)^{s/2}$ is the fractional Laplacian of order $s\in \mathbb{R}$. Global existence of weak solutions is shown  for  initial data in $L_2 (\mathbb{R}^d )$ when $\alpha$, $\beta$ satisfy $1/2<\alpha<(d+1)/2$, $\beta >0$, and $\min\{\alpha+\beta,2\alpha+\beta-1\}>d/2$. Moreover, weak solutions are   unique  if $\beta \geq 1$ and $\min \{\alpha+\beta,2\alpha+\beta-1\}\geq d/2+1$, in addition. This talk  is based on a paper with Hyunwoo Kwon at Brown University.

The governing set of fluid dynamics equations are derived to simplify the complexity of fluid flow systems. In fluid dynamics, this comes in the form of the Euler equations and the Navier-Stokes equations. Also fluid motions have been playing a very important role in the study of meteorology and astrophysics--Boussinesq equations and magnetohydrodynamics equations, respectively.

In this talk, I will give a brief review of Liouville type Theorems of the Navier-Stokes equations-mostly done by Professor Dongho Chae and give a brief comment for the results. Also I will give a brief review of regularity or singularity studies of the incompressible fluids-mostly done by Professor Dongho Chae.

It is proved by the method of partial fraction expansions and Sturm's oscillation theory that the zeros of certain Hankel transforms are all real and distributed regularly between consecutive zeros of Bessel functions. As an application, the sufficient or necessary conditions on parameters for which ${}_1F_2$ hypergeometric functions belong to the Laguerre-P\'olya class are investigated in a constructive manner.

Contact discontinuities for the ideal compressible magnetohydrodynamics (MHD) are most typical interfacial waves for astrophysical plasmas and prototypical fundamental waves for systems of hyperbolic conservations laws. Such waves are characteristic discontinuities for which there is no flow across the discontinuity surface while the magnetic field crosses transversally, which lead to a two-phase free boundary problem where the pressure, velocity and magnetic field are continuous across the interface whereas the entropy and density may have discontinuities. Some of the major difficulties for the existence of the Multi-dimensional ideal MHD contact discontinuities are  the possible nonlinear Rayleigh-Taylor instability and loss of derivatives due to the non-ellipticity of the associated linearized problem. In this talk, I will present the recent work  where we have proved the local existence and uniqueness of MHD contact discontinuities in both 2D and 3D in Sobolev spaces  without any additional constraints such as Rayleigh-Taylor sign condition or with surface tensions. The key ingredients of our  analysis are  the Cauchy formula for MHD, the transversality of the magnetic field, and an elaborate viscous approximation. This talk is based on a joint work with Professor Yanjin Wang of Xiamen University. 

 In this talk,  we consider the self-dual equations arising from the Maxwell-Chern-Simons-Higgs model  in a curved space with a background metric $(1,-b(x),-b(x))$.

We  assume that $b(x)$ is not a constant and decays like $|x|^{-\ga}$ with $\ga \in (0,2)$.  Let $\kappa$  and $q$ be the Maxwell and the Chern-Simons coupling constants, respectively. We show that there exists a constant $\beta_*$ such that  if $\kappa  q >\beta_*$, then  there exists a topological solution.

We study the existence of entire radial solutions of the Liouville-type system

$$ \Delta \begin{pmatrix} u_1 \\ u_2 \end{pmatrix} =-\begin{pmatrix} 2 & -a \\ -b & 2 \end{pmatrix} \begin{pmatrix} e^{u_1} \\ e^{u_2} \end{pmatrix} +4\pi\begin{pmatrix} N_1\delta_{\mathbf{0}} \\ N_2\delta_{\mathbf{0}} \end{pmatrix} \quad\mbox{in }~ \mathbb{R}^2, $$

where $a$ and $b$ are positive constants satisfying $ab<4$. $N_1$ and $N_2$ are nonnegative integers, and $\delta_{\mathbf{0}}$ is the Dirac measure concentrated at the origin. This system is closely related to an elliptic system arising from the Chern-Simons gauge theory of rank two in $\mathbb{R}^2$.

We use the shooting argument and the bubbling analysis to find pairs $(\beta_1,\beta_2)\in\mathbb{R}^2$ for which this system has an entire radial solution $(u_1,u_2)$ subject to the boundary condition $u_k(r)= -2\beta_k\ln r+O(1)$ as $r\to\infty$ ($k=1,2$).

We consider axisymmetric incompressible inviscid flows without swirl in $\R^3$ under the assumption that the axial vorticity is non-positive in the upper half space and odd in the last coordinate, which corresponds to the flow setup for head-on collision of anti-parallel vortex rings. For any such data, we establish monotonicity and infinite growth of the vorticity impulse on the upper half-space. As an application, we achieve infinite growth of Sobolev norms for certain classical/smooth and compactly supported vorticity solutions in $\R^3$. This is joint work with In-Jee Jeong(SNU).

In this talk, we present a systematic algebraic approach for the weak coupling of Cauchy problems to multiple Lohe tensor models. For this, we identify an admissible Cauchy problem to the Lohe tensor (LT) model with a characteristic symbol consisting of four tuples in terms of a size vector, a natural frequency tensor, a coupling strength tensor and admissible initial configuration. In this way, the collection of all admissible Cauchy problems to the LT models is equivalent to the space of characteristic symbols. On the other hand, we introduce a binary operation, namely fusion operation" as a binary operation between characteristic symbols. It turns out that the fusion operation satisfies an associativity and admits the identity element in the space of characteristic symbols which naturally forms a monoid. By virtue of the fusion operation, the weakly coupled system of multi tensor models can be obtained by applying the fusion operation of multiple characteristic symbols corresponding to the Lohe tensor models. As a concrete example, we consider a weak coupling of the swarm sphere model and the Lohe matrix model, and provide sufficient framework leading to emergent dynamics to the proposed weakly coupled model. This is a joint work with Dohyun Kim (Sungshin Women’s Univ.) and Hansol Park (Simon Fraser Univ.)