Title & Abstract
Kiselev, Alexander (Duke Univ.)
Regularity of vortex and SQG patches
I will review some recent progress on regularity properties of vortex and SQG patches. In particular, I will present an example of a vortex patch with continuous initial curvature that immediately becomes infinite but returns to C^2 class at all integer times only without being time periodic. The proof involves derivation of a new system describing the patch evolution in terms of arc-length and curvature. The talk is based on a work joint with Xiaoyutao Luo.
Kim, Sun-Chul (Chung-Ang University)
Stability and time-evolution of vortex patch on a rotating sphere
We study the stability and time evolution of a barotropic vortex patch on a rotating sphere. In particular, we construct two basic mathematical models. The first one is a polar vortex cap approximating the piecewise-continuous vorticity distribution by zonal bands of uniform vorticity. The second model is a vortex strip which is a further complicated model for jet streams. We perform some linear stability analysis and proper numerical computations. The results present some interesting features of vortex phenomena on the surface of a sphere. This is a Joint work with Sung-Ik Sohn (KWNU) & Takashi Sakajo (Kyoto U).
Kang, Kyungkeun(Yonsei University)
Local boundary regularity of weak solutions for the Stokes and Navier-Stokes equations in the half space
We study local boundary regularity of weak solutions for Stokes system. We construct weak solutions of the Stokes system whose normal derivatives are unbounded near boundary in a half space with no-slip boundary conditions. Such singular behavioes can be extended to the Navier-Stokes equations as well. This is a joint work with Dr. Tongkeun Chang.
García, Claudia (Universidad de Granada)
Self-similar spirals for the generalized surface quasi-geostrophic equations
In this talk, we will construct a large class of non-trivial (non radial) self-similar solutions of the generalized surface quasi-geostrophic equation. To the best of our knowledge, this is the first rigorous construction of any self-similar solution for these equations. Moreover, the solutions are of spiral type, locally integrable and may have a change of sign. This is a joint work with Javier Gómez-Serrano.
Bae, Myoungjean (KAIST)
Smooth transonic flows of compressible fluids
The existence and stability analysis of de Laval nozzle flow is one of long standing open problems in mathematical fluid dynamics. The main difficulties occur due to a quasi-linear mixed-type feature of governing equations, which is the compressible Euler system, and the lack of a known background solution. In this talk, I will represent a recent result on a possible approach to solve the de Laval nozzle flow problem. This talk is based on several joint works with B. Duan, H, Park, Y. Park, S.-K. Weng, J.-J. Xiao and C. Xie.
Joerg Wolf (Chung-Ang University)
Existence of weak solutions for the equations of a non-Newtonian fluid with non-standard growth
We consider the equations of a non-Newtonian incompressible fluid in a general time-space cylinder Q. We assume that the rheology of the fluid is changing with respect to time and space and satisfies, for each (x,t) in Q, 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}.
Kim, Junha (Korea Institute for Advanced Study)
On the wellposednee of $\alpha$-SQG equation
In this talk, we consider the $\alpha$-SQG equation in a half-plane. When $\alpha=0$ and $\alpha=1$, this equation corresponds to the 2D Euler equations and the SQG equation, respectively. For the range $\alpha\in(0,1/2]$, we prove that the equation is locally well-posed on the anisotropic Lipschitz space $X_c^{\beta}$ for $\beta \in [\alpha,1-\alpha]$, where $X^{\beta}$ is continuously embedded in $C^{\beta}$. Moreover, if the initial data does not vanish on the boundary, then the $C^{\beta}$-norm with $\beta>1-\alpha$ of the solution blows up instantaneously. For the range $\alpha\in(1/2,1]$, we prove that there is no $C^{\alpha}$ solution to the $\alpha$-SQG equation for any given smooth initial data that does not vanish on the boundary. This is a joint work with In-Jee Jeong(SNU) and Yao Yao(NUS).
Lim, Deokwoo (UNIST)
Stability of monotone vorticities in the half cylinder and infinite perimeter growth for patches
We consider the incompressible Euler equations in the half cylinder $ mathbb{R}_{>0}\times\mathbb{T} $. In this domain, any vorticity which is independent of $x_2$ defines a stationary solution. We prove that such a stationary solution is nonlinearly stable in a weighted $L^{1}$ norm involving the horizontal impulse, if the vorticity is non-negative and non-increasing in $x_1$. This includes stability of a cylindrical patch $ \lbrace x_{1}<1\rbrace $. The stability result is based on the fact that such a profile is the unique minimizer of the horizontal impulse among all functions with the same distribution function. Based on stability, we prove the existence of vortex patches in the half cylinder that exhibit infinite perimeter growth in infinite time.This is a joint work with Kyudong Choi(UNIST) and In-Jee Jeong(SNU).