ABE, Ken(Osaka Metropolitan University)
Title : Stability of Chandrasekhar's nonlinear force-free fields
Abstract : Mathematical research on Taylor's conjecture began with the work of Caflisch et al. in the late 90s. It is known that the total energy of weak solutions to ideal MHD is conserved provided that they have regularity better than 1/3 Holder continuous. As for magnetic helicity, Kang and Lee showed that weak solutions conserve it provided that they have regularity better than L³-integrable in space and time. Faraco and Lindberg recently provided mathematical proof to Taylor's conjecture in terms of weak ideal limits of Leray--Hopf solutions to viscous and resistive MHD. In this talk, I will discuss the Taylor state stability in ideal MHD in terms of weak ideal limits of Leray--Hopf solutions based on Taylor's relaxation theory. Then I will further consider Chandrasekhar's solutions as rare examples of nonlinear force-free fields and discuss their orbital stability in the axisymmetric setting.
BAE, Hantaek (UNIST)
Title : Mathematical Analysis of Active models
Abstract : Active matter includes living and nonliving systems with emergent dynamic properties over a wide range of length and time scales. In this talk. I will introduce several models of active matter and will provide some mathematical results.
CHAE, Dongho (Chung-Ang University)
Title : Some problems in the mathematical fluid mechanics
Abstract : In this lecture we present some challenging mathematical problems in the incompressible fluid flows. After brief introduction to the Euler/Navier-Stokes equations, 2D surface quasi-strophic equations and the Hall-MHD system, we introduce a new active vector system, which 'unifies' them. We may view the new system as a singularized version for the 3D Euler equations, and a part of Hall-MHD system(E-MHD) correspond to the order two more singular one than the 3D Euler equations. The generalized surface quasi-geostrophic equation (gSQG) can be also embedded into a special case of our system when the unknown functions are constant in one coordinate direction. We investigate some basic properties of this system as well as the conservation laws. In the case when the system corresponds up to order one more singular than the 3D Euler equations, we prove local well-posedness in the standard Sobolev spaces. The proof crucially depends on a sharp commutator estimate similar to the one previously used for (gSQG). Since the system covers many areas of both physically and mathematically interesting cases, one can expect that there are various related problems to be investigated, parts of which are discussed here.
CHOI, Young-Pil (Yonsei University)
Title : Critical thresholds in pressureless Euler-Poisson equations: a new method based on Lyapunov functions
Abstract : We investigate the critical thresholds phenomena in a large class of pressureless Euler-Poisson equations in one dimension. We propose a new method based on Lyapunov functions to construct the supercritical region with finite-time breakdown and the supercritical region with global regularity of C^1 solutions. By employing our new method, we provide the analysis of critical thresholds in the pressureless Euler-Poisson system with variable background states.
KAGEI, Yoshiyuki (Tokyo Institute of Technology)
Title : Stability of the compressible Taylor vortices under axisymmetric perturbations
Abstract : This talk is concerned with the bifurcation and stability of the compressible Taylor vortex. Consider the compressible Navier-Stokes equations in a domain between two concentric infinite cylinders. If the outer cylinder is at rest and the inner one rotates with sufficiently small angular velocity, a laminar flow, called the Couette flow, is stable. When the angular velocity of the inner cylinder increases, beyond a certain value of the angular velocity, the Couette flow becomes unstable and a vortex pattern, called the Taylor vortex, bifurcates and is observed stably. This phenomenon is mathematically formulated as a bifurcation and stability problem. In this talk, the compressible Taylor vortices are shown to bifurcate near the criticality for the incompressible problem when the Mach number is sufficiently small. The localized stability of the compressible Taylor vortices is considered and it is shown that the Eckhaus instability of compressible Taylor vortices occurs as in the case of the incompressible ones.
KANG, Kyeongkeun (Yonsei University)
Title : Singular weak solutions of the Stokes system and Navier-Stokes equations near boundary
Abstract : Weak solutions of the Stokes system are constructed in a half space of dimensions three and higher so that its normal derivatives are singular near boundary. Given localized boundary data or external forces may cause, via non-local effect, singular behaviors of the solution near boundary. The method of perturbation is applicable to the Navier-Stokes equations to construct weak solutions whose normal derivatives are unbounded near boundaries.
KIM, Sun-Chul (Chung-Ang University)
Title : Stability and evolution of vortex patch on a rotating sphere
Abstract : 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 nuemrical computations. The results present some interesting features of vortex phenomena on the surface of a sphere.
LEE, Jihoon (Chung-Ang University)
Title : The quasi-geostrophic approximation for rotating stratified fluids
Abstract : The quasi-geostrophic approximation for rotating stratified fluids has been conjectured to be invalid in the sense that the convergence rate becomes singular near certain rotation-stratification ratio while the original Boussinesq dynamics changes continuously in that ratio. Such a paradox, called Devil’s staircase convergence results, was originally cast by Babin-Mahalov-Nicolaenko-Zhou. In this talk, we provide a result for convergence and nonconvergence of the quasi-geostrophic approximation. This is the joint work with Junha Kim(KIAS) and Min Jun Jo(U. Britisch Columbia).
MIURA, Hideyuki (Tokyo Institute of Technology)
Title : Local regularity conditions on initial data for local energy solutions of the Navier-Stokes equations
Abstract : We study the regular sets of local energy solutions to the Navier-Stokes equations in terms of conditions on the initial data. It is shown that if a weighted L^2 norm of the initial data is finite, then all local energy solutions are regular in a region confined by space-time hypersurfaces determined by the weight. This result refines and generalizes Theorems C and D of Caffarelli, Kohn and Nirenberg (1982). This is a joint work with Kyungkeun Kang and Tai-Peng Tsai.
SUZUKI, Masahiro (Nagoya Institute of Technology)
Title : Stability and instability of plasma boundary layers
Abstract : We investigate mathematically a plasma boundary layer near the surface of materials immersed in a plasma, called a sheath. From a kinetic point of view, Boyd--Thompson proposed a kinetic Bohm criterion which is required for the formation of sheaths. Then Riemann pointed out (although without a rigorous proof) that the criterion is a necessary condition for the solvability of the stationary Vlasov--Poisson system. Recently, Suzuki--Takayama analyzed rigorously the solvability of the stationary Vlasov--Poisson system, and clarified in all possible cases whether or not there is a stationary solution. It was concluded that the Bohm criterion is necessary but not sufficient for the solvability. In this talk, we study the nonlinear stability and instability of the stationary solutions of the Vlasov--Poisson system. The location of the support of the initial data is a major factor leading to stability/instability. This talk is based on a joint work with Professor M. Takayama (Keio Univ.) and Professor K. Z. Zhang (New York Univ.).
YONEDA, Tsuyoshi (Hitotsubashi University)
Title : Mathematical reformulation of the Kolmogorov-Richardson energy cascade in terms of vortex stretching and related topics
Abstract : In this talk, with the aid of direct numerical simulations (DNS) of forced turbulence in a periodic domain, we mathematically reformulate the Kolmogorov-Richardson energy cascade in terms of vortex stretching. More precisely, we mathematically construct the hierarchy of tubular vortices, which is statistically self-similar in the inertial range. Under the assumptions of the scale-locally of the vortex stretching/compressing (i.e. energy cascade) process and the statistical independence between vortices that are not directly stretched or compressed, we can derive the -5/3 power law of the energy spectrum of statistically stationary turbulence without directly using the Kolmogorov hypotheses. This is a joint work with Susumu Goto and Tomonori Tsuruhashi.