최규동 교수

Abstract In connection with the recent proposal for possible singularity formation at the boundary for solutions of three‐dimensional axisymmetric incompressible Euler's equations (Luo and Hou, Proc. Natl. Acad. Sci. USA (2014)), we study models for the dynamics at the boundary and show that they exhibit a finite‐time blowup from smooth data. © 2017 Wiley Periodicals, Inc.

In connection with the recent proposal for possible singularity formation at the boundary for solutions of 3d axi-symmetric incompressible Euler's equations (Luo and Hou, 2013), we study models for the dynamics at the boundary and show that they exhibit a finite-time blow-up from smooth data.

We consider inviscid limits to shocks for viscous scalar conservation laws in one space dimension, with strict convex fluxes. We show that we can obtain sharp estimates in $L^2$ for a class of large perturbations and for any bounded time interval. Those perturbations can be chosen big enough to destroy the viscous layer. This shows that the fast convergence to the shock does not depend on the fine structure of the viscous layers. This is the first application of the relative entropy method devel

We study weak solutions of the 3D Navier–Stokes equations with L^{2} initial data. We prove that \mathrm{∇}^{\alpha }u is locally integrable in space–time for any real α such that 1 < \alpha < 3 . Up to now, only the second derivative \mathrm{∇}^{2}u was known to be locally integrable by standard parabolic regularization. We also present sharp estimates of those quantities in weak- L_{\mathrm{loc}}^{4/ (\alpha + 1)} . These estimates depend only on the L^{2} -norm of the initial data and o

Abstract We study stability of a spherical vortex introduced by M. Hill in 1894, which is an explicit solution of the three‐dimensional incompressible Euler equations. The flow is axi‐symmetric with no swirl, the vortex core is simply a ball sliding on the axis of symmetry with a constant speed, and the vorticity in the core is proportional to the distance from the symmetry axis. We use the variational setting introduced by A. Friedman and B. Turkington ( Trans. Amer. Math. Soc ., 1981), which p

For the axi-symmetric incompressible Euler equations, we prove linear in time filamentation near Hill’s vortex: there exists an arbitrary small outward perturbation growing linearly for all times. This is based on combining the recent nonlinear orbital stability obtained by the first author with a dynamical bootstrapping scheme for particle trajectories. These results rigorously confirm numerical simulations by Pozrikidis in 1986.

We consider inviscid limits to shocks for viscous scalar conservation laws in one space dimension, with strict convex fluxes. We show that we can obtain sharp estimates in $L^2$, for a class of large perturbations. Those perturbations can be chosen big enough to destroy the viscous layer. This shows that the fast convergence to the shock does not depend on the fine structure of the viscous layers. This is the first application of the relative entropy method developed in [22], [23] to the study o

We consider a hyperbolic–parabolic system arising from a chemotaxis model in tumor angiogenesis, which is described by a Keller–Segel equation with singular sensitivity. It is known to allow viscous shocks (so-called traveling waves). We introduce a relative entropy of the system, which can capture how close a solution at a given time is to a given shock wave in almost [Formula: see text]-sense. When the shock strength is small enough, we show the functional is non-increasing in time for any lar

Abstract The Sadovskii vortex patch is a traveling wave for the two-dimensional incompressible Euler equations consisting of an odd symmetric pair of vortex patches touching the symmetry axis. Its existence was first suggested by numerical computations of Sadovskii in [J. Appl. Math. Mech., 1971], and has gained significant interest due to its relevance in inviscid limit of planar flows via Prandtl–Batchelor theory and as the asymptotic state for vortex ring dynamics. In this work, we prove the