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Dongho Chae

Publications and source records attributed to Dongho Chae.

85 records · Page 5Linked to original sources
arXiv2007

Remarks on regularity conditions of the Navier-Stokes equations

Let $v$ and $ø$ be the velocity and the vorticity of the a suitable weak solution of the 3D Navier-Stokes equations in a space-time domain containing $z_0 =(x_0, t_0)$, and $Q_{z_0, r} =B_{x_0, r}\times (t_0-r^2, t_0)$ be a parabolic cylinder in the domain. We show that if $v\times \fracø{|ø|}\in L^{γ, α}_{x,t} (Q_{z_0, r})$ or $ø\times \frac{v}{|v|}\in L^{γ, α}_{x,t} (Q_{z_0, r})$, where $L^{γ, α}_{x,t}$ denotes the Serrin type of class, then $z_0$ is a regular point for $v$. This refines previous local regularity criteria for the suitable weak solutions.

math.AP↗
arXiv2007

Nonexistence of self-similar singularities in the viscous magnetohydrodynamics with zero resistivity

We are concerned on the possibility of finite time singularity in a partially viscous magnetohydrodynamic equations in $\Bbb R^n$, $n=2,3$, namely the MHD with positive viscosity and zero resistivity. In the special case of zero magnetic field the system reduces to the Navier-Stokes equations in $\Bbb R^n$. In this paper we exclude the scenario of finite time singularity in the form of self-similarity, under suitable integrability conditions on the velocity and the magnetic field. We also prove the nonexistence of asymptotically self-similar singularity. This provides us information on the behavior of solutions near possible singularity of general type as described in Corollary 1.1 below.

math.AP↗
arXiv2007

Nonexistence of self-similar singularities in the ideal magnetohydrodynamics

In this paper we exclude the scenario of apparition of finite time singularity in the form of self-similar singularities in the ideal magnetohydrodynamic equations, assuming suitable integrability conditions on the vorticity and the magnetic field. We also consider more sophisticated possibility of asymptotically self-similar singularities, which means that the local classical solution converges to the self-similar profile as we approaches to the possible time of singularity. The scenario of asymptotically self-similar singularity is also excluded under suitable conditions on the profile. In the 2D magnetohydrodynamics the magnetic field evolution equations reduce to a divergence free transport equation for a scalar stream function. This helps us to improve the above nonexistence theorems on the self-similar singularities, in the sense that we only need weaker integrability conditions on the profile to prove the results.

math.AP↗
arXiv2007

Incompressible Euler Equations: the blow-up problem and related results

The question of spontaneous apparition of singularity in the 3D incompressible Euler equations is one of the most important and challenging open problems in mathematical fluid mechanics. In this survey article we review some of recent approaches to the problem. We first review Kato's classical local well-posedness result in the Sobolev space and derive the celebrated Beale-Kato-Majda criterion for finite time blow-up. Then, we discuss recent refinements of the criterion as well as geometric type of theorems on the sufficiency condition for the regularity of solutions. After that we review results excluding some of the scenarios leading to finite time singularities. We also survey studies of various simplified model problems. A dichotomy type of result between the finite time blow-up and the global in time regular dynamics is presented, and a spectral dynamics approach to study local in time behaviors of the enstrophy is also reviewed. Finally, progresses on the problem of optimal regularity for solutions to have conserved quantities are presented.

math.AP↗
arXiv2006

Nonexistence of asymptotically self-similar singularities in the Euler and the Navier-Stokes equations

In this paper we rule out the possibility of asymptotically self-similar singularities for both of the 3D Euler and the 3D Navier-Stokes equations. The notion means that the local in time classical solutions of the equations develop self-similar profiles as $t$ goes to the possible time of singularity $T$. For the Euler equations we consider the case where the vorticity converges to the corresponding self-similar voriticity profile in the sense of the critical Besov space norm, $\dot{B}^0_{1, \infty}(\Bbb R^3)$. For the Navier-Stokes equations the convergence of the velocity to the self-similar singularity is in $L^q(B(z,r))$ for some $q\in [2, \infty)$, where the ball of radius $r$ is shrinking toward a possible singularity point $z$ at the order of $\sqrt{T-t}$ as $t$ approaches to $T$. In the $L^q (\Bbb R^3)$ convergence case with $q\in [3, \infty)$ we present a simple alternative proof of the similar result in \cite{hou}.

math.AP↗
arXiv2006

Nonexistence of self-similar singularities for the 3D incompressible Euler equations

We prove that there exists no self-similar finite time blowing up solution to the 3D incompressible Euler equations. By similar method we also show nonexistence of self-similar blowing up solutions to the divergence-free transport equation in $\Bbb R^n$. This result has direct applications to the density dependent Euler equations, the Boussinesq system, and the quasi-geostrophic equations, for which we also show nonexistence of self-similar blowing up solutions.

math.AP↗
arXiv2005

Existence of Multistring Solutions of the Self-Gravitating Massive $W-$Boson

We consider a semilinear elliptic system which include the model system of the $W-$strings in the cosmology as a special case. We prove existence of multi-string solutions and obtain precise asymptotic decay estimates near infinity for the solutions. As a special case of this result we solve an open problem posed in \cite{yan}

math.AP↗
arXiv2005

On the spectral dynamics of the deformation tensor and new a priori estimates for the 3D Euler equations

In this paper we study the dynamics of eigenvalues of the deformation tensor for solutions of the 3D incompressible Euler equations. Using the evolution equation of the $L^2$ norm of spectra, we deduce new a priori estimates of the $L^2$ norm of vorticity. As an immediate corollary of the estimate we obtain a new sufficient condition of $L^2$ norm control of vorticity. We also obtain decay in time estimates of the ratios of the eigenvalues. In the remarks we discuss what these estimates suggest in the study of searching initial data leading to a possible finite time singularities. We find that the dynamical behaviors of $L^2$ norm of vorticity are controlled completely by the second largest eigenvalue of the deformation tensor.

math.AP↗
arXiv1997

Vortex Condensates in the Relativistic Self-Dual Maxwell-Chern-Simons-higgs System

The existence of vortex condensates in the self-dual Maxwell-Chern-Simons-Higgs System on a flat torus is proved by the super-sub solution method under the assumption that the total vortex number in a given periodic domain is not too large. We also study the limiting behaviors of the solutions as the Chern-Simons coupling constant goes to some limits. In the Abelian-Higgs limit we find that our solutions strongly converges to the corresponding vortex condensates of the Abelian-Higgs system, while in the Chern-Simons limit the solutions strongly converges to the corresponding vortex condensates of Chern-Simons system.

cond-mat.supr-con↗
arXiv1997

Topological Multivortex Solutions of the Self-Dual Maxwell-Chern-Simons-Higgs System

We study existence and various behaviors of topological multivortices solutions of the relativistic self-dual Maxwell-Chern-Simons-Higgs system. We first prove existence of general topological solutions by applying variational methods to the newly discovered minimizing functional. Then, by an iteration method we prove existence of topological solutions satisfying some extra conditions, which we call admissible solutions. We establish asymptotic exponential decay estimates for these topological solutions. We also investigate the limiting behaviors of the admissible solutions as parameters in our system goes to some limits. For the Abelian Higgs limit we obtain strong convergence result, while for the Chern-Simons limit we only obtained that our admissible solutions are weakly approximating one of the Chern-Simons solutions.

cond-mat.supr-con↗