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Dongho Chae

Publications and source records attributed to Dongho Chae.

At least 37 records · Page 2Linked to original sources
arXiv2018

On Liouville type theorems for the stationary MHD and Hall-MHD systems

In this paper we prove a Liouville type theorem for the stationary magnetohydrodynamics(MHD) system in $\Bbb R^3$. Let $(v, B, p)$ be a smooth solution to the stationary MHD equations in $\Bbb R^3$. We show that if there exist smooth matrix valued potential functions ${\bf Φ}$, ${\bf Ψ}$ such that $ \nabla \cdot {\bf Φ} =v$ and $\nabla \cdot {\bf Ψ}= B$, whose $L^6$ mean oscillations have certain growth condition near infinity, namely $$-\!\!\!\!\!\int_{B(r)} |\mathbfΦ - \mathbfΦ_{ B(r)} |^6 dx + -\!\!\!\!\!\int_{B(r)} |\mathbfΨ- \mathbfΨ_{ B(r)} |^6 dx\le C r\quad \forall 1< r< +\infty,$$ then $v=B= 0$ and $p=$constant. With additional assumption of $$r^{-8}\int_{B(r)}|B-B_{B(r)}|^6dx\to 0\quad \mathrm{as}\quad r\to+\infty,$$ similar result holds also for the Hall-MHD system.

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arXiv2018

Notes on the Liouville type problem for the stationary Navier-Stokes equations in $\Bbb R^3$

In this paper we study the Liouville type problem for the stationary Navier-Stokes equations in $\Bbb R^3$. We deduce an asymptotic formula for an integral involving the head pressure, $Q=\frac12 |v|^2 +p$, and its derivative over domains enclosed by level surfaces of $Q$. This formula provides us with new sufficient condition for the triviality of solution to the Navier-Stokes equations.

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arXiv2018

Removing Type II singularities off the axis for the 3D axisymmetric Euler equations

We prove local blow-up criterion for smooth axisymmetric solutions to the 3D incompressible Euler equation. If the vorticity satisfies $ \intl_{0}^{t_*} (t_*-t) \| ω(t)\|_{ L^\infty(B(x_{ \ast}, R_0))} dt <+\infty$ for a ball $B(x_{ \ast}, R_0)$ away from the axis of symmetry, then there exists no singularity at $t=t_*$ in the torus $T(x_*, R)$ generated by rotation of the ball $B(x_{ \ast}, R_0)$ around the axis. This implies that possible singularity at $t=t_*$ in the torus $T(x_*, R)$ is excluded if the vorticity satisfies the blow-up rate $ \|ø(t)\|_{L^\infty (T(x_*, R))}= O\left(\frac{1}{(t_*-t)^γ}\right)$ as $t\to t_*$, where $γ<2$ and the torus $T(x_*, R)$ does not touch the axis.

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arXiv2017

On the local Type I conditions for the 3D Euler equations

We prove local non blow-up theorems for the 3D incompressible Euler equations under local Type I conditions. More specifically, for a classical solution $v\in L^\infty (-1,0; L^2 ( B(x_0,r)))\cap L^\infty_{\rm loc} (-1,0; W^{1, \infty} (B(x_0, r)))$ of the 3D Euler equations, where $B(x_0,r)$ is the ball with radius $r$ and the center at $x_0$, if the limiting values of certain scale invariant quantities for a solution $v(\cdot, t)$ as $t\to 0$ are small enough, then $ \nabla v(\cdot,t) $ does not blow-up at $t=0$ in $B(x_0, r)$.

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arXiv2016

On the Liouville type theorems for self-similar solutions to the Navier-Stokes equations

We prove Liouville type theorems for the self-similar solutions to the Navier-Stokes equations. One of our results generalizes the previous ones by Nečas-Ružička-Šverak and Tsai. Using the Liouville type theorem we also remove a scenario of asymtotically self-similar blow-up for the Navier-Stokes equations with the profile belonging to $L^{p, \infty} (\Bbb R^3)$ with $p> \frac{3}{2}$.

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arXiv2016

On the Geometric Regularity Conditions for the 3D Navier-Stokes Equations

We prove geometrically improved version of Prodi-Serrin type blow-up criterion. Let $v$ and $ω$ be the velocity and the vorticity of solutions to the 3D Navier-Stokes equations and denote $\{f\}_+=\max\{f, 0\}$ , $Q_T=\Bbb R^3\times (0, T)$. If $\left\{\left( v \times \fracω{|ω|} \right)\cdot \frac{Λ^βv}{|Λ^βv|}\right\}_+ \in L^{γ, α}_{x,t} (Q_T)$ with $3/γ+2/α\leq 1$ for some $γ>3$ and $1 \leq β\leq 2$, then the local smooth solution $v$ of the Navier-Stokes equations on $(0,T)$ can be continued to $(0, T+δ)$ for some $δ>0$. We also prove localized version of a special case of this. Let $v$ be a suitable weak solution to the Navier-tokes equations in a space-time domain containing $z_0= (x_0, t_0)$, let $Q_{z_0, r}=B_{x_0, r} \times (t_0-r^2, t_0)$ be a parabolic cylinder in the domain. We show that if either $\left\{\left( v \times \fracω{|ω|}\right) \cdot \frac{\nabla \times ω}{|\nabla \times ω|}\right\}_{+} \in L^{γ, α}_{x,t}(Q_{z_0, r})$ with $\frac{3}γ+\frac{2}α \leq 1$, or $\left\{\left(\frac{v}{|v|} \times ω\right) \cdot \frac{\nabla \times ω}{|\nabla \times ω|}\right\}_{+} \in L^{γ, α}_{x,t}(Q_{z_0, r})$ with $\frac{3}γ+\frac{2}α \leq 2$, ($γ\geq 2$, $α\geq 2$), then $z_0$ is a regular point for $v$. This improves previous local regularity criteria for the suitable weak solutions.

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arXiv2016

On the Liouville theorem for weak Beltrami flows

We study Beltrami flows in the setting of weak solution to the stationary Euler equations in $\Bbb R^3$. For this weak Beltrami flow we prove the regularity and the Liouville property. In particular, we show that if tangential part of the velocity has certain decay property at infinity, then the solution becomes trivial. This decay condition of of the velocity is weaker than the previously known sufficient conditions for the Liouville property of the Betrami flows. For the proof we establish a mean value formula and other various formula for the tangential and the normal components of the weak solutions to the stationary Euler equations.

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arXiv2016

On Liouville type theorems for the steady Navier-Stokes equations in $\Bbb R^3$

In this paper we prove three different Liouville type theorems for the steady Navier-Stokes equations in $\Bbb R^3$. In the first theorem we improve logarithmically the well-known $L^{\frac92} (\Bbb R^3)$ result. In the second theorem we present a sufficient condition for the trivially of the solution($v=0$) in terms of the head pressure, $Q=\frac12 |v|^2 +p$. The imposed integrability condition here has the same scaling property as the Dirichlet integral. In the last theorem we present Fubini type condition, which guarantee $v=0$.

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arXiv2016

Liouville type theorems for the steady axially symmetric Navier-Stokes and magnetohydrodynamic equations

In this paper we study Liouville properties of smooth steady axially symmetric solutions of the Navier-Stokes equations. First, we provide another version of the Liouville theorem of \cite{kpr15} in the case of zero swirl, where we replaced the Dirichlet integrability condition by mild decay conditions. Then we prove some Liouville theorems under the assumption $\|\f{u_r}{r}{\bf 1}_{\{u_r< -\f 1r\}}\|_{L^{3/2}(\mbR^3)}< C_{\sharp}$ where $C_{\sharp}$ is a universal constant to be specified. In particular, if $u_r(r,z)\geq -\f1r$ for $\forall (r,z)\in[0,\oo)\times\mbR$, then ${\bf u}\equiv 0$. Liouville theorems also hold if $\displaystyle\lim_{|x|\to \oo}\Ga =0$ or $\Ga\in L^q(\mbR^3)$ for some $q\in [2,\oo)$ where $\Ga= r u_þ$. We also established some interesting inequalities for $\Om\co \f{\p_z u_r-\p_r u_z}{r}$, showing that $\na\Om$ can be bounded by $\Om$ itself. All these results are extended to the axially symmetric MHD and Hall-MHD equations with ${\bf u}=u_r(r,z){\bf e}_r +u_þ(r,z) {\bf e}_þ + u_z(r,z){\bf e}_z, {\bf h}=h_þ(r,z){\bf e}_þ$, indicating that the swirl component of the magnetic field does not affect the triviality. Especially, we establish the maximum principle for the total head pressure $Φ=\f {1}{2} (|{\bf u}|^2+|{\bf h}|^2)+p$ for this special solution class.

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arXiv2015

Global regularity for a model Navier-Stokes equations on $\Bbb R^3$

We study a nonlinear parabolic system for a time dependent solenoidal vector field on $\Bbb R^3$. The nonlinear term of this new model equations is obtained slightly modifying that of the Navier-Stokes equations. The system has the same scaling property and the Galileian invariance as the Navier-Stokes equations. For such system we prove the global regularity for a smooth initial data.

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arXiv2015

Remarks on the asymptotically discretely self-similar solutions of the Navier-Stokes and the Euler equations

We study scenarios of self-similar type blow-up for the incompressible Navier-Stokes and the Euler equations. The previous notions of the discretely (backward) self-similar solution and the asymptotically self-similar solution are generalized to the locally asymptotically discretely self-similar solution. We prove that there exists no such locally asymptotically discretely self-similar blow-up for the 3D Navier-Stokes equations if the blow-up profile is a time periodic function belonging to $C^1(\Bbb R ; L^3(\Bbb R^3)\cap C^2 (\Bbb R^3))$. For the 3D Euler equations we show that the scenario of asymptotically discretely self-similar blow-up is excluded if the blow-up profile satisfies suitable integrability conditions.

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arXiv2015

On partial regularity for the steady Hall magnetohydrodynamics system

We study partial regularity of suitable weak solutions of the steady Hall magnetohydrodynamics equations in a domain $Ω\subset \Bbb R^3$. In particular we prove that the set of possible singularities of the suitable weak solution has Hausdorff dimension at most one. Moreover, in the case $Ω=\Bbb R^3$, we show that the set of possible singularities is compact.

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