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Hantaek Bae

Publications and source records attributed to Hantaek Bae.

18 recordsLinked to original sources
arXiv2026

On a generalized incompressible model in two dimensions

We analyze a generalized incompressible model proposed by Ohkitani [19]. This model is based on the observation that the two-dimensional Burgers' equation can be related to the incompressible Navier-Stokes equations by rotating the velocity gradient by 90 degrees. We present several results with initial data in $H^{3}$. First of all, we examine the inviscid model and show the existence and uniqueness of a local-in-time solution that blows up in finite time if the initial vorticity contains a negative part. In the presence of viscosity, we show the existence of a unique global-in-time solution without requiring a sign condition on the initial vorticity, establish the long-time behavior of the difference between two solutions, and derive temporal decay rates for the velocity field when the initial vorticity is non-positive.

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arXiv2025

Long-time existence for the 2D ideal Boussinesq and the 2D density-dependent Euler equations

We establish long-time existence of smooth solutions to the 2D ideal Boussinesq equations and to the 2D non-homogeneous incompressible Euler equations for initial data consisting of small temperature perturbations, or small density perturbations, of smooth initial flows which are not necessarily small. Both results are known (see Danchin and Fanelli 2013, Danchin 2011 in the references) but the technique we develop to prove them is at the same time elementary and has broad potential applicability.

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arXiv2025

On the double Beltrami states in Hall magnetohydrodynamics

In this paper, we investigate double Beltrami states in the Hall magnetohydrodynamic (Hall MHD) equations. Initially, we examine the double Beltrami states as a special class of steady solutions to the ideal Hall MHD equations, which are closely related to Beltrami flows in incompressible fluid dynamics. Specifically, we classify the double Beltrami states and show that they can be derived by using the variational method as energy minimizers, subject to the conservation of two helicities. We then extend our analysis to time-dependent double Beltrami states in the viscous and resistive Hall MHD equations, exploring their exact form and stability properties.

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arXiv2024

On the regularity of magneto-vorticity field and the global existence for the Hall magnetohydrodynamic equations

In this paper, we investigate the incompressible viscous and resistive Hall magnetohydrodynamic equations (Hall MHD in short). We first study the regularity of the magneto-vorticity field $B+\omega$. In three dimensions, we derive some bounds of $B+\omega$ under a condition of the velocity field $u$. Moreover, if we consider the Hall MHD with 2D variables, the uniform-in-time bounds of $B+\omega$ come from the three dimensional case. The regularity of $B+\omega$ gives us a crucial clue of blow-up scenario and provides conditions of the existence of global-in-time solutions. In particular, we prove the global well-posedness of the Hall MHD (also the electron MHD) with 2D variables when the third component of the initial current density $J_0=\nabla\times B_0$ is sufficiently small. We also derive temporal decay rate of $B+\omega$.

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arXiv2024

On the temporal estimates for the incompressible Navier-Stokes equations and the Hall-magnetohydrodynamic equations

In this paper, we derive decay rates for solutions to the incompressible Navier-Stokes equations and Hall-magnetohydrodynamic equations. We first improve the decay rate of weak solutions to these equations by refining the Fourier splitting method with initial data in the space of pseudo-measures. Additionally, we investigate these equations with initial data in the Lei-Lin spaces and establish decay rates for those solutions.

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arXiv2024

Global existence and long time behavior of solutions to some Oldroyd type models in hybrid Besov spaces

In this paper, we deal with some Oldroyd type models, which describe incompressible viscoelastic fluids. There are 3 parameters in these models: the viscous coefficient of fluid $\nu_{1}$, the viscous coefficient of the elastic part of the stress tensor $\nu_{2}$, and the damping coefficient of the elastic part of the stress tensor $\alpha$. In this paper, we assume at least one of the parameters is zero: $(\nu_{1}, \nu_{2},\alpha)=(+,0,+), (+,0,0), (0,+,+), (0,+,0), (0,0,+)$ and prove the global existence of unique solutions to all these 5 cases in the framework of hybrid Besov spaces. We also derive decay rates of the solutions except for the case $(\nu_{1}, \nu_{2},\alpha)=(+,0,0)$. To the best of our knowledge, decay rates in this paper are the first results in this framework, and can improve some previous works.

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arXiv2023

Global solutions to Stokes-Magneto equations with fractional dissipations

In this paper, we investigate a Stokes-Magneto system with fractional diffusions. We first deal with the non-resistive case in $\mathbb{T}^{d}$ and establish the local and global well-posedness with initial magnetic field $\mathbf{b}_0\in H^{s}(\mathbb{T}^d)$. We also show the existence of a unique mild solution of the resistive case with initial data $\mathbf{b}_0$ in the critical $L^{p}(\mathbb{R}^d)$ space. Moreover, we show that $\|\mathbf{b}(t)\|_{L^{p}}$ converges to zero as $t\rightarrow\infty$ when the initial data is sufficiently small.

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arXiv2021

On the existence and temporal asymptotics of solutions for the two and half dimensional Hall MHD

In this paper, we deal with the $2\frac{1}{2}$ dimensional Hall MHD by taking the velocity field $u$ and the magnetic field $B$ of the form $u(t,x,y)=\left(\nabla^{\perp}ϕ(t,x,y), W(t,x,y)\right)$ and $B(t,x,y)=\left(\nabla^{\perp}ψ(t,x,y), Z(t,x,y)\right)$. We begin with the Hall equations (without the effect of the fluid part). We first show the long time behavior of weak solutions and weak-strong uniqueness. We then proceed to prove the existence of unique strong solutions locally in time and to derive a blow-up criterion. We also demonstrate that the strong solution exists globally in time and decay algebraically if some smallness conditions are imposed. We further improve the decay rates of $ψ$ using the structure of the equation of $ψ$. As a consequence of the decay rates of $(ψ,Z)$, we find the asymptotic profiles of $(ψ,Z)$. We finally show that a small perturbation of initial data near zero can be extended to a small perturbations near harmonic functions. In the presence of the fluid filed, the results, by comparison, fall short of the previous ones in the absence of the fluid part. We prove two results: the existence of unique strong solutions locally in time and a blow-up criterion, and the existence of unique strong solutions globally in time with some smallness condition on initial data.

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arXiv2021

Singularity formation for the Serre-Green-Naghdi equations and applications to abcd-Boussinesq systems

In this work we prove that the solution of the Serre-Green-Naghdi equation cannot be globally defined when the interface reaches the impervious bottom tangentially. As a consequence, our result complements the paper \emph{Camassa, R., Falqui, G., Ortenzi, G., Pedroni, M., \& Thomson, C. Hydrodynamic models and confinement effects by horizontal boundaries. Journal of Nonlinear Science, 29(4), 1445-1498, 2019.} Furthermore, we also prove that the solution to the $abcd-$Boussinesq system can change sign in finite time. Finally, we provide with a proof of a scenario of finite time singularity for the $abcd-$Boussinesq system. These latter mathematical results are related to the numerics in \emph{Bona, \& Chen, Singular solutions of a Boussinesq system for water waves. J. Math. Study, 49(3), 205-220, 2016}.

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arXiv2020

Global existence and exponential decay to equilibrium for DLSS-type equations

In this paper, we deal with two logarithmic fourth order differential equations: the extended one-dimensional DLSS equation and its multi-dimensional analog. We show the global existence of solution in critical spaces, its convergence to equilibrium and the gain of spatial analyticity for these two equations in a unified way.

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arXiv2018

Striated Regularity for the Euler Equations

In 1993, Chemin proved that vorticity possessing negative Holder regularity in directions given by a sufficient family of vector fields (striated regularity) maintains such regularity for all time when measured against the push-forward of those vector fields. Later work of Gamblin and Saint Raymond, and of Danchin, established analogous results in higher dimension. We give an alternative proof of these results, and establish the propagation of striated regularity of the Lagrangian velocity in a positive Holder space.

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arXiv2018

Global existence of weak solutions to dissipative transport equations with nonlocal velocity

We consider 1D dissipative transport equations with nonlocal velocity field: \[ θ_t+uθ_x+δu_{x} θ+Λ^γθ=0, \quad u=\mathcal{N}(θ), \] where $\mathcal{N}$ is a nonlocal operator given by a Fourier multiplier. Especially we consider two types of nonlocal operators: $\mathcal{N}=\mathcal{H}$, the Hilbert transform, $\mathcal{N}=(1-\partial_{xx} )^{-α}$. In this paper, we show several global existence of weak solutions depending on the range of $γ$ and $δ$. When $0<γ<1$, we take initial data having finite energy, while we take initial data in weighted function spaces (in the real variables or in the Fourier variables), which have infinite energy, when $γ\in (0,2)$.

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arXiv2018

Uniqueness of solutions for Keller-Segel system of porous medium type coupled to fluid equations

We prove the uniqueness of Hölder continuous weak solutions via duality argument and vanishing viscosity method for the Keller-Segel system of porous medium type equations coupled to the Stokes system in dimensions three. An important step is the estimate of the Green function of parabolic equations with lower order terms of variable coefficients, which seems to be of independent interest.

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arXiv2014

The vortex patches of Serfati

In 1993, two proofs of the persistence of regularity of the boundary of a classical vortex patch for the 2D Euler equations were published, one by Chemin (announced in 1991) the other by Bertozzi and Constantin. Chemin, in fact, proved a more general result, extending it further in 1995 showing, roughly, that vorticity initially having discontinuities only in directions normal to a family of vector fields that together foliate the plane continue to be so characterized by the time-evolved vector fields. A different, four-page "elementary" proof of Chemin's 1993 result was published in 1994 by Ph. Serfati, who also gave a fuller characterization of the velocity gradient's regularity. We give a detailed version of Serfati's proof along with an extension of it to a family of vector fields that reproduces the 1995 result of Chemin.

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arXiv2014

Global existence for some transport equations with nonlocal velocity

In this paper, we study transport equations with nonlocal velocity fields with rough initial data. We address the global existence of weak solutions of an one dimensional model of the surface quasi-geostrophic equation and the incompressible porous media equation, and one dimensional and $n$ dimensional models of the dissipative quasi-geostrophic equations and the dissipative incompressible porous media equation.

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arXiv2014

Gevrey regularity for a class of dissipative equations with analytic nonlinearity

In this paper, we establish Gevrey class regularity of solutions to a class of dissipative equations with an analytic nonlinearity in the whole space. This generalizes the results of Ferrari and Titi in the periodic space case with initial data in $L^2-$based Sobolev spaces to the $L^p$ setting and in the whole space. Our generalization also includes considering rougher initial data, in negative Sobolev spaces in some cases including the Navier-Stokes and the subcritical quasi-geostrophic equations, and allowing the dissipation operator to be a fractional Laplacian. Moreover, we derive global (in time) estimates in Gevrey norms which yields decay of higher order derivatives which are optimal. Applications include (temporal) decay of solutions in higher Sobolev norms for a large class of equations including the Navier-Stokes equations, the subcritical quasi-geostrophic equations, nonlinear heat equations with fractional dissipation, a variant of the Burgers' equation with a cubic or higher order nonlinearity, and the generalized Cahn-Hilliard equation. The decay results for the last three cases seem to be new while our approach provides an alternate proof for the recently obtained $L^p\, (1<p<2)$ decay result for the Navier-Stokes equations by Bae, Biswas and Tadmor. These applications follow from our global Gevrey regularity result for initial data in \emph{critical spaces} with low regularity.

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arXiv2013

Analyticity of the Subcritical and Critical Quasi-Geostrophic equations in Besov Spaces

We establish analyticity of the subcritical and critical quasi-geostrophic equations in critical Besov spaces. The main method is so-called Gevrey estimates, which is motivated by the work of Foias and Temam. We show that mild solutions θ(t), are Gevrey regular, i.e. they satisfy the estimate \sup_{t>0}\|e^{αt^{1/γ}Λ_1}θ(t)\|_{\cap L}<\infty. for a suitably chosen α>0 and a scaling invariant Besov space {\cap L}.

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