arXiv subjects

Kyungkeun Kang

Publications and source records attributed to Kyungkeun Kang.

At least 19 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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arXiv2026

Existence of weak solutions for fast diffusion equation with a divergence type of drift term

We construct non-negative weak solutions of fast diffusion equations with a divergence type of drift term satisfying the $L^q$-energy inequality and speed estimate in Wasserstein spaces under some integrability conditions on the drift term. Furthermore, in the case that the drift term has a divergence-free structure, it turns out that its integrability conditions can be relaxed, which is also applicable to porous medium equations, thereby improving previous results. As an application, the existence of weak solutions is also discussed for a viscous Boussinesq system of the fast diffusion type.

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arXiv2026

Bounded weak solutions for Keller-Segel equations with generalized diffusion and logistic source via an unbalanced Optimal Transport splitting scheme

We consider a parabolic-elliptic type of Keller-Segel equations with generalized diffusion and logistic source under homogeneous Neumann-Neumann boundary conditions. We construct bounded weak solutions globally in time in an unbalanced optimal transport framework, provided that the magnitude of the chemotactic sensitivity can be restricted depending on parameters. In the case of subquadratic degradation of the logistic source, we quantify the chemotactic sensitivity, in particular, in terms of the power of degradation and the pointwise bound of the initial density.

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arXiv2026

Existence of weak solutions for nonlinear drift-diffusion equations with measure data

We consider nonlinear drift-diffusion equations (both porous medium equations and fast diffusion equations) with measure data. We establish the existence of nonnegative weak solutions satisfying gradient estimates, provided that the drift term belongs to a sub-scaling class relevant to the $L^1$ space. When the drift is divergence-free, this requirement can be relaxed: the drift may belong to a class that is supercritical with respect to $L^1$-scaling class, and the admissible range of the diffusion exponent $m$ is enlarged as well. By handling both the measure data and the drift, we obtain a new type of energy estimate. We also discuss sharpness by constructing counterexamples showing that the general-drift range cannot be improved under the corresponding integrability scale without the divergence-free cancellation. As an application, we construct weak solutions for a specific type of nonlinear diffusion equation with measure data coupled to the incompressible Navier-Stokes equations.

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arXiv2026

On the Existence of Boundary Layer Separation for Incompressible Fluid Flow in the Half-Space

We consider the Stokes system in the half-space with localized boundary data. We prove that a boundary layer separation point exists provided that a certain singular integral determined by the boundary data is negative. On the other hand, if this integral is strictly positive, then boundary layer separation does not occur. When boundary layer separation occurs, we also investigate the dynamics of the separation point and the sign of the pressure gradient. Furthermore, by a perturbation argument, we construct solutions to the Navier--Stokes equations in the half-space that exhibit the same qualitative behavior as in the Stokes case.

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arXiv2026

Flow reversal of the Stokes system with localized boundary data in the half space

We consider the unsteady Stokes system in the half-space with zero initial data and nonzero, space-time localized boundary data. We show that there exist boundary influxes for which the induced flow exhibits flow reversal, in the sense that at least one component of the velocity field changes its sign in the half-space. This phenomenon is demonstrated by a careful analysis of the representation formula for the Stokes system in the half-space, including pointwise estimates, based on the Green tensor with nonzero boundary data. We construct solutions of the Stokes system such that the tangential components of the velocity field exhibit at least one sign change, while the normal component exhibits at least two sign changes. Moreover, the normal component of the constructed velocity field has the opposite sign to the tangential components near the boundary, whereas it has the same sign as the tangential components sufficiently far from the boundary.

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arXiv2025

On the local well-posedness of strong solutions to the unsteady flows of shear-thinning non-Newtonian fluids with a concentration-dependent power-law index

We investigate a system of nonlinear partial differential equations modeling the unsteady flow of a shear-thinning non-Newtonian fluid with a concentration-dependent power-law index. The system consists of the generalized Navier-Stokes equations coupled with a convection-diffusion equation describing the evolution of chemical concentration. This model arises from the mathematical description of the behavior of synovial fluid in the cavities of articulating joints. We prove the existence of a local-in-time strong solution in a three-dimensional spatially periodic domain, assuming that $\frac{7}{5} < p^- \le p(\cdot) \le p^+ \le 2$, where $p(\cdot)$ denotes the variable power-law index and $p^-$ and $p^+$ are its lower and upper bounds, respectively. Furthermore, we prove the uniqueness of the solution under the additional condition $p^+ < \frac{28}{15}$. In particular, our three-dimensional analysis directly implies the existence and uniqueness of solutions in the two-dimensional case under the less restrictive condition $1 < p^- \le p(\cdot) \le p^+ \le 2$.

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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 $B+ω$. In three dimensions, we derive some bounds of $B+ω$ under a condition of the velocity field $u$. Moreover, if we consider the Hall MHD with 2D variables, the uniform-in-time bounds of $B+ω$ come from the three dimensional case. The regularity of $B+ω$ 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 $B+ω$.

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arXiv2024

Well-posedness, magnetic helicity conservation, inviscid limit and asymptotic stability for the generalized Navier-Stokes-Maxwell equations with the Hall effect

This paper is devoted to studying the well-posedness, (conditional) conservation of magnetic helicity, inviscid limit and asymptotic stability of the generalized Navier-Stokes-Maxwell equations (NSM) under the Hall effect in two and three dimensions. More precisely, in the viscous case we prove the global well-posedness of NSM for small initial data, which allows us to establish a connection with either the Hall-magnetohydrodynamics (H-MHD) system as the speed of light tends to infinity or NSM without the Hall coefficient as this constant goes to zero. In addition, in the inviscid case the local well-posedness of NSM is also obtained for possibly large initial data. Moreover, under suitable conditions on the initial data and additional assumptions of solutions to NSM in three dimensions, the magnetic helicity is conserved as the electric conductivity goes to infinity. It is different to the case of the fractional H-MHD with critical fractional Laplacian exponents for both the velocity and magnetic fields, where the conservation of magnetic helicity can be provided for smooth initial data without any further conditions on the solution. Furthermore, the asymptotic stability of NSM around a constant magnetic field is established in the case of having a velocity damping term.

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arXiv2024

Global boundedness and blow-up in a repulsive chemotaxis-consumption system in higher dimensions

This paper investigates the repulsive chemotaxis-consumption model \begin{align*} \partial_t u &= \nabla \cdot (D(u) \nabla u) + \nabla \cdot (u \nabla v), \\ 0 &= Δv - uv \end{align*} in an $n$-dimensional ball, $n \ge 3$, where the diffusion coefficient $D$ is an appropriate extension of the function $0\leξ\mapsto(1+ξ)^{m-1}$ for some $m>0$. Under the boundary conditions \begin{equation*} ν\cdot (D(u) \nabla u + u \nabla v) = 0 \quad\text{ and }\quad v = M>0,\end{equation*} we first demonstrate that for $m > 1$, or $m = 1$ with $0 < M < 2/(n-2)$, the system admits globally defined classical solutions that are uniformly bounded in time for any choice of sufficiently smooth radial initial data. This result is further extended to the case $0<m<1$ when $M$ is chosen to be sufficiently small, depending on the initial conditions. In contrast, it is shown that for $0 < m < \frac{2}{n}$, the system exhibits blow-up behavior for sufficiently large $M$.

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arXiv2024

Global well-posedness and stability of the 2D Boussinesq equations with partial dissipation near a hydrostatic equilibrium

The paper is devoted to investigating the well-posedness, stability and large-time behavior near the hydrostatic balance for the 2D Boussinesq equations with partial dissipation. More precisely, the global well-posedness is obtained in the case of partial viscosity and without thermal diffusion for the initial data belonging to $H^δ(\mathbb{R}^2) \times H^{s}(\mathbb{R}^2)$ for $δ\in [s-1,s+1]$ if $s \in \mathbb{R}, s > 2$, for $δ\in (1,s+1]$ if $s \in (0,2]$ and for $δ\in [0,1]$ if $s = 0$. In addition, if one has either horizontal or vertical thermal diffusion then the stability and large-time behavior are provided in $H^m(\mathbb{R}^2)$, $m \in \mathbb{N}$ and in $\dot{H}^{m-1}(\mathbb{R}^2)$ with $m \in \mathbb{N}$, $m \geq 2$, respectively.

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arXiv2024

On the existence of strong solutions for unsteady motions of incompressible chemically reacting generalized Newtonian fluids

We consider a system of nonlinear partial differential equations modeling the unsteady motion of an incompressible generalized Newtonian fluid with chemical reactions. The system consists of the generalized Navier-Stokes equations with power-law type viscosity with a power-law index depending on the concentration, and the convection-diffusion equation which describes chemical concentration. This system of partial differential equations arises in the mathematical models describing the synovial fluid which can be found in the cavities of movable joints. We prove the existence of a global strong solution for the two and three-dimensional spatially periodic domain, provided that the power-law index is greater than or equal to $(d+2)/2$ where $d$ is the dimension of the spatial domain. Moreover, we also prove that such a solution is unique under the further assumption that $p^+ < \frac{3}{2} p^-$ for the two-dimensional case and $p^+ < \frac{7}{6}p^-$ for the three-dimensional case, where $p^-$ and $p^+$ are the lower and upper bounds of the power-law index $p(\cdot)$ respectively.

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arXiv2024

Applications of the Green tensor estimates of the nonstationary Stokes system in the half space

In this paper, we present a series of applications of the pointwise estimates of the (unrestricted) Green tensor of the nonstationary Stokes system in the half space, established in our previous work [CMP 2023]. First, we show the $L^1$-$L^q$ estimates for the Stokes flow with possibly non-solenoidal $L^1$ initial data, generalizing the results of Giga-Matsui-Shimizu [Math. Z. 1999] and Desch-Hieber-Prüss [J. Evol. Equ. 2001]. Second, we construct mild solutions of the Navier-Stokes equations in the half space with mixed-type pointwise decay or with pointwise decay alongside boundary vanishing. Finally, we explore various coupled fluid systems in the half space including viscous resistive magnetohydrodynamics equations, a coupled system for the flow and the magnetic field of MHD type, and the nematic liquid crystal flow. For each of these systems, we construct mild solutions in $L^q$, pointwise decay, and uniformly local $L^q$ spaces.

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arXiv2024

Well-posedness, magnetic helicity conservation, inviscid limit and asymptotic stability for the generalized Navier-Stokes-Maxwell equations

This paper is devoted to studying the well-posedness, conservation of magnetic helicity, inviscid limit and asymptotic stability of the generalized Navier-Stokes-Maxwell (NSM) equations with the standard Ohm's law in $\mathbb{R}^d$ for $d \in \{2,3\}$. More precisely, the global well-posedness is established in case of fractional Laplacian velocity $(-Δ)^αv$ with $α= \frac{d}{2}$ for suitable data. In addition, the local well-posedness in the inviscid case is also provided for sufficient smooth data, which allows us to study the inviscid limit of associated positive viscosity solutions in the case $α= 1$, where an explicit bound on the difference is given. Furthermore, in three dimensions if the initial data satisfies futher suitable conditions then magnetic helicity is conserved as the electric conductivity goes to infinity. On the other hand, in the case $α= 0$ the stability near a magnetohydrostatic equilibrium with a constant (or equivalently bounded) magnetic field is also obtained in which nonhomogeneous Sobolev norms of the velocity and electric fields, and for $p \in (2,\infty]$ the $L^p$ norm of the magnetic field converge to zero as time goes to infinity with an implicit rate. In this velocity damping case, the situation is different both in case of the two and a half, and three-dimensional (Hall)-magnetohydrodynamics ((H)-MHD) system, where an explicit rate of convergence in infinite time is computed for both the velocity and magnetic fields in nonhomogeneous Sobolev norms. Therefore, it seems that there is a gap between NSM and MHD in terms of the norm convergence of the magnetic field and the rate of decaying in time, even the latter equations can be proved as a limiting system of the former one in the sense of distributions as the speed of light tends to infinity.

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arXiv2024

Singular velocity of the Stokes and Navier-Stokes equations near boundary in the half space

Local behaviors near boundary are analyzed for solutions of the Stokes and Navier-Stoke equations in the half space with localized non-smooth boundary data. We construct solutions of Stokes equations whose velocity field is not bounded near boundary away from the support of boundary data, although velocity and gradient velocity of solutions are locally square integrable. This is an improvement compared to known results in the sense that velocity field is unbounded itself, since previously constructed solutions were bounded near boundary, although their normal derivatives are singular. We also establish singular solutions and their derivatives that do not belong to $L^q_{\rm{loc}}$ near boundary with $q> 1$. For such examples, there corresponding pressures turn out not to be locally integrable. Similar construction via a perturbation argument is available to the Navier-Stokes equations near boundary as well.

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arXiv2024

Global existence and asymptotic stability for the Toner-Tu model of flocking

This paper deals with the Toner-Tu (TT) model, which is a hydrodynamic model describing the collective motion of numerous self-propelled agents. We analytically study the global-in-time well-posedness of the TT model near the steady-state solution in the ordered phase. We also show the large-time behavior of solutions showing that the steady-state solution is polynomially stable in a Sobolev space in the sense that solutions that are initially close to that steady state converge to that at least polynomially fast as time tends to infinity. Moreover, we investigate the variant of the TT model which describes the dynamics of the actin filament.

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arXiv2023

Local and global regularity for the Stokes and Navier-Stokes equations with the localized boundary data in the half-space

We study the Stokes system with the localized boundary data in the half-space. We are concerned with the local regularity of its solution near the boundary away from the support of the given boundary data which are product forms of each spatial variable and the temporal variable. We first show that if the boundary data are smooth in time, the corresponding solutions are also smooth in space and time near the boundary, even if the boundary data are only spatially integrable. Secondly, if the normal component of the boundary data is absent, we are able to construct a solution such that its second normal derivatives of the tangential components become singular near the boundary. Perturbation argument enables us to construct solutions of the Navier-Stokes equations with similar singular behaviors near the boundary in the half-space as the case of Stokes system. Lastly, we provide specific types of the localized boundary data to obtain the pointwise bounds of the solutions up to second derivatives. It turns out that such solutions are globally strong, and the second normal derivatives are, however, unbounded near the boundary. These results can be compared to the previous works in which only the normal component is present. In fact, the temporally non-smooth tangential boundary data can also cause spatially singular behaviors near the boundary, although such behaviors are milder than those caused by the normal boundary data of the same type.

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