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Kyungkeun Kang

Publications and source records attributed to Kyungkeun Kang.

At least 37 records · Page 2Linked to original sources
arXiv2023

Existence of weak solutions for Porous medium equation with a divergence type of drift term in a bounded domain

We study porous medium equations with a divergence form of drift terms in a bounded domain with no-flux lateral boundary conditions. We establish $L^q$-weak solutions for $ 1\leq q < \infty$ in Wasserstein space under appropriate conditions on the drift, which is an extension of authors' previous works done in the whole space into the case of bounded domains. Applying existence results to a certain Keller-Segel equation of consumption type, construction of $L^q$-weak solutions is also made, in case that the equation of a biological organism is of porous medium type.

math.AP↗
arXiv2023

Regular solutions of chemotaxis-consumption systems involving tensor-valued sensitivities and Robin type boundary conditions

This paper deals with a parabolic-elliptic chemotaxis-consumption system with tensor-valued sensitivity $S(x,n,c)$ under no-flux boundary conditions for $n$ and Robin-type boundary conditions for $c$. The global existence of bounded classical solutions is established in dimension two under general assumptions on tensor-valued sensitivity $S$. One of main steps is to show that $\nabla c(\cdot,t)$ becomes tiny in $L^{2}(B_{r}(x)\cap Ω)$ for every $x\in\overlineΩ$ and $t$ when $r$ is sufficiently small, which seems to be of independent interest. On the other hand, in the case of scalar-valued sensitivity $S=χ(x,n,c)\mathbb{I}$, there exists a bounded classical solution globally in time for two and higher dimensions provided the domain is a ball with radius $R$ and all given data are radial. The result of the radial case covers scalar-valued sensitivity $χ$ that can be singular at $c=0$.

math.AP↗
arXiv2023

Singular weak solutions near boundaries in a half space away from localized force for the Stokes and Navier-Stokes equations

We prove that there exists a weak solution of the Stokes system with a non-zero external force and no-slip boundary conditions in a half space of dimensions three and higher so that its normal derivatives are unbounded near boundary. A localized and divergence free singular force causes, via non-local effect, singular behaviors of normal derivatives for the solution near boundary, although such boundary is away from the support of the external force. The constructed one is a weak solution that has finite energy globally, and it can be comparable to the one in \cite{Seregin-Sverak10} as a form of a shear flow that is of only locally finite energy. Similar construction is performed for the Navier-Stokes equations as well.

math.AP↗
arXiv2023

Existence of weak solutions for Porous medium equation with a divergence type of drift term

We consider degenerate porous medium equations with a divergence type of drift terms. We establish the existence of $L^{q}$-weak solutions (satisfying energy estimates or even further with moment and speed estimates in Wasserstein spaces), in case the drift term belongs to a sub-scaling (including scaling invariant) class depending on $q$ and $m$ caused by the nonlinear structure of diffusion, which is a major difference compared to that of a linear case. It is noticeable that the classes of drift terms become wider if the drift term is divergence-free. Similar conditions of gradients of drift terms are also provided to ensure the existence of such weak solutions. Uniqueness results follow under an additional condition on the gradients of the drift terms with the aid of methods developed in Wasserstein spaces. One of our main tools is so called the splitting method to construct a sequence of approximated solutions, which implies, bypassing to the limit, the existence of weak solutions satisfying not only an energy inequality but also moment and speed estimates. One of the crucial points in the construction is uniform Hölder continuity up to initial time for homogeneous porous medium equations, which seems to be of independent interest. As an application, we improve a regularity result for solutions of a repulsive Keller-Segel system of porous medium type.

math.AP↗
arXiv2022

The Green tensor of the nonstationary Stokes system in the half space

We prove the first ever pointwise estimates of the (unrestricted) Green tensor and the associated pressure tensor of the nonstationary Stokes system in the half-space, for every space dimension greater than one. The force field is not necessarily assumed to be solenoidal. The key is to find a suitable Green tensor formula which maximizes the tangential decay, showing in particular the integrability of Green tensor derivatives. With its pointwise estimates, we show the symmetry of the Green tensor, which in turn improves pointwise estimates. We also study how the solutions converge to the initial data, and the (infinitely many) restricted Green tensors acting on solenoidal vector fields. As applications, we give new proofs of existence of mild solutions of the Navier-Stokes equations in $L^q$, pointwise decay, and uniformly local $L^q$ spaces in the half-space.

math.AP↗
arXiv2022

Local regularity criteria in terms of one velocity component for the Navier-Stokes equations

This paper is devoted to presenting new interior regularity criteria in terms of one velocity component for weak solutions to the Navier-Stokes equations in three dimensions. It is shown that the velocity is regular near a point $z$ if its scaled $L^p_tL^q_x$-norm of some quantities related to the velocity field is finite and the scaled $L^p_tL^q_x$-norm of one velocity component is sufficiently small near $z$.

math.AP↗
arXiv2022

Local regularity near boundary for the Stokes and Navier-Stokes equations

We are concerned with local regularity of the solutions for the Stokes and Navier-Stokes equations near boundary. Firstly, we construct a bounded solution but its normal derivatives are singular in any $L^p$ with $1<p$ locally near boundary. On the other hand, we present criteria of solutions of the Stokes equations near boundary to imply that the gradients of solutions are bounded (in fact, even further Hölder continuous). Finally, we provide examples of solutions whose local regularity near boundary is optimal.

math.AP↗
arXiv2021

Well-posedness and singularity formation for inviscid Keller-Segel-fluid system of consumption type

We consider the Keller-Segel system of consumption type coupled with an incompressible fluid equation. The system describes the dynamics of oxygen and bacteria densities evolving within a fluid. We establish local well-posedness of the system in Sobolev spaces for partially inviscid and fully inviscid cases. In the latter, additional assumptions on the initial data are required when either the oxygen or bacteria density touches zero. Even though the oxygen density satisfies a maximum principle due to consumption, we prove finite time blow-up of its $C^{2}$--norm with certain initial data.

math.AP↗
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.

math.AP↗
arXiv2021

Local well-posedness in the Wasserstein space for a chemotaxis model coupled to Navier-Stokes equations

We consider a coupled system of Keller-Segel type equations and the incompressible Navier-Stokes equations in spatial dimension two and three. In the previous work [19], we established the existence of a weak solution of a Fokker-Plank equation in the Wasserstein space using the optimal transportation technique. Exploiting this result, we constructed solutions of Keller-Segel-Navier-Stokes equations such that the density of biological organism belongs to the absolutely continuous curves in the Wasserstein space. In this work, we refine the result on the existence of a weak solution of a Fokker-Plank equation in the Wasserstein space. As a result, we construct solutions of Keller-Segel-Navier-Stokes equations under weaker assumptions on the initial data.

math.AP↗
arXiv2021

Finite energy Navier-Stokes flows with unbounded gradients induced by localized flux in the half-space

For the Stokes system in the half space, Kang [Math.~Ann.~2005] showed that a solution generated by a compactly supported, Hölder continuous boundary flux may have unbounded normal derivatives near the boundary. In this paper we first prove explicit global pointwise estimates of the above solution, showing in particular that it has finite global energy and its derivatives blow up everywhere on the boundary away from the flux. We then use the above solution as a profile to construct solutions of the Navier-Stokes equations which also have finite global energy and unbounded normal derivatives due to the flux. Our main tool is the pointwise estimates of the Green tensor of the Stokes system proved by us in \cite{Green} (arXiv:2011.00134). We also examine the Stokes flows generated by dipole bumps boundary flux, and identify the regions where the normal derivatives of the solutions tend to positive or negative infinity near the boundary.

math.AP↗
arXiv2021

Local regularity conditions on initial data for local energy solutions of the Navier-Stokes equations

We study the regular sets of local energy solutions to the Navier-Stokes equations in terms of conditions on the initial data. It is shown that if a weighted $L^2$ norm of the initial data is finite, then all local energy solutions are regular in a region confined by space-time hypersurfaces determined by the weight. This result refines and generalizes Theorems C and D of Caffarelli, Kohn and Nirenberg (Comm. Pure Appl. Math. 35; 1982) and our recent paper [15] as well.

math.AP↗
arXiv2021

Global well-posedness of logarithmic Keller-Segel type systems

We consider a class of logarithmic Keller-Segel type systems modeling the spatio-temporal behavior of either chemotactic cells or criminal activities in spatial dimensions two and higher. Under certain assumptions on parameter values and given functions, the existence of classical solutions is established globally in time, provided that initial data are sufficiently regular. In particular, we enlarge the range of chemotatic sensitivity $χ$, compared to known results, in case that spatial dimensions are between two and eight. In addition, we provide new type of small initial data to obtain global classical solution, which is also applicable to the urban crime model. We discuss long-time asymptotic behaviors of solutions as well.

math.AP↗
arXiv2020

An $ε$-regularity criterion and estimates of the regular set for Navier-Stokes flows in terms of initial data

We prove an $ε$-regularity criterion for the 3D Navier-Stokes equations in terms of initial data. It shows that if a scaled local $L^2$ norm of initial data is sufficiently small around the origin, a suitable weak solution is regular in a set enclosed by a paraboloid started from the origin. The result is applied to the estimate of the regular set for local energy solutions with initial data in weighted $L^2$ spaces. We also apply this result to studying energy concentration near a possible blow-up time and regularity of forward discretely self-similar solutions.

math.AP↗
arXiv2020

Temporal decays and asymptotic behaviors for a Vlasov equation with a flocking term coupled to incompressible fluid flow

We are concerned with large-time behaviors of solutions for Vlasov--Navier--Stokes equations in two dimensions and Vlasov-Stokes system in three dimensions including the effect of velocity alignment/misalignment. We first revisit the large-time behavior estimate for our main system and refine assumptions on the dimensions and a communication weight function. In particular, this allows us to take into account the effect of the misalignment interactions between particles. We then use a sharp heat kernel estimate to obtain the exponential time decay of fluid velocity to its average in $L^\infty$-norm. For the kinetic part, by employing a certain type of Sobolev norm weighted by modulations of averaged particle velocity, we prove the exponential time decay of the particle distribution, provided that local particle distribution function is uniformly bounded. Moreover, we show that the support of particle distribution function in velocity shrinks to a point, which is the mean of averaged initial particle and fluid velocities, exponentially fast as time goes to infinity. This also provides that for any $p \in [1,\infty]$, the $p$-Wasserstein distance between the particle distribution function and the tensor product of the local particle distributions and Dirac measure at that point in velocity converges exponentially fast to zero as time goes to infinity.

math.AP↗
arXiv2019

Uniqueness and characterization of local minimizers for the interaction energy with mildly repulsive potentials

In this paper, we are concerned with local minimizers of an interaction energy governed by repulsive-attractive potentials of power-law type in one dimension. We prove that sum of two Dirac masses is the unique local minimizer under the $λ-$Wasserstein metric topology with $1\le λ<\infty$, provided masses and distance of Dirac deltas are equally half and one, respectively. In addition, in case of $\infty$-Wasserstein metric, we characterize stability of steady-state solutions depending on powers of interaction potentials.

math.PR↗
arXiv2019

Global Well-posedness and Long Time Behaviors of Chemotaxis-Fluid System Modeling Coral Fertilization

We consider generalized models on coral broadcast spawning phenomena involving diffusion, advection, chemotaxis, and reactions when egg and sperm densities are different. We prove the global-in-time existence of the regular solutions of the models as well as their temporal decays in two and three dimensions. We also show that the total masses of egg and sperm density have positive lower bounds as time tends to infinity in three dimensions.

math.AP↗