• Bulletin of the Korean Mathematical Society
• /
• v.58 no.6
• /
• pp.1445-1461
• /
• 2021

The aim of this paper is to show the local well-posedness of 2 dimensional Dirac equations with power type and Hartree type nonlin-earity derived from honeycomb structure in H^s for s > $\frac{7}{8}$ and s > $\frac{3}{8}$, respectively. We also provide the smoothness failure of flows of Dirac equations.

• Journal of the Korean Mathematical Society
• /
• v.38 no.6
• /
• pp.1205-1234
• /
• 2001

We consider the time local well-posedness of the Benjamin equation. Like the result due to Keing-Ponce-Vega [10], [12], we show that the initial value problem is time locally well posed in the Sobolev space H$^{s}$ (R) for s>-3/4.

• Communications of the Korean Mathematical Society
• /
• v.37 no.3
• /
• pp.735-748
• /
• 2022

The global well-posedness for the fourth-order modified Zakharov equations in 1-D, which is a system of PDE in two variables describing interactions between quantum Langmuir and quantum ionacoustic waves is studied. In this paper, it is proven that the system is globally well-posed in (u, n) ∈ L² × L² by making use of Bourgain restriction norm method and L² conservation law in u, and controlling the growth of n via appropriate estimates in the local theory. In particular, this improves on the well-posedness results for this system in [9] to lower regularity.

• Bulletin of the Korean Mathematical Society
• /
• v.48 no.6
• /
• pp.1291-1302
• /
• 2011

We establish the local well-posedness for the Cauchy problem of the nonlinear Schr$\ddot{o}$odinger equation with harmonic potential in $H^s(\mathbb{R}^n)$, where $s{\in}\mathbb{R}$, s > 0.

• Bulletin of the Korean Mathematical Society
• /
• v.56 no.6
• /
• pp.1601-1615
• /
• 2019

We show the existence of ground state and orbital stability of standing waves of nonlinear $Schr{\ddot{o}}dinger$ equations with singular linear potential and essentially mass-subcritical power type nonlinearity. For this purpose we establish the existence of ground state in $H^1$. We do not assume symmetry or monotonicity. We also consider local and global well-posedness of Strichartz solutions of energy-subcritical equations. We improve the range of inhomogeneous coefficient in [5, 12] slightly in 3 dimensions.

• Transactions of the Korean Society of Mechanical Engineers A
• /
• v.21 no.5
• /
• pp.850-858
• /
• 1997

A new a posteriori error estimator is introduced and applied to variational inequalities occurring in problems of flow through porous media. In order to construct element-wise a posteriori error estimates the global error is localized by a special mixed formulation in which continuity conditions at interfaces are treated as constraints. This approach leads to error indicators which provide rigorous upper bounds of the element errors. A discussion of a compatibility condition for the well-posedness of the local error analysis problem is given. Two numerical examples are solved to check the compatibility of the local problems and convergence of the effectivity index both in a local and a global sense with respect to local refinements.

• Bulletin of the Korean Mathematical Society
• /
• v.55 no.1
• /
• pp.267-296
• /
• 2018

This paper studies the Cauchy problem and blow-up phenomena for a new generalized Camassa-Holm equation with cubic nonlinearity in the nonhomogeneous Besov spaces. First, by means of the Littlewood-Paley decomposition theory, we investigate the local well-posedness of the equation in $B^s_{p,r}$ with s > $max\{{\frac{1}{p}},\;{\frac{1}{2}},\;1-{\frac{1}{p}}\},\;p,\;r{\in}[0,{\infty}]$. Second, we prove that the equation is locally well-posed in $B^s_{2,r}$ with the critical index $s={\frac{1}{2}}$ by virtue of the logarithmic interpolation inequality and the Osgood's Lemma, and it is shown that the data-to-solution mapping is $H{\ddot{o}}lder$ continuous. Finally, we derive two kinds of blow-up criteria for the strong solution by using induction and the conservative property of m along the characteristics.

• Journal of the Chungcheong Mathematical Society
• /
• v.37 no.1
• /
• pp.27-40
• /
• 2024

In this paper, we consider the higher-order HartreeFock equations. The higher-order linear Schrödinger equation was introduced in [5] as the formal finite Taylor expansion of the pseudorelativistic linear Schrödinger equation. In [13], the authors established global-in-time Strichartz estimates for the linear higher-order equations which hold uniformly in the speed of light c ≥ 1 and as their applications they proved the convergence of higher-order Hartree-Fock equations to the corresponding pseudo-relativistic equation on arbitrary time interval as c goes to infinity when the Taylor expansion order is odd. To achieve this, they not only showed the existence of solutions in L² space but also proved that the solutions stay bounded uniformly in c. We address the remaining question on the convergence of higherorder Hartree-Fock equations when the Taylor expansion order is even. The distinguished feature from the odd case is that the group velocity of phase function would be vanishing when the size of frequency is comparable to c. Owing to this property, the kinetic energy of solutions is not coercive and only weaker Strichartz estimates compared to the odd case were obtained in [13]. Thus, we only manage to establish the existence of local solutions in H^s space for s > $\frac{1}{3}$ on a finite time interval [-T, T], however, the time interval does not depend on c and the solutions are bounded uniformly in c. In addition, we provide the convergence result of higher-order Hartree-Fock equations to the pseudo-relativistic equation with the same convergence rate as the odd case, which holds on [-T, T].