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Youngwoo Koh

Publications and source records attributed to Youngwoo Koh.

At least 19 recordsLinked to original sources
arXiv2024

Time splitting method for nonlinear Schrödinger equation with rough initial data in $L^2$

We establish convergence results related to the operator splitting scheme on the Cauchy problem for the nonlinear Schrödinger equation with rough initial data in $L^2$, $$ \left\{ \begin{array}{ll} i\partial_t u +Δu = λ|u|^{p} u, & (x,t) \in \mathbb{R}^d \times \mathbb{R}_+, u (x,0) =ϕ(x), & x\in\mathbb{R}^d, \end{array} \right. $$ where $λ\in \{-1,1\}$ and $p >0$. While the Lie approximation $Z_L$ is known to converge to the solution $u$ when the initial datum $ϕ$ is sufficiently smooth, the convergence result for rough initial data is open to question. In this paper, for rough initial data $ϕ\in L^2 (\mathbb{R}^d)$, we prove the $L^2$ convergence of the filtered Lie approximation $Z_{flt}$ to the solution $u$ in the mass-subcritical range, $0< p < \frac{4}{d}$. Furthermore, we provide a precise convergence result for radial initial data $ϕ\in L^2 (\mathbb{R}^d)$.

math.NA↗
arXiv2024

Convergence in the incompressible limit of the corner singularities

In this paper, we treat the corner singularity expansion and its convergence result regarding the penalized system obtained by eliminating the pressure variable in the Stokes problem of incompressible flow. The penalized problem is a kind of the Lamé system, so we first discuss the corner singularity theory of the Lamé system with inhomogeneous Dirichlet boundary condition on a non-convex polygon. Considering the inhomogeneous condition, we show the decomposition of its solution, composed of singular parts and a smoother remainder near a re-entrant corner, and furthermore, we provide the explicit formulae of coefficients in singular parts. In particular, these formulae can be used in the development of highly accurate numerical scheme. In addition, we formulate coefficients in singular parts regarding the Stokes equations with inhomogeneous boundary condition and non-divergence-free property of velocity field, and thus we show the convergence results of coefficients in singular parts and remainder regarding the concerned penalized problem.

math.AP↗
arXiv2024

Adhesion and volume filling in one-dimensional population dynamics under no-flux boundary condition

We study the (generalized) one-dimensional population model developed by Anguige \& Schmeiser [1], which reflects cell-cell adhesion and volume filling under no-flux boundary condition. In this generalized model, depending on the adhesion and volume filling parameters $α,β\in[0,1],$ the resulting equation is classified into six types. Among these, we focus on the type exhibiting strong effects of both adhesion and volume filling, which results in a class of advection-diffusion equations of the forward-backward-forward type. For five distinct cases of initial maximum, minimum and average population densities, we derive the corresponding patterns for the global behavior of weak solutions to the initial and no-flux boundary value problem. Due to the presence of a negative diffusion regime, we indeed prove that the problem is ill-posed and admits infinitely many global-in-time weak solutions, with the exception of one specific case of the initial datum. This nonuniqueness is inherent in the method of convex integration that we use to solve the Dirichlet problem of a partial differential inclusion arising from the ill-posed problem.

math.AP↗
arXiv2024

Adhesion and volume filling in one-dimensional population dynamics under Dirichlet boundary condition

We generalize the one-dimensional population model of Anguige \& Schmeiser [1] reflecting the cell-to-cell adhesion and volume filling and classify the resulting equation into the six types. Among these types, we fix one that yields a class of advection-diffusion equations of forward-backward-forward type and prove the existence of infinitely many global-in-time weak solutions to the initial-Dirichlet boundary value problem when the maximum value of an initial population density exceeds a certain threshold. Such solutions are extracted from the method of convex integration by Müller \& \v Sverák [12]; they exhibit fine-scale density mixtures over a finite time interval, then become smooth and identical, and decay exponentially and uniformly to zero as time approaches infinity.

math.AP↗
arXiv2023

On local well-posedness of nonlinear dispersive equations with partially regular data

We revisit the local well-posedness theory of nonlinear Schrödinger and wave equations in Sobolev spaces $H^s$ and $\dot{H}^s$, $0< s\leq 1$. The theory has been well established over the past few decades under Sobolev initial data regular with respect to all spatial variables. But here, we reveal that the initial data do not need to have complete regularity like Sobolev spaces, but only partially regularity with respect to some variables is sufficient. To develop such a new theory, we suggest a refined Strichartz estimate which has a different norm for each spatial variable. This makes it possible to extract a different integrability/regularity of the data from each variable.

math.AP↗
arXiv2023

Convergence analysis of the splitting method to the nonlinear heat equation

In this paper, we analyze an operator splitting scheme of the nonlinear heat equation in $Ω\subset\mathbb{R}^d$ ($d\geq 1$): $\partial_t u = Δu + λ|u|^{p-1} u$ in $Ω\times(0,\infty)$, $u=0$ in $\partialΩ\times(0,\infty)$, $u ({\bf x},0) =ϕ({\bf x})$ in $Ω$. where $λ\in\{-1,1\}$ and $ϕ\in W^{1,q}(Ω)\cap L^{\infty} (Ω)$ with $2\leq p < \infty$ and $d(p-1)/2 0$. Finally, we give some numerical examples to confirm the reliability of the analyzed result.

math.NA↗
arXiv2022

Pointwise convergence of sequential Schrödinger means

We study pointwise convergence of the fractional Schrödinger means along sequences $t_n$ which converge to zero. Our main result is that bounds on the maximal function $\sup_{n} |e^{it_n(-Δ)^{α/2}} f| $ can be deduced from those on $\sup_{0<t\le 1} |e^{it(-Δ)^{α/2}} f|$ when $\{t_n\}$ is contained in the Lorentz space $\ell^{r,\infty}$. Consequently, our results provide seemingly optimal results in higher dimensions, which extend the recent work of Dimou-Seeger, and Li-Wang-Yan to higher dimensions. Our approach based on a localization argument also works for other dispersive equations and provides alternative proofs of previous results on sequential convergence.

math.CA↗
arXiv2021

A global space-time estimate for dispersive operators through its local estimate

We will show that a local space-time estimate implies a global space-time estimate for dispersive operators. In order for this implication we consider a Littlewood-Paley type square function estimate for dispersive operators in a time variable and a generalization of Tao's epsilon removal lemma in mixed norms. By applying this implication to the fractional Schrodinger equation in R^{2+1} we obtain the sharp global space-time estimates with optimal regularity from the previous known local ones.

math.AP↗
arXiv2021

Strichartz estimates in Wiener amalgam spaces and applications to nonlinear wave equations

In this paper we obtain some new Strichartz estimates for the wave propagator $e^{it\sqrt{-Δ}}$ in the context of Wiener amalgam spaces. While it is well understood for the Schrödinger case, nothing is known about the wave propagator. This is because there is no such thing as an explicit formula for the integral kernel of the propagator unlike the Schrödinger case. To overcome this lack, we instead approach the kernel by rephrasing it as an oscillatory integral involving Bessel functions and then by carefully making use of cancellation in such integrals based on the asymptotic expansion of Bessel functions. Our approach can be applied to the Schrödinger case as well. We also obtain some corresponding retarded estimates to give applications to nonlinear wave equations where Wiener amalgam spaces as solution spaces can lead to a finer analysis of the local and global behavior of the solution.

math.AP↗
arXiv2021

On the integrability of the wave propagator arising from the Liouville-von Neumann equation

The Liouville-von Neumann equation describes the change in the density matrix with time. Interestingly, this equation was recently regarded as a wave equation for wave functions but not a equation for density functions. This setting leads to an extended form of the Schrödinger wave equation governing the motion of a quantum particle. In this paper we obtain the integrability of the wave propagator arising from the Liouville-von Neumann equation in this setting.

math.AP↗
arXiv2019

Strichartz and smoothing estimates in weighted $L^2$ spaces and their applications

The primary objective in this paper is to give an answer to an open question posed by J. A. Barceló, J. M. Bennett, A. Carbery, A. Ruiz and M. C. Vilela concerning the problem of determining the optimal range on $s\geq0$ and $p\geq1$ for which the following Strichartz estimate with time-dependent weights $w$ in Morrey-Campanato type classes $\mathfrak{L}^{2s+2,p}_2$ holds: \begin{equation}\label{absset} \|e^{itΔ}f\|_{L_{x,t}^2(w(x,t))}\leq C\|w\|_{\mathfrak{L}^{2s+2,p}_2}^{1/2}\|f\|_{\dot{H}^s}. \end{equation} Beyond the case $s\geq0$, we further ask how much regularity we can expect on this setting. But interestingly, it turns out that this estimate is false whenever $s<0$, which shows that the smoothing effect cannot occur in this time-dependent setting and the dispersion in the Schrödinger flow $e^{itΔ}$ is not strong enough to have the effect. This naturally leads us to consider the possibility of having the effect at best in higher-order versions of this estimate with $e^{-it(-Δ)^{γ/2}}$ ($γ>2$) whose dispersion is more strong. We do obtain a smoothing effect exactly for these higher-order versions. In fact, we will obtain the estimates where $γ\geq1$ in a unified manner and also their corresponding inhomogeneous estimates to give applications to the global well-posedness for Schrödinger and wave equations with time-dependent perturbations. This is our secondary objective in this paper.

math.AP↗
arXiv2019

Strichartz estimates for the Schrödinger propagator in Wiener amalgam spaces

In this paper we study the Strichartz estimates for the Schrödinger propagator in the context of Wiener amalgam spaces which, unlike the Lebesgue spaces, control the local regularity of a function and its decay at infinity separately. This separability makes it possible to perform a finer analysis of the local and global behavior of the propagator. Our results improve some of the classical ones in the case of large time.

math.AP↗
arXiv2017

Unique continuation for the Schrödinger equation with gradient term

We obtain a unique continuation result for the differential inequality $| (i\partial_t +Δ)u | \leq |Vu| + | W\cdot\nabla u |$ by establishing $L^2$ Carleman estimates. Here, $V$ is a scalar function and $W$ is a vector function, which may be time-dependent or time-independent. As a consequence, we give a similar result for the magnetic Schrödinger equation.

math.AP↗
arXiv2017

Strichartz estimates for the magnetic Schrödinger equation with potentials $V$ of critical decay

We study the Strichartz estimates for the magnetic Schrödinger equation in dimension $n\geq3$. More specifically, for all Schrödinger admissible pairs $(r,q)$, we establish the estimate $$ \|e^{itH}f\|_{L^{q}_{t}(\mathbb{R}; L^{r}_{x}(\mathbb{R}^n))} \leq C_{n,r,q,H} \|f\|_{L^2(\mathbb{R}^n)} $$ when the operator $H= -Δ_A +V$ satisfies suitable conditions. In the purely electric case $A\equiv0$, we extend the class of potentials $V$ to the Fefferman-Phong class. In doing so, we apply a weighted estimate for the Schrödinger equation developed by Ruiz and Vega. Moreover, for the endpoint estimate of the magnetic case in $\mathbb{R}^3$, we investigate an equivalence $$ \| H^{\frac{1}{4}} f \|_{L^r(\mathbb{R}^3)} \approx C_{H,r} \big\| (-Δ)^{\frac{1}{4}} f \big\|_{L^r(\mathbb{R}^3)} $$ and find sufficient conditions on $H$ and $r$ for which the equivalence holds.

math.AP↗
arXiv2017

Strichartz estimates for Schrödinger equations in weighted $L^2$ spaces and their applications

We obtain weighted $L^2$ Strichartz estimates for Schrödinger equations $i\partial_tu+(-Δ)^{a/2}u=F(x,t)$, $u(x,0)=f(x)$, of general orders $a>1$ with radial data $f,F$ with respect to the spatial variable $x$, whenever the weight is in a Morrey-Campanato type class. This is done by making use of a useful property of maximal functions of the weights together with frequency-localized estimates which follow from using bilinear interpolation and some estimates of Bessel functions. As consequences, we give an affirmative answer to a question posed in \cite{BBCRV} concerning weighted homogeneous Strichartz estimates, and improve previously known Morawetz estimates. We also apply the weighted $L^2$ estimates to the well-posedness theory for the Schrödinger equations with time-dependent potentials in the class.

math.AP↗
arXiv2016

On the splitting method for the nonlinear Schr\"odinger equation with initial data in $H^1$

In this paper, we establish a convergence result for the operator splitting scheme $Z_{\tau}$ introduced by Ignat, with initial data in $H^1$, for the nonlinear Schr\"odinger equation : $$ \partial_t u = i \Delta u + i\lambda |u|^{p} u,\qquad u (x,0) =\phi (x), $$ where $(x,t) \in \mathbb{R}^d \times [0,\infty)$, with $0< p < 4/(d-2)$ for $d\geq3$ and $0< p<\infty$ for $d=1,2$. We prove the $L^2$ convergence of order $\mathcal{O}(\tau^{1/2})$ for this scheme with initial data in the space $H^1 (\mathbb{R}^d)$.

math.AP↗
arXiv2016

Inhomogeneous Strichartz estimates for Schrödinger's equation

Foschi and Vilela in their independent works (\cite{F},\cite{V}) showed that the range of $(1/r,1/\widetilde{r})$ for which the inhomogeneous Strichartz estimate $ \big\|\int_{0}^{t}e^{i(t-s)Δ}F(\cdot,s)ds\big\|_{L^{q}_tL^{r}_x} \lesssim \|F\|_{L^{\widetilde{q}'}_tL^{\widetilde{r}'}_x} $ holds for some $q,\widetilde{q}$ is contained in the closed pentagon with vertices $A,B,B',P,P'$ except the points $P,P'$ (see Figure 1). We obtain the estimate for the corner points $P,P'$.

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