arXiv subjects
Publications and source records attributed to Ildoo Kim.
We study the second-order quasi-linear stochastic partial differential equations (SPDEs) defined on $C^1$ domains. The coefficients are random functions depending on $t,x$ and the unknown solutions. We prove the uniqueness and existence of solutions in appropriate Sobolev spaces, and in addition, we obtain $L_p$ and Hölder estimates of both the solution and its gradient.
We measure the Marangoni elasticity of a flowing soap film to be 22 dyne/cm irrespective of its width, thickness, flow speed, or the bulk soap concentration. We perform this measurement by generating an oblique shock in the soap film and measuring the shock angle, flow speed and thickness. We postulate that the elasticity is constant because the film surface is crowded with soap molecules. Our method allows non-destructive measurement of flowing soap film elasticity, and the value 22 dyne/cm is likely applicable to other similarly constructed flowing soap films.
In this article we present an $L_p$-theory ($p\geq 2$) for the time-fractional quasi-linear stochastic partial differential equations (SPDEs) of type $$ \partial^α_tu=L(ω,t,x)u+f(u)+\partial^β_t \sum_{k=1}^{\infty}\int^t_0 ( Λ^k(ω,t,x)u+g^k(u))dw^k_t, $$ where $α\in (0,2)$, $β<α+\frac{1}{2}$, and $\partial^α_t$ and $\partial^β_t$ denote the Caputo derivative of order $α$ and $β$ respectively. The processes $w^k_t$, $k\in \mathbb{N}=\{1,2,\cdots\}$, are independent one-dimensional Wiener processes defined on a probability space $Ω$, $L$ is a second order operator of either divergence or non-divergence type, and $Λ^k$ are linear operators of order up to two. The coefficients of the equations depend on $ω(\in Ω), t,x$ and are allowed to be discontinuous. This class of SPDEs can be used to describe random effects on transport of particles in medium with thermal memory or particles subject to sticking and trapping.
We introduce an $L_q(L_p)$-theory for the quasi-linear fractional equations of the type $$ \partial^α_t u(t,x)=a^{ij}(t,x)u_{x^i x^j}(t,x)+f(t,x,u), \quad t>0, \,x\in \mathbf{R}^d. $$ Here, $α\in (0,2)$, $p,q>1$, and $\partial^α_t$ is the Caupto fractional derivative of order $α$. Uniqueness, existence, and $L_q(L_p)$-estimates of solutions are obtained. The leading coefficients $a^{ij}(t,x)$ are assumed to be piecewise continuous in $t$ and uniformly continuous in $x$. In particular $a^{ij}(t,x)$ are allowed to be discontinuous with respect to the time variable. Our approach is based on classical tools in PDE theories such as the Marcinkiewicz interpolation theorem, the Calderon-Zygmund theorem, and perturbation arguments.
In this paper we present a Calderón-Zygmund approach for a large class of parabolic equations with pseudo-differential operators $\mathcal{A}(t)$ of arbitrary order $γ\in(0,\infty)$. It is assumed that $\cA(t)$ is merely measurable with respect to the time variable. The unique solvability of the equation $$ \frac{\partial u}{\partial t}=\cA u-λu+f, \quad (t,x)\in \fR^{d+1} $$ and the $L_{q}(\fR,L_{p})$-estimate $$ \|u_{t}\|_{L_{q}(\fR,L_{p})}+\|(-Δ)^{γ/2}u\|_{L_{q}(\fR,L_{p})} +λ\|u\|_{L_{q}(\fR,L_{p})}\leq N\|f\|_{L_{q}(\fR,L_{p})} $$ are obtained for any $λ> 0$ and $p,q\in (1,\infty)$.
In this paper we prove a parabolic version of the Littlewood-Paley inequality for a class of time-dependent local and non-local operators of arbitrary order, and as an application we show this inequality gives a fundamental estimate for the $L_p$-theory of the stochastic partial differential equations.
In this article we prove the BMO-$L_{\infty}$ estimate $$ \|(-Δ)^{γ/2} u\|_{BMO(\mathbf{R}^{d+1})}\leq N \|\frac{\partial}{\partial t}u-A(t)u\|_{L_{\infty}(\mathbf{R}^{d+1})}, \quad \forall\, u\in C^{\infty}_c(\mathbf{R}^{d+1}) $$ for a wide class of pseudo-differential operators $A(t)$ of order $γ\in (0,\infty)$. The coefficients of $A(t)$ are assumed to be merely measurable in time variable. As an application to the equation $$ \frac{\partial}{\partial t}u=A(t)u+f,\quad t\in \mathbf{R} $$ we prove that for any $u\in C^{\infty}_c(\mathbf{R}^{d+1})$ $$ \|u_t\|_{L_p(\mathbf{R}^{d+1})}+\|(-Δ)^{γ/2}u\|_{L_p(\mathbf{R}^{d+1})}\leq N\|u_t-A(t)u\|_{L_p(\mathbf{R}^{d+1})}, $$ where $p\ in (1,\infty)$ and the constant $N$ is independent of $u$.
We study the integro-differential operators $L$ with kernels $K(y) = a(y) J(y)$, where $J(y)dy$ is a Lévy measure on $\bR^d$ (i.e. $\int_{\bR^d}(1\wedge |y|^2)J(y)dy<\infty$) and $a(y)$ is an only measurable function with positive lower and upper bounds. Under few additional conditions on $J(y)$, we prove the unique solvability of the equation $Lu-λu=f$ in $L_p$-spaces and present some $L_p$-estimates of the solutions.
In this paper we prove a parabolic version of the Littlewood-Paley inequality for the operators of the type $ϕ(-Δ)$, where $ϕ$ is a Bernstein function. As an application, we construct an $L_p$-theory for the stochastic integro-differential equations of the type $du=(-ϕ(-Δ)u+f)dt +gdW_t$.
We prove a parabolic version of the Littlewood-Paley inequality for the fractional Laplacian $(-Δ)^{α/2}$, where $α\in (0,2)$.