arXiv subjects

Ningyao Zhang

Publications and source records attributed to Ningyao Zhang.

3 recordsLinked to original sources

arXiv2013

Homogenization of elliptic equations with large random potential

We consider an elliptic equation with purely imaginary, highly heterogeneous, and large random potential with a sufficiently rapidly decaying correlation function. We show that its solution is well approximated by the solution to a homogeneous equation with a real-valued homogenized potential as the correlation length of the random medium $\varepsilon\rightarrow 0$ and estimate the size of the random fluctuations in the setting $d\geq3$.

math.AP↗

arXiv2012

Convergence to SPDE of the Schrodinger equation with large, random potential

We study the asymptotic behavior of solutions to the Schr{ö}dinger equation with large-amplitude, highly oscillatory, random potential. In dimension $d<\mathfrak{m}$, where $\mathfrak{m}$ is the order of the leading operator in the Schrödinger equation, we construct the heterogeneous solution by using a Duhamel expansion and prove that it converges in distribution, as the correlation length $\varepsilon$ goes to 0, to the solution of a stochastic differential equation, whose solution is represented as a sum of iterated Stratonovich integral, over the space $C([0,+\infty),\mathcal{S}')$. The uniqueness of the limiting solution in a dense space of $L^2(Ω\times\mathbb{R}^d)$ is shown by verifying the property of conservation of mass for the Schrödinger equation. In dimension $d>\mathfrak{m}$, the solution to the Schr{ö}dinger equation is shown to converge in $L^2(Ω\times\mathbb{R}^d)$ to a deterministic Schr{ö}dinger solution in \cite{ZB-12}.

math.AP↗

arXiv2012

Homogenization of the Schroedinger equation with large, random potential

We study the behavior of solutions to a Schr{ö}dinger equation with large, rapidly oscillating, mean zero, random potential with Gaussian distribution. We show that in high dimension $d>\mathfrak{m}$, where $\mathfrak{m}$ is the order of the spatial pseudo-differential operator in the Schr{ö}dinger equation (with $\mathfrak{m}=2$ for the standard Laplace operator), the solution converges in the $L^2$ sense uniformly in time over finite intervals to the solution of a deterministic Schr{ö}dinger equation as the correlation length $\varepsilon$ tends to 0. This generalizes to long times the convergence results obtained for short times and for the heat equation. The result is based on a careful decomposition of multiple scattering contributions. In dimension $d<\mathfrak{m}$, the random solution converges to the solution of a stochastic partial differential equation.

math.AP↗