论文标题
随机二次非线性schr {Ö} dinger方程的多线性平滑和局部良好
Multilinear smoothing and local well-posedness of a stochastic quadratic nonlinear Schr{ö}dinger equation
论文作者
论文摘要
在本文中，我们研究了一个时空白噪声的分数衍生物（$-α<0 $的订单$-α<0 $）驱动的A $ d $维二次二次非线性schrödinger方程（SNLS）：$ $ \ \ weft \ weft \ weak {\ webt {array} {array} {array} \ nabla \ rangle^{ - α} \ dot {w} \，，\ quad t \ in [0，t] \，，\，\，x \ in \ Mathbb {r}^d \，，，，\\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ u_0 = ϕ = ϕ \ there en of end end end end end end en \ ,, ,, aray {array} where white where white where white where bert。 \ rightarrow \ mathbb {r} $是一个平滑的紧凑型功能。当$α<\ frac {d} {2} $时，随机卷积是时间的函数，其值在负顺序的sobolev空间中，并且必须通过时间依赖性的重态化来以智慧的方式来解释该模型。当$ 1 \ leq d \ leq 3 $结合了经典的strichartz估计和确定性的本地平滑时，我们就以\ cite {schaeffer1}的精神建立了（SNLS）的本地良好（SNL）的本地良好性。然后，我们重新审视我们的论点，并在第二阶随机术语上建立多线性平滑。这使我们能够以$α$的价格改善本地良好的结果。我们指出，这是关于$ \ mathbb {r}^d $schrödinger方程的第一个结果，由这种不规则的噪声驱动，其本地良好性均来自随机的多内性平滑性和确定性的本地局部效果，结合了strichartz的局部性。
In this article, we study a $d$-dimensional stochastic quadratic nonlinear Schrödinger equation (SNLS), driven by a fractional derivative (of order $-α<0$) of a space-time white noise: $$\left\{ \begin{array}{l}i\partial_t u-Δu= ρ^2 |u|^2 + \langle \nabla \rangle^{-α}\dot{W} \, , \quad t\in [0,T] \, , \, x\in \mathbb{R}^d \, ,\\ u_0 = ϕ\, ,\end{array}\right.$$ where $ρ:\mathbb{R}^d \rightarrow \mathbb{R}$ is a smooth compactly-supported function. When $α< \frac{d}{2}$, the stochastic convolution is a function of time with values in a negative-order Sobolev space and the model has to be interpreted in the Wick sense by means of a time-dependent renormalization. When $1\leq d \leq 3$, combining both the classical Strichartz estimates and a deterministic local smoothing, we establish the local well-posedness of (SNLS) for a small range of $α$, in the spirit of \cite{Schaeffer1}. Then, we revisit our arguments and establish multilinear smoothing on the second order stochastic term. This allows us to improve our local well-posedness result for some $α$. We point out that this is the first result concerning a Schrödinger equation on $\mathbb{R}^d$ driven by such an irregular noise and whose local well-posedness results from both a stochastic multilinear smoothing and a deterministic local one combined with Strichartz inequalities.
