论文标题
关于与白噪声分散的周期性非线性schrödinger方程的良好性
On the wellposedness for periodic nonlinear Schrödinger equations with white noise dispersion
论文作者
论文摘要
我们考虑具有白噪声分散体的周期性非线性schrödinger方程，并给出的功率非线性 \ begin {equation*} idu =ΔU\ circ dw_t + | u |^{p-1} u \; dt \ end {equation*} 通过证明随机的Strichartz估计值，我们能够证明该方程式几乎可以通过$ 1 <p \ p \ p \ leq 3 $的非线性使用$ l^2 $初始数据进行证明。通过将傅立叶限制空间$ x^{s，b} $推广到随机设置，我们还证明了我们的解决方案与Chouk和Gubinelli使用粗糙路径技术构建的解决方案一致。我们还考虑了五重方程（$ p = 5 $），并表明它在$ l^1_Ωc_tl^2_x $中分析不正确。
We consider a periodic nonlinear Schrödinger equation with white noise dispersion and a power nonlinearity given by \begin{equation*} idu = Δu \circ dW_t + |u|^{p-1}u\;dt \end{equation*} By proving stochastic Strichartz estimates, we are able to prove almost sure global wellposedness of this equation with $L^2$ initial data for nonlinearities with exponent $1 < p \leq 3$. By generalizing the Fourier restriction spaces $X^{s,b}$ to the stochastic setting, we also prove that our solutions agree with the ones constructed by Chouk and Gubinelli using rough path techniques. We also consider the quintic equation ($p=5$), and show that it is analytically illposed in $L^1_ωC_t L^2_x$.
