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arXiv 1810.04076math.AP
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分数阶Hartree方程在调制空间和Fourier代数上的整体适定性

Global well-posedness for fractional Hartree equation on modulation spaces and Fourier algebra

Divyang G. Bhimani

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15 pages
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摘要：

在调制空间$M^{p,q}(\mathbb R^{d}).$中研究了具有三次卷积非线性项的分数阶薛定谔方程（$i\partial_t u - (-Δ)^{\fracα{2}}u\pm (K\ast |u|^2) u =0$）的Cauchy问题对于$K(x)= |x|^{-γ}$ $ (0< γ< \text{min} \{α, d/2\})$，当$α=2, d\geq 1$时，我们在$M^{p,q}(\mathbb R^{d}) (1\leq p \leq 2, 1\leq q < 2d/ (d+γ))$中得到了整体适定性结果，当$d\geq 2, \frac{2d}{2d-1}< α< 2. $时，我们在Fourier代数$\mathcal{F}L^1(\mathbb R^d) \cap L^2(\mathbb R^d).$中得到了类似的结果

英文摘要：

We study the Cauchy problem for fractional Schrödinger equation with cubic convolution nonlinearity ($i\partial_t u - (-Δ)^{\fracα{2}}u\pm (K\ast |u|^2) u =0$) with Cauchy data in the modulation spaces $M^{p,q}(\mathbb R^{d}).$ For $K(x)= |x|^{-γ}$ $ (0< γ< \text{min} \{α, d/2\})$, we establish global well-posedness results in $M^{p,q}(\mathbb R^{d}) (1\leq p \leq 2, 1\leq q < 2d/ (d+γ))$ when $α=2, d\geq 1$, and with radial Cauchy data when $d\geq 2, \frac{2d}{2d-1}< α< 2. $ Similar results are proven in Fourier algebra $\mathcal{F}L^1(\mathbb R^d) \cap L^2(\mathbb R^d).$

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