论文标题
在关键案例中，关于不均匀的哈特里方程的适应性良好
On well-posedness for inhomogeneous Hartree equations in the critical case
论文作者
论文摘要
我们研究了不均匀的hartree方程$ i \ partial_t u +Δu=λ（I_α\ ast | \ cdot | \ cdot |^{ - b} | u |^p）| x |^{ - b} |直到最近，它的体积良好的理论一直在深入研究，重点是解决关键指数$ p = 1+\ frac {2-2b+α} {N-2S} $的问题，并使用$ 0 \ le s \ le 1 $，但是情况$ 1/2 \ leq s \ leq s \ leq s \ leq s \ leq 1 $仍然是一个开放的问题。在本文中，在这种情况下，我们发展了体系良好的理论，尤其是包括关键的案例。为此，我们基于Sobolev-Lorentz空间来解决此问题，这可以使我们对此方程进行更精细的分析。这是因为它可以控制涉及奇异性$ | x |^{ - b} $的非线性以及Riesz潜力$I_α$更有效。
We study the well-posedness for the inhomogeneous Hartree equation $i\partial_t u + Δu = λ(I_α\ast |\cdot|^{-b}|u|^p)|x|^{-b}|u|^{p-2}u$ in $H^s$, $s\ge0$. Until recently, its well-posedness theory has been intensively studied, focusing on solving the problem for the critical index $p=1+\frac{2-2b+α}{n-2s}$ with $0\le s \le 1$, but the case $1/2\leq s \leq 1$ is still an open problem. In this paper, we develop the well-posedness theory in this case, especially including the energy-critical case. To this end, we approach to the matter based on the Sobolev-Lorentz space which can lead us to perform a finer analysis for this equation. This is because it makes it possible to control the nonlinearity involving the singularity $|x|^{-b}$ as well as the Riesz potential $I_α$ more effectively.
