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We characterize the dynamics of the finite time blow up solutions withminimal mass for the focusing mass critical Hartree equation with$H^1(\mathbb{R}^4)$ data and $L^2(\mathbb{R}^4)$ data, where we make use of therefined Gagliardo-Nirenberg inequality of convolution type and the profiledecomposition. Moreover, we also analyze the mass concentration phenomenon ofsuch blow up solutions.

On the blow up phenomenon for the $L^2$-critical focusing Hartree equation in $\Bbb R^4$