On the focusing mass-critical nonlinear fourth-order Schrodinger equation below the energy space

International audienceIn this paper, we consider the focusing mass-critical nonlinear fourth-order Schrödinger equation. We prove that blowup solutions to this equation with initial data in $H^\gamma(\mathbb{R}^d), 5 \leq d \leq 7, \frac{56−3d+ \sqrt{137d^2 +1712d+3136}}{2(2d+32)} < \gamma < 2$ concentrate at least the mass of the ground state at the blowup time. This extends the work in [35] where Zhu-Yang-Zhang studied the formation of singularity for the equation with rough initial data in $\mathbb{R}^4$. We also prove that the equation is globally well-posed with initial data $u_0 \in H^\gamma(\mathbb{R}^d), 5 \leq d \leq 7, \frac{8d}{3d+8} < \gamma < 2$ satisfying $\|u_0\| _{L^2 (\mathbb{R}^d)} < \|Q\|_{L^2 (\mathbb{R}^d)}$, where $Q$ is the solution to the ground state equation