We consider NLS $$i\partial_t u + \Delta u + |u|^{p-1}u=0, u(x,0)=u_0 \in H^1(\mathbb R^d)$$ where $u:\mathbb R^{d+1} \to \mathbb C, u_0:\mathbb R^d \to \mathbb C, 1<p<\infty.$
Assume that $\int_{\mathbb R^d} |\nabla u_0(x)|^2 dx <b$ for some fix $b\in (0, \infty).$ Assume that there exists $p, d$ and the interal $0\in I $ such that solution $u(x,t)$ of the NLS exists such that $\int_{\mathbb R^{d}} |\nabla_xu(x,t)|^2 dx < \infty$ for all $t\in I.$
My Question is: Can we expect that $\int_{\mathbb R^{d}} |\nabla_xu(x,t)|^2 dx < b$ for all $t\in I$?