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We study the existence of left invariant closed $G_2$-structures defining aRicci soliton metric on simply connected nonabelian nilpotent Lie groups.  For each one of these $G_2$-structures, we show long time existence anduniqueness of solution for the Laplacian flow on the noncompact manifold.Moreover, considering the Laplacian flow on the associated Lie algebra as abracket flow on $\R^7$ in a similar way as in [23] we prove that the underlyingmetrics $g(t)$ of the solution converge smoothly, up to pull-back bytime-dependent diffeomorphisms, to a flat metric, uniformly on compact sets inthe nilpotent Lie group, as $t$ goes to infinity.

Laplacian flow of closed $G_2$-structures inducing nilsolitons