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We consider the energy supercritical defocusing nonlinear Schr\"odingerequation $i\partial_tu+\Delta u-u|u|^{p-1}=0$ in dimension $d\ge 5$. In asuitable range of energy supercritical parameters $(d,p)$, we prove theexistence of $\mathcal C^\infty$ well localized spherically symmetric initialdata such that the corresponding unique strong solution blows up in finitetime. Unlike other known blow up mechanisms, the singularity formation does notoccur by concentration of a soliton or through a self similar solution, whichare unknown in the defocusing case, but via a front mechanism. Blow up isachieved by compression in the associated hydrodynamical flow which in turnproduces a highly oscillatory singularity. The front blow up profile is chosenamong the countable family of $\mathcal C^\infty$ spherically symmetric selfsimilar solutions to the compressible Euler equation whose existence andproperties in a suitable range of parameters are established in the companionpaper \cite{MRRSprofile}.

On blow up for the energy super critical defocusing non linear Schr\"odinger equations