jats:titleAbstract</jats:title>jats:pWe consider the energy supercritical jats:italicdefocusing</jats:italic> nonlinear Schrödinger equation jats:disp-formulajats:alternativesjats:tex-math$$\begin{aligned} i\partial _tu+\Delta u-u|u|^{p-1}=0 \end{aligned}$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> mml:mrow mml:mtable mml:mtr mml:mtd mml:mrow mml:mii</mml:mi> mml:msub mml:mi∂</mml:mi> mml:mit</mml:mi> </mml:msub> mml:miu</mml:mi> mml:mo+</mml:mo> mml:miΔ</mml:mi> mml:miu</mml:mi> mml:mo-</mml:mo> mml:miu</mml:mi> mml:msup mml:mrow mml:mo|</mml:mo> mml:miu</mml:mi> mml:mo|</mml:mo> </mml:mrow> mml:mrow mml:mip</mml:mi> mml:mo-</mml:mo> mml:mn1</mml:mn> </mml:mrow> </mml:msup> mml:mo=</mml:mo> mml:mn0</mml:mn> </mml:mrow> </mml:mtd> </mml:mtr> </mml:mtable> </mml:mrow> </mml:math></jats:alternatives></jats:disp-formula>in dimension jats:inline-formulajats:alternativesjats:tex-math$$d\ge 5$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> mml:mrow mml:mid</mml:mi> mml:mo≥</mml:mo> mml:mn5</mml:mn> </mml:mrow> </mml:math></jats:alternatives></jats:inline-formula>. In a suitable range of energy supercritical parameters (jats:italicd</jats:italic>, jats:italicp</jats:italic>), we prove the existence of jats:inline-formulajats:alternativesjats:tex-math$${\mathcal {C}}^\infty $$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> mml:msup mml:mrow mml:miC</mml:mi> </mml:mrow> mml:mi∞</mml:mi> </mml:msup> </mml:math></jats:alternatives></jats:inline-formula> well localized spherically symmetric initial data such that the corresponding unique strong solution blows up in finite time. Unlike other known blow up mechanisms, the singularity formation does not occur by concentration of a soliton or through a self similar solution, which are unknown in the defocusing case, but via a jats:italicfront mechanism</jats:italic>. Blow up is achieved by jats:italiccompression</jats:italic> for the associated hydrodynamical flow which in turn produces a highly oscillatory singularity. The front blow up profile is chosen among the countable family of jats:inline-formulajats:alternativesjats:tex-math$${\mathcal {C}}^\infty $$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> mml:msup mml:mrow mml:miC</mml:mi> </mml:mrow> mml:mi∞</mml:mi> </mml:msup> </mml:math></jats:alternatives></jats:inline-formula> spherically symmetric self similar solutions to the compressible Euler equation whose existence and properties in a suitable range of parameters are established in the companion paper (Merle et al. in Preprint (2019)) under a non degeneracy condition which is checked numerically.</jats:p>