On blow up for the energy super critical defocusing nonlinear Schrödinger equations

Abstract

<jats:title>Abstract</jats:title><jats:p>We consider the energy supercritical <jats:italic>defocusing</jats:italic> nonlinear Schrödinger equation <jats:disp-formula><jats:alternatives><jats: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:mi>i</mml:mi> <mml:msub> <mml:mi>∂</mml:mi> <mml:mi>t</mml:mi> </mml:msub> <mml:mi>u</mml:mi> <mml:mo>+</mml:mo> <mml:mi>Δ</mml:mi> <mml:mi>u</mml:mi> <mml:mo>-</mml:mo> <mml:mi>u</mml:mi> <mml:msup> <mml:mrow> <mml:mo>|</mml:mo> <mml:mi>u</mml:mi> <mml:mo>|</mml:mo> </mml:mrow> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:mtd> </mml:mtr> </mml:mtable> </mml:mrow> </mml:math></jats:alternatives></jats:disp-formula>in dimension <jats:inline-formula><jats:alternatives><jats:tex-math>$$d\ge 5$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>d</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>5</mml:mn> </mml:mrow> </mml:math></jats:alternatives></jats:inline-formula>. In a suitable range of energy supercritical parameters (<jats:italic>d</jats:italic>, <jats:italic>p</jats:italic>), we prove the existence of <jats:inline-formula><jats:alternatives><jats:tex-math>$${\mathcal {C}}^\infty $$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow> <mml:mi>C</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:italic>front mechanism</jats:italic>. Blow up is achieved by <jats:italic>compression</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-formula><jats:alternatives><jats:tex-math>$${\mathcal {C}}^\infty $$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow> <mml:mi>C</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>