## Blowing up time-dependent solutions of the planar, Chern-Simons gauged nonlinear Schrödinger equation.(English)Zbl 0822.35125

Summary: Time-dependent solutions of the two-dimensional Chern-Simons gauged nonlinear Schrödinger equation are investigated in terms of an initial- value problem. We prove that this Cauchy problem is locally well posed in $$H^ 2 (\mathbb{R}^ 2)$$, and that global solutions exist in $$H^ 1 (\mathbb{R}^ 2)$$ provided that the initial data are small enough in $$L^ 2 (\mathbb{R}^ 2)$$. On the other hand, under certain conditions ensuring, for example a negative Hamiltonian, solutions blow up in a finite time which only depends on the initial data. The diverging shape of collapsing structure is finally discussed throughout a self-similar analysis.

### MSC:

 35Q55 NLS equations (nonlinear Schrödinger equations) 35B40 Asymptotic behavior of solutions to PDEs 81T13 Yang-Mills and other gauge theories in quantum field theory 81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics

### Keywords:

Cauchy problem; blow up; self-similar analysis
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