Revisit of the Faddeev Model in Dimension Two

Abstract

The Faddeev model is a fundamental model in relativistic quantum field theory used to model elementary particles. The Faddeev model can be regarded as a system of non-linear wave equations with both quasi-linear and semi-linear non-linearities, which is particularly challenging in two space dimensions. A key feature of the system is that there exist undifferentiated wave components in the non-linearities, which somehow causes extra difficulties. Nevertheless, the Cauchy problem in two space dimenions was tackled by Lei-Lin-Zhou (2011) with small, regular, and compactly supported initial data, using Klainerman’s vector field method enhanced by a novel angular-radial anisotropic technique. In the present paper, the authors revisit the Faddeev model and remove the compactness assumptions on the initial data by Lei-Lin-Zhou (2011). The proof relies on an improved L ² norm estimate of the wave components in Theorem 3.1 and a decomposition technique for non-linearities of divergence form.

Keywords

Faddeev model in ℝ¹⁺² / Global existence / Null condition

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