Decay of solutions of nonlinear Dirac equations
A talk in the Oberseminar Analysis series by
Christopher Maulén from Bielefeld
| Abstract: | In this talk, we explore the long-time dynamics of solutions to nonlinear Dirac-type equations across several regimes. In one dimension, we show that every global $L^2$
solution of the massless Dirac equation decays to zero in time within a region expanding at a rate proportional to $log^{-2}t$
, without any smallness assumptions on the initial data or restrictions on the power of the nonlinearity. This rules out the existence of localized, oscillatory, or breather-like structures in this setting. These methods extend to three dimensions, yielding decay under boundedness of the $H^1$ norm on certain partial wave subspace. Finally, in arbitrary dimensions, we establish $L^2$ decay in the exterior of the light cone, confirming that nonlinear Dirac fields cannot support superluminal propagation. The analysis is driven by a family of weighted virial identities that reveal the dispersive structure of the Dirac operator. If time permits, we will also discuss recent progress toward the two-dimensional case. This talk is based on joint work with S. Herr and C. Muñoz. Within the CRC this talk is associated to the project(s): A1 |