# Conormal Regularity after Nonlinear Interaction

## Abstract

When more than a pair of characteristic hypersurfaces carrying conormal singularities for a solution to a nonlinear problem intersect transversally in a lower dimensional manifold, new singularities can form, even for a second order equation. An example exhibiting this phenomenon was constructed in Rauch-Reed [60], with singularities as indicated in Figure 4.1. The solution *u* to a semilinear wave equation □*u* = *f(t,x,y,u)* in two space dimensions is conormal in the past with respect to a triple of characteristic hyperplanes which intersect transversally at the origin. A new singularity is present at later times on the surface of the light cone over the origin. The nature of this singularity will be analyzed in detail below. Its presence is not surprising: any definition of conormal space for this geometry would include functions with wavefront sets over the origin having (τ, ξ,η) projections which include three linearly independent directions. An algebra of such functions would include *u* for which *WF(u)*⊃ {(0,0,0, τ, ξ,η): (τ, ξ,η) ≠ 0}. Hörmander’s Theorem would then allow for the propagation of such singularities onto the surface of the forward light cone. These are the only such singularities which arise, and they are also conormal, as shall also be established.

## Keywords

Vector Field Nonlinear Interaction Light Cone Degree Zero Triple Intersection## Preview

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