Conormal and piecewise smooth solutions to quasilinear wave equations

Abstract

In this paper, we show first that if a solution u u of the equation P 2 ( t , x , u , D u , D ) u = f ( t , x , u , D u ) {P_2}(t,x,u,Du,D)u = f(t,x,u,Du) , where P 2 ( t , x , u , D u , D ) {P_2}(t,x,u,Du,D) is a second order strictly hyperbolic quasilinear operator, is conormal with respect to a single characteristic hypersurface Σ \Sigma of P 2 {P_2} in the past and Σ \Sigma is smooth in the past, then Σ \Sigma is smooth and u u is conormal with respect to Σ \Sigma for all time. Second, let Σ 0 {\Sigma _0} and Σ 1 {\Sigma _1} be characteristic hypersurfaces of P 2 {P_2} which intersect transversally and let Γ = Σ 0 ∩ Σ 1 \Gamma = {\Sigma _0} \cap {\Sigma _1} . If Σ 0 {\Sigma _0} and Σ 1 {\Sigma _1} are smooth in the past and u u is conormal with repect to { Σ 0 , Σ 1 } \{ {\Sigma _0},{\Sigma _1}\} in the past, then Γ \Gamma is smooth, and u u is conormal with respect to { Σ 0 , Σ 1 } \{ {\Sigma _0},{\Sigma _1}\} locally in time outside of Γ \Gamma , even though Σ 0 {\Sigma _0} and Σ 1 {\Sigma _1} are no longer necessarily smooth across Γ \Gamma . Finally, we show that if u ( 0 , x ) u(0,x) and ∂ t u ( 0 , x ) {\partial _t}u(0,x) are in an appropriate Sobolev space and are piecewise smooth outside of Γ \Gamma , then u u is piecewise smooth locally in time outside of Σ 0 ∪ Σ 1 {\Sigma _0} \cup {\Sigma _1} .