Gevrey micro-regularity for solutions to first order nonlinear PDE

Let (x,t)∈Rm×R(x,t)∈Rm×R and u∈C2(Rm×R)u∈C2(Rm×R). We study the Gevrey micro-regularity of solutions u of the nonlinear equationut=f(x,t,u,ux),ut=f(x,t,u,ux), where f(x,t,ζ0,ζ)f(x,t,ζ0,ζ) is a Gevrey function of order s>1s>1 and holomorphic in (ζ0,ζ)(ζ0,ζ). We show that the Gevrey wave-front set of any C2C2 solution u is contained in the characteristic set of the linearized operatorLu=∂∂t−∑j=1m∂f∂ζj(x,t,u,ux)∂∂xj. To achieve this, we study the notion of Gevrey approximate solutions, a concept which we believe is of independent interest and could be applied to much more general situations.