Gevrey micro-regularity for solutions to first order nonlinear PDE

Let and . We study the Gevrey micro-regularity of solutions u of the nonlinear equationut=f(x,t,u,ux), where f(x,t,ζ0,ζ) is a Gevrey function of order s>1 and holomorphic in (ζ0,ζ). We show that the Gevrey wave-front set of any C2 solution u is contained in the characteristic set of the linearized operator 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.