Existence of Gevrey approximate solutions for certain systems of linear vector fields applied to involutive systems of first-order nonlinear PDEs (Q549755)

The authors consider a \(G^s\)-involutive structure of rank \(n\) in a manifold of dimension \(m+n\). In local coordinates: \[ L_j= \partial/\partial t_j+ \sum^m_{k=1} a_{jk}(x, t)\partial/\partial x_k,\quad j= 1,\dots, n, \] where \(a_{jk}\) belong to the Gevrey space \(G^s\), \(s> 1\), and \([L_i,L_j]= 0\) for \(0\leq i\), \(j\leq n\). A formal solution \(u(x,t)\in G^s\) is constructed for the local Gevrey problem \(u(x,0)= f(x)\in G^s\), \(L_ju= 0\), \(j= 1,\dots, n\). Other interesting results concern the related nonlinear systems: \[ u_{t_j}= F_j(x,t,u,u_x),\quad j= 1,\dots, n, \] for which the authors prove results of propagation of Gevrey wave front sets. For a counterpart of these results in the analytic category see [\textit{N. Hanges} and \textit{F. Treves}, Trans. Am. Math. Soc. 331, No. 2, 627--638 (1992; Zbl 0758.35018)].