Sharp estimates for analytic pseudodifferential operators and application to Cauchy problems
This paper deals with sharp estimates for analytic pseudodifferential operators in complex tubular neighbourhoods of a compact analytic manifold \(\Gamma\). The operators mentioned above are defined by means of their distribution kernels and \(L^{\infty}\) and Hölder estimates in the complex domain are established. In order to study some classes of nonlinear Cauchy problems for analytic pseudodifferential operators the authors use Nishida's theorem in the framework of decreasing scale of Banach spaces as well as the inequalities just obtained. Results concerning existence and uniqueness of the analytic solutions are proved. Theorem III.1 can be considered as a specific generalization of the nonlinear Cauchy-Kovalevsky theorem for analytic pseudodifferential operators P. To illustrate the main statements consider the following simple example (III.5) \(\partial u_ 1/\partial t = f(t,x,u_1,u_2)\), \(\partial u_ 2/\partial t=Pf(t,x,u_ 1,u_ 2), u_ 1(0,x)=u_ 0(x),u_ 2(0,x)=Pu_ 0(x),\) where \(f_ 1,u_ 0\) are holomorphic functions. Then (III.5) possesses a unique analytic solution. - The authors study Euler's equations for nonviscous incompressible fluid at the end of the paper.
scientific article; zbMATH DE number 3849750