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Approximate solutions of vector fields and an application to Denjoy–Carleman regularity of solutions of a nonlinear PDE

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  • Nicholas Braun Rodrigues
  • Antonio Victor da Silva

Abstract

In this paper we study microlocal regularity of a C2‐solution u of the equation ut=f(x,t,u,ux),where f(x,t,ζ0,ζ) is ultradifferentiable in the variables (x,t)∈RN×R and holomorphic in the variables (ζ0,ζ)∈C×CN. We proved that if CM is a regular Denjoy–Carleman class (including the quasianalytic case) then: WFM(u)⊂Char(Lu),where WFM(u) is the Denjoy–Carleman wave‐front set of u and Char(Lu) is the characteristic set of the linearized operator Lu: Lu=∂∂t−∑j=1N∂f∂ζj(x,t,u,ux)∂∂xj.

Suggested Citation

  • Nicholas Braun Rodrigues & Antonio Victor da Silva, 2021. "Approximate solutions of vector fields and an application to Denjoy–Carleman regularity of solutions of a nonlinear PDE," Mathematische Nachrichten, Wiley Blackwell, vol. 294(8), pages 1452-1471, August.
    • Handle: RePEc:bla:mathna:v:294:y:2021:i:8:p:1452-1471
    DOI: 10.1002/mana.201800516
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