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Low regularity solutions for the general quasilinear ultrahyperbolic Schrödinger equation

by B. Pineau and M. Taylor

(Report number 2023-42)

Abstract
We present a novel method for establishing large data local well-posedness in low regularity Sobolev spaces for general quasilinear Schrödinger equations with non-degenerate and nontrapping metrics. Our result represents a definitive improvement over the landmark results of Kenig, Ponce, Rolvung and Vega, as it weakens the regularity and decay assumptions to the same scale of spaces considered by Marzuola, Metcalfe and Tataru, but removes the uniform ellipticity assumption on the metric from their result. Our method has the additional benefit of being relatively simple but also very robust. In particular, it only relies on the use of pseudodifferential calculus for classical symbols.

Keywords: Quasilinear Schrödinger, ultrahyperbolic, local well-posedness.

@Techreport{PT23_1079,
  author = {B. Pineau and M. Taylor},
  title = {Low regularity solutions for the general quasilinear ultrahyperbolic Schr\"odinger equation},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2023-42},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2023/2023-42.pdf },
  year = {2023}
}

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