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On the optimal Sobolev threshold for evolution equations with rough nonlinearities

by B. Pineau and M. Taylor

(Report number 2025-18)

Abstract
In this article we are concerned with evolution equations of the form \begin{equation*} \partial_tu-A(D)u=F(u,\overline{u},\nabla u, \nabla \overline{u}) \end{equation*} where \(A(D)\) is a Fourier multiplier of either dispersive or parabolic type and the nonlinear term \(F\) is of limited regularity. Our objective is to develop a robust set of principles which can be used in many cases to predict the highest Sobolev exponent \(s=s(q,d)\) for which the above evolution is well-posed in \(W_x^{s,q}(\mathbb{R}^d)\) (necessarily restricting to \(q=2\) for dispersive problems). We will confirm the validity of these principles for two of the most important model problems; namely, the nonlinear Schrodinger and heat equations.

Keywords: Nonlinear Schrodinger equation; nonlinear heat equation; well-posedness; rough nonlinearity.

@Techreport{PT25_1139,
  author = {B. Pineau and M. Taylor},
  title = {On the optimal Sobolev threshold for evolution equations with rough nonlinearities},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2025-18},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2025/2025-18.pdf },
  year = {2025}
}

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