Ill-posedness of a quasilinear wave equation in two dimensions for data in H7∕4

We study the ill-posedness of a quasilinear wave equation. It was shown by Smith and Tataru in 2005 that for any $s > \frac{7}{4}$ (or $\frac{11}{4}$ in our situation), the equation is well-posed in $H^{s} \times H^{s - 1}$ . We show a sharpness result by exhibiting a quasilinear wave equation and an initial data such that the Cauchy problem is ill-posed for in $H^{11 ∕ 4} {(\ln H)}^{- β} \times H^{7 ∕ 4} {(\ln H)}^{- β}$ .

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Keywords

PDE, quasilinear wave, wave equation, ill-posedness, well-posedness, Sobolev spaces, analysis, fractional derivatives

Mathematical Subject Classification

Primary: 35A01, 35A30, 35L05

Secondary: 35A21, 35D30

Milestones

Received: 30 July 2021

Revised: 14 January 2023

Accepted: 12 March 2023

Published: 24 August 2023

Authors

Gaspard Ohlmann
University of Basel Basel Switzerland

Vol. 5, No. 3, 2023

Gaspard Ohlmann

Abstract

PDF Access Denied

Keywords

Mathematical Subject Classification

Milestones

Authors