Communications in Mathematical Sciences

Volume 20 (2022)

Number 8

On BV-instability and existence for linearized radial Euler flows

Pages: 2207 – 2230



Helge Kristian Jenssen (Department of Mathematics, Pennsylvania State University, University Park, Penn., U.S.A.)

Yushuang Luo (Department of Mathematics, Pennsylvania State University, University Park, Penn., U.S.A.)


We provide concrete examples of immediate BV-blowup from small and radially symmetric initial data for the 3-dimensional, linearized Euler system. More precisely, we exhibit data arbitrarily close to a constant state, measured in L-infinity and BV (functions of bounded variation), whose solution has unbounded BV-norm at any positive time. Furthermore, this type of BV-instability can occur in the absence of any focusing waves in the solution. We also show that the BV-norm of a solution may well remain bounded while suffering L-infinity blowup due to wave focusing. Finally, we demonstrate how an argument based on scaling of the dependent variables, together with 1-d variation estimates, yields global existence for a class of finite energy, but possibly unbounded, radial solutions.


multi-dimensional systems of hyperbolic PDEs, radial solutions, BV-instability

2010 Mathematics Subject Classification

35B44, 35L45

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This work was supported in part by NSF awards DMS-1311353 and DMS-1813283 (Jenssen).

Received 10 October 2021

Received revised 20 February 2022

Accepted 1 March 2022

Published 29 November 2022