On BV-instability and existence for linearized radial Euler flows

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.

Keywords

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