TY - JOUR

T1 - ON BV-INSTABILITY AND EXISTENCE FOR LINEARIZED RADIAL EULER FLOWS

AU - Jenssen, Helge Kristian

AU - Luo, Yushuang

N1 - Funding Information:
This work was supported in part by NSF awards DMS-1311353 and DMS-1813283 (Jenssen).
Funding Information:
in part by NSF awards DMS-
Publisher Copyright:
© 2022 International Press

PY - 2022

Y1 - 2022

N2 - 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.

AB - 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.

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U2 - 10.4310/CMS.2022.v20.n8.a4

DO - 10.4310/CMS.2022.v20.n8.a4

M3 - Article

AN - SCOPUS:85143863547

SN - 1539-6746

VL - 20

SP - 2207

EP - 2230

JO - Communications in Mathematical Sciences

JF - Communications in Mathematical Sciences

IS - 8

ER -