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.

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Work Title On BV-instability and existence for linearized radial Euler flows
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Open Access
Creators
  1. Helge Kristian Jenssen
  2. Yushuang Luo
Keyword
  1. Multi-dimensional systems of hyperbolic PDEs
  2. Radial solutions
  3. BV-instability
License In Copyright (Rights Reserved)
Work Type Article
Publisher
  1. Communications in Mathematical Sciences
Publication Date January 1, 2022
Publisher Identifier (DOI)
  1. https://doi.org/10.4310/CMS.2022.v20.n8.a4
Deposited January 06, 2023

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    Work Title
    • ON BV-INSTABILITY AND EXISTENCE FOR LINEARIZED RADIAL EULER FLOWS
    • On BV-instability and existence for linearized radial Euler flows
    Keyword
    • Multi-dimensional systems of hyperbolic PDEs, Radial solutions, BV-instability
    Description
    • <p>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.</p>
    • 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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