Plane-wave analysis of a hyperbolic system of equations with relaxation in R^d

Maarten V. de Hoop; Jian-Guo Liu; Peter A. Markowich; Nail S. Ussembayev

doi:10.4310/cms.2019.v17.n1.a3

Plane-wave analysis of a hyperbolic system of equations with relaxation in R^d

Maarten V. de Hoop, Jian-Guo Liu, Peter A. Markowich, Nail S. Ussembayev

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Abstract

We consider a multi-dimensional scalar wave equation with memory corresponding to the viscoelastic material described by a generalized Zener model. We deduce that this relaxation system is an example of a non-strictly hyperbolic system satisfying Majda’s block structure condition. Wellposedness of the associated Cauchy problem is established by showing that the symbol of the spatial derivatives is uniformly diagonalizable with real eigenvalues. A long-time stability result is obtained by plane-wave analysis when the memory term allows for dissipation of energy.

Original language	English (US)
Pages (from-to)	61-79
Number of pages	19
Journal	Communications in Mathematical Sciences
Volume	17
Issue number	1
DOIs	https://doi.org/10.4310/cms.2019.v17.n1.a3
State	Published - 2019

Bibliographical note

KAUST Repository Item: Exported on 2020-10-01
Acknowledgements: M.V.d.H. gratefully acknowledges support from the Simons Foundation under the MATH + X program, the National Science Foundation under grant DMS-1559587, and the corporate members of the GeoMathematical Group at Rice University. J.-G.L. is supported by the National Science Foundation under grant DMS-1812573 and KI-Net RNMS11-07444.

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10.4310/cms.2019.v17.n1.a3

https://repository.kaust.edu.sa/bitstream/10754/655965/1/CMS-2019-0017-0001-a003.pdf

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title = "Plane-wave analysis of a hyperbolic system of equations with relaxation in R^d",

abstract = "We consider a multi-dimensional scalar wave equation with memory corresponding to the viscoelastic material described by a generalized Zener model. We deduce that this relaxation system is an example of a non-strictly hyperbolic system satisfying Majda{\textquoteright}s block structure condition. Wellposedness of the associated Cauchy problem is established by showing that the symbol of the spatial derivatives is uniformly diagonalizable with real eigenvalues. A long-time stability result is obtained by plane-wave analysis when the memory term allows for dissipation of energy.",

author = "{de Hoop}, {Maarten V.} and Jian-Guo Liu and Markowich, {Peter A.} and Ussembayev, {Nail S.}",

note = "KAUST Repository Item: Exported on 2020-10-01 Acknowledgements: M.V.d.H. gratefully acknowledges support from the Simons Foundation under the MATH + X program, the National Science Foundation under grant DMS-1559587, and the corporate members of the GeoMathematical Group at Rice University. J.-G.L. is supported by the National Science Foundation under grant DMS-1812573 and KI-Net RNMS11-07444.",

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doi = "10.4310/cms.2019.v17.n1.a3",

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AU - de Hoop, Maarten V.

AU - Liu, Jian-Guo

AU - Markowich, Peter A.

AU - Ussembayev, Nail S.

N1 - KAUST Repository Item: Exported on 2020-10-01 Acknowledgements: M.V.d.H. gratefully acknowledges support from the Simons Foundation under the MATH + X program, the National Science Foundation under grant DMS-1559587, and the corporate members of the GeoMathematical Group at Rice University. J.-G.L. is supported by the National Science Foundation under grant DMS-1812573 and KI-Net RNMS11-07444.

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Y1 - 2019

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AB - We consider a multi-dimensional scalar wave equation with memory corresponding to the viscoelastic material described by a generalized Zener model. We deduce that this relaxation system is an example of a non-strictly hyperbolic system satisfying Majda’s block structure condition. Wellposedness of the associated Cauchy problem is established by showing that the symbol of the spatial derivatives is uniformly diagonalizable with real eigenvalues. A long-time stability result is obtained by plane-wave analysis when the memory term allows for dissipation of energy.

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JO - Communications in Mathematical Sciences

JF - Communications in Mathematical Sciences

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Plane-wave analysis of a hyperbolic system of equations with relaxation in R^d

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