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On the existence of L2-valued thermodynamic entropy solutions for a hyperbolic system with boundary conditions

dc.contributor.authorMarchesani, Stefano
dc.contributor.authorOlla, Stefano
HAL ID: 18345
ORCID: 0000-0003-0845-1861
dc.date.accessioned2019-10-16T13:58:39Z
dc.date.available2019-10-16T13:58:39Z
dc.date.issued2020
dc.identifier.issn0360-5302
dc.identifier.urihttps://basepub.dauphine.fr/handle/123456789/20188
dc.language.isoenen
dc.subjecthyperbolic conservation laws
dc.subjectboundary conditions
dc.subjectweak solutions
dc.subjectvanishing viscos-ity
dc.subjectquasi-linear wave equation
dc.subjectvanishing viscosity
dc.subjectcompensated compactness
dc.subjectentropy solutions
dc.subjectClausius inequality
dc.subject.ddc515en
dc.titleOn the existence of L2-valued thermodynamic entropy solutions for a hyperbolic system with boundary conditions
dc.typeArticle accepté pour publication ou publié
dc.description.abstractenWe prove existence of L2-weak solutions of a quasilinear wave equation with boundary conditions. This describes the isothermal evolution of a one dimensional non-linear elastic material, attached to a fixed point on one side and subject to a force (tension) applied to the other side. The L2-valued solutions appear naturally when studying the hydrodynamic limit from a microscopic dynamics of a chain of anharmonic springs connected to a thermal bath. The proof of the existence is done using a vanishing viscosity approximation with extra Neumann boundary conditions added. In this setting we obtain a uniform a priori estimate in L2, allowing us to use L2 Young measures, together with the classical tools of compensated compactness. We then prove that the viscous solutions converge to weak solutions of the quasilinear wave equation strongly in Lp , for any p ∈ [1, 2), that satisfy, in a weak sense, the boundary conditions. Furthermore, these solutions satisfy the Clausius inequality: the change of the free energy is bounded by the work done by the boundary tension. In this sense they are the correct thermodynamic solutions, and we conjecture their uniqueness.
dc.publisher.cityParisen
dc.relation.isversionofjnlnameCommunications in Partial Differential Equations
dc.relation.isversionofjnldate2020
dc.relation.isversionofdoi10.1080/03605302.2020.1750426
dc.relation.isversionofjnlpublisherTaylor & Francis
dc.subject.ddclabelAnalyseen
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dc.description.audienceInternational
dc.relation.Isversionofjnlpeerreviewedoui
dc.date.updated2020-06-05T15:54:34Z


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