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Hamiltonian Shocks and Other Singular Fronts in Hyperbolic Systems of Conservation Laws.

Record Type:	Language materials, manuscript : Monograph/item
Title/Author:	Hamiltonian Shocks and Other Singular Fronts in Hyperbolic Systems of Conservation Laws./
Author:	Arnold, Russell.
Description:	1 online resource (105 pages)
Notes:	Source: Dissertations Abstracts International, Volume: 84-11, Section: B.
Contained By:	Dissertations Abstracts International84-11B.
Subject:	Applied mathematics. -
Online resource:	click for full text (PQDT)
ISBN:	9798379552886

Hamiltonian Shocks and Other Singular Fronts in Hyperbolic Systems of Conservation Laws.
Arnold, Russell.

Hamiltonian Shocks and Other Singular Fronts in Hyperbolic Systems of Conservation Laws. - 1 online resource (105 pages)

Source: Dissertations Abstracts International, Volume: 84-11, Section: B.

Thesis (Ph.D.)--The University of North Carolina at Chapel Hill, 2023.

Includes bibliographical references

The nature of wave interaction in a continuum dynamical model may undergo a qualitative change in certain asymptotic regimes, most notably when linearity or complete integrability is introduced. This occurs in particular when the mKdV equation is used to model the unidirectional dispersive dynamics of two layer shallow water fluid flow near a critical interfacial height. Motivated by the symmetric properties of conjugate states which have been observed for the MCC equations in the Boussinesq limit, this work elucidates a more subtle qualitative shift, residing purely in the dispersionless reduction of a 2x2 system, which determines whether a Hamiltonian undercompressive shock, representing a profile connecting two conjugate states, may interact with a continuous background wave without producing a loss of regularity, which would take the form of a classical dispersive shock. The resulting criterion is also related to an infinitude of conservation laws, drawing a further parallel to the integrable case.Then, motivated by the study of shallow water fluid flow, criteria are derived for the splitting of corner points in the initial conditions of a solution to a one dimensional quasilinear hyperbolic system of conservation laws. To this end, a distributional approach to moving singularities is elaborated. Then the class of systems admitting solutions with persisting infinite derivatives is shown to coincide with the class for which genuine nonlinearity does not hold uniformly and fails at such singular points in particular. In both cases the application to problems in fluid flow is demonstrated in the context of explicit solutions.

Electronic reproduction.
Ann Arbor, Mich. :
ProQuest,
2024

Mode of access: World Wide Web

ISBN: 9798379552886Subjects--Topical Terms:

1069907
Applied mathematics.
Subjects--Index Terms:

Asymptotic regimesIndex Terms--Genre/Form:

554714
Electronic books.

Hamiltonian Shocks and Other Singular Fronts in Hyperbolic Systems of Conservation Laws.
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500 $a Source: Dissertations Abstracts International, Volume: 84-11, Section: B.
500 $a Advisor: Camassa, Roberto.
502 $a Thesis (Ph.D.)--The University of North Carolina at Chapel Hill, 2023.
504 $a Includes bibliographical references
520 $a The nature of wave interaction in a continuum dynamical model may undergo a qualitative change in certain asymptotic regimes, most notably when linearity or complete integrability is introduced. This occurs in particular when the mKdV equation is used to model the unidirectional dispersive dynamics of two layer shallow water fluid flow near a critical interfacial height. Motivated by the symmetric properties of conjugate states which have been observed for the MCC equations in the Boussinesq limit, this work elucidates a more subtle qualitative shift, residing purely in the dispersionless reduction of a 2x2 system, which determines whether a Hamiltonian undercompressive shock, representing a profile connecting two conjugate states, may interact with a continuous background wave without producing a loss of regularity, which would take the form of a classical dispersive shock. The resulting criterion is also related to an infinitude of conservation laws, drawing a further parallel to the integrable case.Then, motivated by the study of shallow water fluid flow, criteria are derived for the splitting of corner points in the initial conditions of a solution to a one dimensional quasilinear hyperbolic system of conservation laws. To this end, a distributional approach to moving singularities is elaborated. Then the class of systems admitting solutions with persisting infinite derivatives is shown to coincide with the class for which genuine nonlinearity does not hold uniformly and fails at such singular points in particular. In both cases the application to problems in fluid flow is demonstrated in the context of explicit solutions.
533 $a Electronic reproduction. $b Ann Arbor, Mich. : $c ProQuest, $d 2024
538 $a Mode of access: World Wide Web
650 4 $a Applied mathematics. $3 1069907
650 4 $a Fluid mechanics. $3 555551
653 $a Asymptotic regimes
653 $a Conjugate states
653 $a Dispersive shock
653 $a Hyperbolic system
653 $a Conservation laws
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690 $a 0204
710 2 $a ProQuest Information and Learning Co. $3 1178819
710 2 $a The University of North Carolina at Chapel Hill. $b Mathematics. $3 1180561
773 0 $t Dissertations Abstracts International $g 84-11B.
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