Hyperbolic balance laws with a dissipative non local source

Colombo, Rinaldo M.; Guerra, Graziano

doi:10.3934/cpaa.2008.7.1077

Cited by 18 publications

(17 citation statements)

References 17 publications

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“…The above result is only local in time, mostly due to the source term which is far from being dissipative or diagonally dominant, see for instance [11] or [15,Sect. 13.9].…”

Section: Analytical Study Of the Modelmentioning

confidence: 98%

A hyperbolic model for granular flow

Cattani¹,

Colombo²,

Guerra³

2011

Z Angew Math Mech

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We present a model that describes the motion of some granular material sliding along a slope. During this movement, both erosion and deposition may take place, depending on the speed of the sliding material. Analytically, this model consists of a hyperbolic system of partial differential equations. In the 1D case, the resulting system of balance laws displays interesting behavior. Its convective part gives rise to a 3 × 3 globally well defined Riemann Problem, in spite of the appearance of vacuum and of the lack of strict hyperbolicity. Several numerical integrations show various features of this model.

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“…The above result is only local in time, mostly due to the source term which is far from being dissipative or diagonally dominant, see for instance [11] or [15,Sect. 13.9].…”

Section: Analytical Study Of the Modelmentioning

confidence: 98%

A hyperbolic model for granular flow

Cattani¹,

Colombo²,

Guerra³

2011

Z Angew Math Mech

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“…This is the usual assumption on G, which also covers non-local terms [7,8] as well as real applications [9]. Next we define the Cauchy problem at junctions, which corresponds to our special set of coupling conditions, and weak solutions.…”

Section: The Cauchy Problem At the Junctionmentioning

confidence: 99%

Entropy-Preserving Coupling of Hierarchical Gas Models

Mindt¹,

Lang²,

Domschke³

2019

SIAM J. Math. Anal.

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This paper is concerned with coupling conditions at junctions for transport models which differ in their fidelity to describe transient flow in gas pipelines. It also includes the integration of compressors between two pipes with possibly different models. A hierarchy of three one-dimensional gas transport models is built through the 3 × 3 polytropic Euler equations, the 2 × 2 isentropic Euler equations and a simplified version of it for small velocities. To ensure entropy preservation, we make use of the novel entropy-preserving coupling conditions recently proposed by Lang and Mindt [Netw. Heterog. Media, 13:177-190, 2018] and require the equality of the total enthalpy at the junction and that the specific entropy for pipes with outgoing flow equals the convex combination of all entropies that belong to pipes with incoming flow. We prove the existence and uniqueness of solutions to generalised Riemann problems at a junction in the neighbourhood of constant coupling functions and stationary states which belong to the subsonic region. This provides the basis for the well-posedness of certain Cauchy problems for initial data with sufficiently small total variation.

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“…(45) This is the usual assumption on G, which also covers non-local terms [6,7] as well as real applications [8].…”

Section: The Cauchy Problem At the Junctionmentioning

confidence: 99%

Entropy-preserving coupling conditions for one-dimensional Euler systems at junctions

Lang¹,

Mindt²

2018

Networks &Amp; Heterogeneous Media

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This paper is concerned with a set of novel coupling conditions for the 3 × 3 one-dimensional Euler system with source terms at a junction of pipes with possibly different cross-sectional areas. Beside conservation of mass, we require the equality of the total enthalpy at the junction and that the specific entropy for pipes with outgoing flow equals the convex combination of all entropies that belong to pipes with incoming flow. Previously used coupling conditions include equality of pressure or dynamic pressure. They are restricted to the special case of a junction having only one pipe with outgoing flow direction. Recently, Reigstad [SIAM J. Appl. Math., 75:679-702, 2015] showed that such pressure-based coupling conditions can produce non-physical solutions for isothermal flows through the production of mechanical energy. Our new coupling conditions ensure energy as well as entropy conservation and also apply to junctions connecting an arbitrary number of pipes with flexible flow directions. We prove the existence and uniqueness of solutions to the generalised Riemann problem at a junction in the neighbourhood of constant stationary states which belong to the subsonic region. This provides the basis for the well-posedness of the homogeneous and inhomogeneous Cauchy problems for initial data with sufficiently small total variation.

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Hyperbolic balance laws with a dissipative non local source

Cited by 18 publications

References 17 publications

A hyperbolic model for granular flow

A hyperbolic model for granular flow

Entropy-Preserving Coupling of Hierarchical Gas Models

Entropy-preserving coupling conditions for one-dimensional Euler systems at junctions

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