## A UNIQUENESS CONDITION FOR NONLINEAR CONVECTION-DIFFUSION EQUATIONS WITH DISCONTINUOUS COEFFICIENTS (2008)

Venue: | JOURNAL OF HYPERBOLIC DIFFERENTIAL EQUATIONS |

Citations: | 12 - 1 self |

### BibTeX

@MISC{Diehl08auniqueness,

author = {Stefan Diehl},

title = {A UNIQUENESS CONDITION FOR NONLINEAR CONVECTION-DIFFUSION EQUATIONS WITH DISCONTINUOUS COEFFICIENTS},

year = {2008}

}

### OpenURL

### Abstract

The paper focuses on the uniqueness issue for scalar convection-diffusion equations where both the convective flux and diffusion functions have a spatial discontinuity. An interface entropy condition is proposed at such a spatial discontinuity. It implies the Kruˇzkov-type entropy condition presented by Karlsen et al. [Trans. Royal Norwegian Society Sci. Letters 3, 49 pp, 2003]. They proved uniqueness when the convective flux function satisfies an additional ‘crossing condition’. The crossing condition becomes redundant with the entropy condition proposed here. Thereby, more general flux functions are allowed. Another advantage of the entropy condition is its simple geometrical interpretation, which facilitates the construction of stationary solutions.

### Citations

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Citation Context ...), respectively. The presence of this space-discontinuous diffusion term has been the motivation for the analysis in the present paper. The results of [15] are based on the analyses by Karlsen et al. =-=[34,35]-=- and Bürger et al. [14]. 1.3. Two entropy concepts A general result on uniqueness was obtained by Karlsen et al. [35], who analyzed the initial-value problem for the strongly degenerate parabolic equa... |

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order quasilinear equations in several independent variables
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Citation Context ...bolic equation in which an application-specific nonlinear diffusion term vanishes. With this new flux-connection-adapted-entropy (FCAE) condition at x = 0 and the standard Kruˇzkov entropy condition (=-=[40]-=-) away from x = 0, Bürger et al. [16] establish uniqueness and existence (for the hyperbolic case D = 0) and generalize the results in [2]. There are some advantages with the FCAE condition. The cross... |

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Citation Context ...several other suitable applications except sedimentation). We note that the uniqueness issue for CTE solutions has been widely studied, mostly in the hyperbolic special case when D = 0 in (1.1), e.g. =-=[7,14,15,18,19,20,27,30,31,32,35,36,37,38,39,44,45]-=-. In the notation of (1.1), the crossing condition in [35] imposes conditions on f and g, for example, that the graphs of f and g may intersect at most at one point. The crossing condition has also be... |

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Citation Context ... applications. When D = 0, Equation (1.1) becomes a first-order hyperbolic model, a conservation law with a discontinuous flux, which appears in the modelling of two-phase flow in heterogeneous media =-=[31,33]-=-, traffic flow with abruptly changing surface conditions, or number of lanes, [9,10,41], and continuous sedimentation in a clarifier-thickener unit [13,14,19,21]. In these three areas of application i... |

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Citation Context ...several other suitable applications except sedimentation). We note that the uniqueness issue for CTE solutions has been widely studied, mostly in the hyperbolic special case when D = 0 in (1.1), e.g. =-=[7,14,15,18,19,20,27,30,31,32,35,36,37,38,39,44,45]-=-. In the notation of (1.1), the crossing condition in [35] imposes conditions on f and g, for example, that the graphs of f and g may intersect at most at one point. The crossing condition has also be... |

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Citation Context ...t treat the existence problem, but merely assume that there exists a weak entropy solution of (1.1) (in a way that is specified below). For the equations that are treated by Bürger, Karlsen et al. in =-=[11,14,15,34,35,36]-=-, the proofs of existence are carried out by showing convergence of different numerical methods. We refer to those papers for substantial reviews and references of the previous results on uniqueness, ... |

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Citation Context ...ū]. The definitions of ˆ f, ˇg, P, N and c then yield that (u+, u−) is a fixed point of version 2. Remark 3.3. In the special case f = g, Condition Γ implies the standard entropy condition by Oleinik =-=[43]-=-, which for a stationary discontinuity can be written (u+ − u−) ( f(v) − η ) ≥ 0 ∀v ∈ ch(u−, u+) , where η = f(u−) = f(u+) . (3.9) If u− = u+, then both (3.9) and Condition Γ are satisfied trivially. ... |

38 | An Engquist-Osher-type scheme for conservation laws with discontinuous flux adapted to flux connections - Bürger, Karlsen, et al. |

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Existence and uniqueness of entropy solution of scalar conservation laws with a flux function involving discontinuous coefficients
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Citation Context ...), respectively. The presence of this space-discontinuous diffusion term has been the motivation for the analysis in the present paper. The results of [15] are based on the analyses by Karlsen et al. =-=[34,35]-=- and Bürger et al. [14]. 1.3. Two entropy concepts A general result on uniqueness was obtained by Karlsen et al. [35], who analyzed the initial-value problem for the strongly degenerate parabolic equa... |

30 |
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Citation Context ...r interface entropy inequality involves both a problem-specific ‘flux connection’, introduced by Adimurthi et al. [2], and a single ‘adapted Kruˇzkov entropy’ of the type used by Audusse and Perthame =-=[6]-=-. Audusse and Perthame introduced an infinite family of adapted entropies and showed uniqueness without an interface condition and without any assumption on the traces of the solution at x = 0. In a s... |

26 | Monotone difference approximations of BV solutions to degenerate convectiondiffusion equations
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Citation Context ...ON EQUATIONS 3 The Kruˇzkov-type entropy condition by Karlsen et al. [35] implies partly the standard entropy condition away from the spatial discontinuities (see Wu and Yin [47] and Evje and Karlsen =-=[29]-=-), partly an interface entropy condition at such discontinuities. It yields the physically relevant solution of the clarifier-thickener (CT) problem for given initial data. Let us call such a solution... |

24 |
Well-posedness in BVt and convergence of a difference scheme for continuous sedimentation in ideal clarifier-thickener units
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Citation Context ...ing of two-phase flow in heterogeneous media [31,33], traffic flow with abruptly changing surface conditions, or number of lanes, [9,10,41], and continuous sedimentation in a clarifier-thickener unit =-=[13,14,19,21]-=-. In these three areas of application it is also natural to consider a second-order model with diffusion present: see [28,42] in the case of two-phase flow, [10] in traffic flow, and [15] in continuou... |

24 |
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Citation Context ...ION FOR CONVECTION-DIFFUSION EQUATIONS 3 The Kruˇzkov-type entropy condition by Karlsen et al. [35] implies partly the standard entropy condition away from the spatial discontinuities (see Wu and Yin =-=[47]-=- and Evje and Karlsen [29]), partly an interface entropy condition at such discontinuities. It yields the physically relevant solution of the clarifier-thickener (CT) problem for given initial data. L... |

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21 | Well-posedness for a class of 2 × 2 conservation laws with L∞ data
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Citation Context ...wed uniqueness without an interface condition and without any assumption on the traces of the solution at x = 0. In a simpler setup, the idea of using adapted entropies goes back to Baiti and Jenssen =-=[8]-=-. The problem-specific flux connection (in [16]) should be defined a priori and it corresponds to a stationary solution, which is either an undercompressive or marginally undercompressive wave at x = ... |

21 |
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Citation Context ...t treat the existence problem, but merely assume that there exists a weak entropy solution of (1.1) (in a way that is specified below). For the equations that are treated by Bürger, Karlsen et al. in =-=[11,14,15,34,35,36]-=-, the proofs of existence are carried out by showing convergence of different numerical methods. We refer to those papers for substantial reviews and references of the previous results on uniqueness, ... |

19 | A model of continuous sedimentation of flocculated suspensions in clarifier-thickener units
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Citation Context ...n Section 3. In Section 4 it is extended to include the degenerate parabolic case. Some immediate properties are stated together with the fact that it implies the Kruˇzkov-type CTE inequality used in =-=[15,35]-=-. The proof of uniqueness can be found in Section 5. It follows closely the one in [35]. Modifications are done mainly where the crossing condition and their coupling condition were used, but also to ... |

18 |
Analysis and approximation of conservation laws with source terms
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18 |
Solving the Buckley-Leverett equation with gravity in a heterogeneous porous medium
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Citation Context ... applications. When D = 0, Equation (1.1) becomes a first-order hyperbolic model, a conservation law with a discontinuous flux, which appears in the modelling of two-phase flow in heterogeneous media =-=[31,33]-=-, traffic flow with abruptly changing surface conditions, or number of lanes, [9,10,41], and continuous sedimentation in a clarifier-thickener unit [13,14,19,21]. In these three areas of application i... |

17 |
Veerappa Gowda. Optimal entropy solutions for conservation laws with discontinuous flux-functions
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(Show Context)
Citation Context ...n D = 0, but comment that the analysis could be generalized to include diffusion. Their interface entropy inequality involves both a problem-specific ‘flux connection’, introduced by Adimurthi et al. =-=[2]-=-, and a single ‘adapted Kruˇzkov entropy’ of the type used by Audusse and Perthame [6]. Audusse and Perthame introduced an infinite family of adapted entropies and showed uniqueness without an interfa... |

15 |
Scalar conservation laws with discontinuous flux function: II. On the stability of the viscous profiles
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- 1996
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14 |
An analysis of the traffic on highways with changing road surface conditions
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- 1987
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Citation Context ...onservation law with a discontinuous flux, which appears in the modelling of two-phase flow in heterogeneous media [31,33], traffic flow with abruptly changing surface conditions, or number of lanes, =-=[9,10,41]-=-, and continuous sedimentation in a clarifier-thickener unit [13,14,19,21]. In these three areas of application it is also natural to consider a second-order model with diffusion present: see [28,42] ... |

13 |
On a diffusively corrected kinematic-wave traffic flow model with changing road surface conditions
- Bürger, Karlsen
- 2003
(Show Context)
Citation Context ...onservation law with a discontinuous flux, which appears in the modelling of two-phase flow in heterogeneous media [31,33], traffic flow with abruptly changing surface conditions, or number of lanes, =-=[9,10,41]-=-, and continuous sedimentation in a clarifier-thickener unit [13,14,19,21]. In these three areas of application it is also natural to consider a second-order model with diffusion present: see [28,42] ... |

12 | Effects of capillary forces on immiscible two-phase flow in strongly hetereogeneous porous media
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(Show Context)
Citation Context ...9,10,41], and continuous sedimentation in a clarifier-thickener unit [13,14,19,21]. In these three areas of application it is also natural to consider a second-order model with diffusion present: see =-=[28,42]-=- in the case of two-phase flow, [10] in traffic flow, and [15] in continuous sedimentation. In particular, we highlight the well-posedness analysis of the idealized model of continuous sedimentation w... |

11 |
Veerappa Gowda, Godunov-type methods for conservation laws with a flux function discontinuous in space
- Adimurthi, D
- 2004
(Show Context)
Citation Context ...ic flux connection (in [16]) should be defined a priori and it corresponds to a stationary solution, which is either an undercompressive or marginally undercompressive wave at x = 0. Adimurthi et al. =-=[1,2,3]-=- used such a connection in an interface entropy condition to prove uniqueness for piecewise smooth solutions. The condition implies that the solution contains no undercompressive waves at x = 0 except... |

11 | Operating charts for continuous sedimentation I: Control of steady states
- Diehl
(Show Context)
Citation Context ... problem is that the fluxes f and g vary with time via volume fluxes and a feed inlet. Furthermore, these fluxes are often used as control variables, which means that they depend on the solution, see =-=[22,23,24,25,26]-=-. This suggests that if the FCAE concept should be adapted to the CT problem, the flux connection should depend on the solution instead of being defined a priori. Today it is not clear how this should... |

9 | On conservation laws with discontinuous flux
- Bürger, Karlsen, et al.
- 2005
(Show Context)
Citation Context ...phase flow in heterogeneous media, see [3,33], and in the modelling of traffic flow, see [9]. We emphasize that different solutions may be picked by the CTE and the FCAE conditions, see Bürger et al. =-=[12]-=-. The possibility of adapting the entropy condition to different physical problems is then an advantage. However, this has so far only been done for these two physical problems (two-phase flow and tra... |

9 | Operating charts for continuous sedimentation II: Step responses
- Diehl
- 2005
(Show Context)
Citation Context ... problem is that the fluxes f and g vary with time via volume fluxes and a feed inlet. Furthermore, these fluxes are often used as control variables, which means that they depend on the solution, see =-=[22,23,24,25,26]-=-. This suggests that if the FCAE concept should be adapted to the CT problem, the flux connection should depend on the solution instead of being defined a priori. Today it is not clear how this should... |

8 |
Veerappa Gowda. Existence and stability of entropy solutions for a conservation law with discontinuous non-convex fluxes
- Adimurthi, D
(Show Context)
Citation Context ...monotone or both unimodal. The CT problem is in general more complex where f may have both a local minimum and a local maximum (in the interior of the range of u). In a recent paper, Adimurthi et al. =-=[4]-=- generalize their previous flux-connection entropy condition to the case of several extrema of f and g. A vector of flux connections should be defined a priori and their coupling condition is rather i... |

8 | Operating charts for continuous sedimentation III: Control of step inputs
- Diehl
- 2006
(Show Context)
Citation Context ... problem is that the fluxes f and g vary with time via volume fluxes and a feed inlet. Furthermore, these fluxes are often used as control variables, which means that they depend on the solution, see =-=[22,23,24,25,26]-=-. This suggests that if the FCAE concept should be adapted to the CT problem, the flux connection should depend on the solution instead of being defined a priori. Today it is not clear how this should... |

7 |
On a model for continuous sedimentation in vessels with discontinuously varying crosssectional area
- Bürger, Karlsen, et al.
- 2003
(Show Context)
Citation Context ...ing of two-phase flow in heterogeneous media [31,33], traffic flow with abruptly changing surface conditions, or number of lanes, [9,10,41], and continuous sedimentation in a clarifier-thickener unit =-=[13,14,19,21]-=-. In these three areas of application it is also natural to consider a second-order model with diffusion present: see [28,42] in the case of two-phase flow, [10] in traffic flow, and [15] in continuou... |

5 |
Difference schemes, entropy solutions, and speedup impulse for an inhomogeneous kinematic traffic flow model, Netw
- Bürger, Garćıa, et al.
(Show Context)
Citation Context ...onservation law with a discontinuous flux, which appears in the modelling of two-phase flow in heterogeneous media [31,33], traffic flow with abruptly changing surface conditions, or number of lanes, =-=[9,10,41]-=-, and continuous sedimentation in a clarifier-thickener unit [13,14,19,21]. In these three areas of application it is also natural to consider a second-order model with diffusion present: see [28,42] ... |

5 |
Continuous sedimentation of multi-component particles
- Diehl
- 1997
(Show Context)
Citation Context ...ing of two-phase flow in heterogeneous media [31,33], traffic flow with abruptly changing surface conditions, or number of lanes, [9,10,41], and continuous sedimentation in a clarifier-thickener unit =-=[13,14,19,21]-=-. In these three areas of application it is also natural to consider a second-order model with diffusion present: see [28,42] in the case of two-phase flow, [10] in traffic flow, and [15] in continuou... |

4 | Analysis of a Nonlinear ParabolicHyperbolic Problem
- Aguilar, Lisbona, et al.
- 1999
(Show Context)
Citation Context ...more direct than the Kruˇzkov-type condition in [35]. This makes the construction of stationary solutions easier, see the last section here and the stationary solutions in [15,17] by Bürger et al. In =-=[5]-=-, Aguilar et al. considered the case of (1.1) where the flux function has no spatial discontinuity (f = g) but there is a discontinuity in the diffusion function corresponding to A = 0 and B ′ > 0. Th... |

4 | Entropy conditions for heterogeneity induced shocks in two-phase problems
- Molenaar
- 1995
(Show Context)
Citation Context ...9,10,41], and continuous sedimentation in a clarifier-thickener unit [13,14,19,21]. In these three areas of application it is also natural to consider a second-order model with diffusion present: see =-=[28,42]-=- in the case of two-phase flow, [10] in traffic flow, and [15] in continuous sedimentation. In particular, we highlight the well-posedness analysis of the idealized model of continuous sedimentation w... |

3 | Operating charts for continuous sedimentation IV: Limitations for control of dynamic behaviour
- Diehl
- 2007
(Show Context)
Citation Context |

3 | A regulator for continuous sedimentation in ideal clarifier-thickener units
- Diehl
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Citation Context |

2 |
Veerappa Gowda. Convergence of Godunov type methods for a conservation law with a spatially varying discontinuous flux function
- Adimurthi, D
- 2007
(Show Context)
Citation Context ...ic flux connection (in [16]) should be defined a priori and it corresponds to a stationary solution, which is either an undercompressive or marginally undercompressive wave at x = 0. Adimurthi et al. =-=[1,2,3]-=- used such a connection in an interface entropy condition to prove uniqueness for piecewise smooth solutions. The condition implies that the solution contains no undercompressive waves at x = 0 except... |

2 |
Steady-state, control, and capacity calculations for flocculated suspensions in clarifier-thickeners
- Bürger, Narváez
- 2007
(Show Context)
Citation Context ...pretation of Condition Γ is more direct than the Kruˇzkov-type condition in [35]. This makes the construction of stationary solutions easier, see the last section here and the stationary solutions in =-=[15,17]-=- by Bürger et al. In [5], Aguilar et al. considered the case of (1.1) where the flux function has no spatial discontinuity (f = g) but there is a discontinuity in the diffusion function corresponding ... |

1 | L 1 stability for entropy solutions of 15 - Karlsen, Risebro, et al. - 2003 |