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## On general balance laws with boundary (2010)

Venue: | Journal of Differential Equations |

Citations: | 7 - 6 self |

### Citations

436 | Hyperbolic Conservation Laws in Continuum Physics, - Dafermos - 2010 |

76 |
Boundary conditions for nonlinear hyperbolic systems of conservation laws.
- Dubois, Floch
- 1988
(Show Context)
Citation Context ... preliminary to the study of (1.1), is that of the purely convective system ⎧ ⎪⎨ ⎪⎩ ∂tu ( + ∂xf(u) = 0 x > γ(t) b u ( t,γ(t) )) = g(t) t ≥ 0 u(0,x) = uo(x) x ≥ γ(0) (1.2) considered, for instance, in =-=[1, 2, 12, 15, 16]-=-. Below, we provide results on (1.2) that are not contained in these papers. In particular, the present estimates explicitly blow up as the boundary tends to be characteristic. The choice of the Glimm... |

44 | Well-posedness of the Cauchy problem for n× n systems of conservation laws - Bressan, Crasta, et al. |

41 |
Hyperbolic systems of conservation laws, volume 20 of Oxford
- Bressan
- 2000
(Show Context)
Citation Context ...hose precise value is not relevant. This section is devoted to the homogeneous initial boundary value problem (1.2) and proves Theorem 2.2. Our general reference on the theory of conservation laws is =-=[6]-=-. Let σ → Rj(σ)(u), respectively σ → Sj(σ)(u), be the j-rarefaction curve, respectively the j-shock curve, exiting u. If the j-th field is linearly degenerate, then the parameter σ above is the arc-le... |

35 |
Hyperbolic systems of balance laws with inhomogeneity and dissipation,”
- Dafermos, Hsiao
- 1982
(Show Context)
Citation Context ..., 2) with ω = 0 and 3) implies its uniqueness through standard computations, see for instance [5, Section 6, Corollary 1]. □ 5 The Source Term This section is devoted to the source term, similarly to =-=[3, 7, 8, 14, 18]-=- but following the general metric space technique in [10], applied to L 1 equipped with the L 1 -distance d. The key point is to show that the map ˇF(t,to)u = P(t,to)u + t G ( P(t,to)u ) χ [γ(to+t),+∞... |

34 |
Quasilinear hyperbolic systems,”
- Liu
- 1979
(Show Context)
Citation Context ..., 2) with ω = 0 and 3) implies its uniqueness through standard computations, see for instance [5, Section 6, Corollary 1]. □ 5 The Source Term This section is devoted to the source term, similarly to =-=[3, 7, 8, 14, 18]-=- but following the general metric space technique in [10], applied to L 1 equipped with the L 1 -distance d. The key point is to show that the map ˇF(t,to)u = P(t,to)u + t G ( P(t,to)u ) χ [γ(to+t),+∞... |

20 |
Initial-boundary value problems for nonlinear systems of conservation laws
- Amadori
- 1997
(Show Context)
Citation Context ... preliminary to the study of (1.1), is that of the purely convective system ⎧ ⎪⎨ ⎪⎩ ∂tu ( + ∂xf(u) = 0 x > γ(t) b u ( t,γ(t) )) = g(t) t ≥ 0 u(0,x) = uo(x) x ≥ γ(0) (1.2) considered, for instance, in =-=[1, 2, 12, 15, 16]-=-. Below, we provide results on (1.2) that are not contained in these papers. In particular, the present estimates explicitly blow up as the boundary tends to be characteristic. The choice of the Glimm... |

20 |
Continuous dependence for 22 conservation laws with boundary
- Amadori, Colombo
- 1997
(Show Context)
Citation Context ...t admit a lower semicontinuous extension, due to the presence of the boundary. 11We now construct ε-approximate solutions to (1.2) by means of the classical wave front tracking technique, see [6] or =-=[1, 2]-=- for the case with boundary. Let ε > 0 be fixed and approximate the initial and boundary data in (1.2) by means of piecewise constant functions u ε o and g ε such that (see [9, formula (3.1)]) ∥uε o −... |

14 |
Initial Boundary Value Problems for Hyperbolic Systems of Conservation Laws
- Goodman
- 1982
(Show Context)
Citation Context ...ee [13, Chapter 14]. We define below the domain Dγ = { (t,x) ∈ R + } × R: x ≥ γ(t) and extend to [0,+∞[ × R any function defined on Dγ to vanish outside Dγ. We slightly modify the definition given in =-=[17]-=- of solution to (1.2) in the non characteristic case, see also [1, 2, 15] and [12, Definition 2.1]. Indeed, here we require the boundary condition to be satisfied by the solution only almost everywher... |

12 |
Uniqueness and continuous dependence for systems of balance laws with dissipation
- Amadori, Guerra
- 2002
(Show Context)
Citation Context ... present non homogeneous case, due to the non local nature of the source term here considered. Indeed, let t γ(t) ˜γ(t) u(t,x) ⎧⎪ 2 ( ∫ 1 u = 0 ⎨ ∂tu + ∂xu = 0 1 u = 1 ⎪⎩ 0 1 2 3 4 x u(t,ξ)dξ ) χ (x) =-=[3,4]-=- u(t,0) = 0 (3.1) u(0,x) = χ (x). [0,1] It is immediate to verify that the assumptions of Theorem 3.2 hold. The solution u, shown above, is non zero in the delimited area above and, in particular, for... |

12 | Optimal control in networks of pipes and canals
- Colombo, Guerra, et al.
(Show Context)
Citation Context ...r the case with a non local source but no boundary and [12] for the case of a Temple type f. 1Examples of physical models that fit into this class are found, besides in the cited references, also in =-=[11]-=-. There, a model describing the flow of a fluid in a simple pipeline is based on a system essentially of the form (1.1). As is well known, preliminary to the study of (1.1), is that of the purely conv... |

11 | Hyperbolic balance laws with a non local source.
- Colombo, Guerra
- 2007
(Show Context)
Citation Context ..., 2) with ω = 0 and 3) implies its uniqueness through standard computations, see for instance [5, Section 6, Corollary 1]. □ 5 The Source Term This section is devoted to the source term, similarly to =-=[3, 7, 8, 14, 18]-=- but following the general metric space technique in [10], applied to L 1 equipped with the L 1 -distance d. The key point is to show that the map ˇF(t,to)u = P(t,to)u + t G ( P(t,to)u ) χ [γ(to+t),+∞... |

9 | Differential equations in metric spaces with applications
- Colombo, Guerra
(Show Context)
Citation Context ...tions, see for instance [5, Section 6, Corollary 1]. □ 5 The Source Term This section is devoted to the source term, similarly to [3, 7, 8, 14, 18] but following the general metric space technique in =-=[10]-=-, applied to L 1 equipped with the L 1 -distance d. The key point is to show that the map ˇF(t,to)u = P(t,to)u + t G ( P(t,to)u ) χ [γ(to+t),+∞[ (5.1) is a local flow in the sense of [10, Definition 2... |

9 |
Stability of front tracking solutions to the initial and boundary value problem for systems of conservation laws. NoDEA
- Donadello, Marson
(Show Context)
Citation Context ... preliminary to the study of (1.1), is that of the purely convective system ⎧ ⎪⎨ ⎪⎩ ∂tu ( + ∂xf(u) = 0 x > γ(t) b u ( t,γ(t) )) = g(t) t ≥ 0 u(0,x) = uo(x) x ≥ γ(0) (1.2) considered, for instance, in =-=[1, 2, 12, 15, 16]-=-. Below, we provide results on (1.2) that are not contained in these papers. In particular, the present estimates explicitly blow up as the boundary tends to be characteristic. The choice of the Glimm... |

6 | Hyperbolic balance laws with a dissipative non local source.
- Colombo, Guerra
- 2008
(Show Context)
Citation Context ... by the boundary data g. Above and in what follows, we assume that all BV functions are right continuous. Systems belonging to this class were already considered in the literature. See, for instance, =-=[7, 8]-=- for the case with a non local source but no boundary and [12] for the case of a Temple type f. 1Examples of physical models that fit into this class are found, besides in the cited references, also ... |

5 | Lower semicontinuity of weighted path lengths in BV
- Baiti, Bressan
- 1997
(Show Context)
Citation Context ... present non homogeneous case, due to the non local nature of the source term here considered. Indeed, let t γ(t) ˜γ(t) u(t,x) ⎧⎪ 2 ( ∫ 1 u = 0 ⎨ ∂tu + ∂xu = 0 1 u = 1 ⎪⎩ 0 1 2 3 4 x u(t,ξ)dξ ) χ (x) =-=[3,4]-=- u(t,0) = 0 (3.1) u(0,x) = χ (x). [0,1] It is immediate to verify that the assumptions of Theorem 3.2 hold. The solution u, shown above, is non zero in the delimited area above and, in particular, for... |

5 |
On the stability functional for conservation laws. Nonlinear Anal
- Colombo, Guerra
- 2008
(Show Context)
Citation Context ... + ∑ n∑ x≥γ(t) i=ℓ+1 ∣ ∣σx,i Υt(u) = V t(u) + H2 Q t(u) + H1 TV { g,[t,+∞[ } ∣ (4.9) the set A of approaching waves being defined as usual, see [6] and the constant K,H1,H2 to be defined later. As in =-=[9]-=-, using Lemma 4.2, the Glimm functional Υ can be extended in a lower semicontinuous way to all functions with small total variation in L 1 (R; R n ) that vanish for x ≤ γ(t). On the contrary, the inte... |

2 |
Well posedness of balance laws with noncharacteristic boundary
- Colombo, Rosini
(Show Context)
Citation Context ...at all BV functions are right continuous. Systems belonging to this class were already considered in the literature. See, for instance, [7, 8] for the case with a non local source but no boundary and =-=[12]-=- for the case of a Temple type f. 1Examples of physical models that fit into this class are found, besides in the cited references, also in [11]. There, a model describing the flow of a fluid in a si... |