Analysis of a system of nonlocal conservation laws for multicommodity flow on networks
Pages: 749  785,
Issue 4,
December
2015
doi:10.3934/nhm.2015.10.749 Abstract
References
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Martin Gugat  FriedrichAlexanderUniversität ErlangenNürnberg (FAU), Department Mathematik, Chair of Applied Mathematics 2, Cauerstraße 11, 91058 Erlangen, Germany (email)
Alexander Keimer  FriedrichAlexanderUniversität ErlangenNürnberg (FAU), Department Mathematik, Chair of Applied Mathematics 2, Cauerstraße 11, 91058 Erlangen, Germany (email)
Günter Leugering  FriedrichAlexanderUniversität ErlangenNürnberg (FAU), Department Mathematik, Chair of Applied Mathematics 2, Cauerstraße 11, 91058 Erlangen, Germany (email)
Zhiqiang Wang  School of Mathematical Sciences and Shanghai Key Laboratory for Contemporary Applied Mathematics, Fudan University, Shanghai 200433, China (email)
1 
R. A. Adams and J. J. Fournier, Sobolev Spaces, vol. 140 of Pure and Applied Mathematics (Amsterdam), 2nd edition, Elsevier/Academic Press, Amsterdam, 2003. 

2 
A. Aggarwal, R. M. Colombo and P. Goatin, Nonlocal systems of conservation laws in several space dimensions, SIAM Journal on Numerical Analysis, 53 (2015), 963983. 

3 
L. Ambrosio, N. Fusco and D. Pallara, Functions Of Bounded Variation And Free Discontinuity Problems, Oxford Mathematical Monographs, The Clarendon Press Oxford University Press, New York, 2000. 

4 
D. Armbruster, P. Degond and C. Ringhofer, A model for the dynamics of large queuing networks and supply chains, SIAM J. Appl. Math., 66 (2006), 896920. 

5 
D. Armbruster, D. E. Marthaler, C. A. Ringhofer, K. G. Kempf and T.C. Jo, A continuum model for a reentrant factory, Operations Research, 54 (2006), 933950. 

6 
A. A. Assad, Multicommodity network flows  a survey, Networks, 8 (1978), 3791. 

7 
H. Attouch, G. Buttazzo and G. Michaille, Variational Analysis in Sobolev and BV Spaces, vol. 6 of MPS/SIAM Series on Optimization, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2006. 

8 
S. Blandin and P. Goatin, Wellposedness of a conservation law with nonlocal flux arising in traffic flow modeling, Numerische Mathematik, Springer Berlin Heidelberg, (2015), 125. 

9 
H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Universitext, Springer, New York, 2011. 

10 
R. M. Colombo, M. Herty and M. Mercier, Control of the continuity equation with a non local flow, ESAIM Control Optim. Calc. Var., 17 (2011), 353379. 

11 
J.M. Coron, M. Kawski and Z. Wang, Analysis of a conservation law modeling a highly reentrant manufacturing system, Discrete Contin. Dyn. Syst. Ser. B, 14 (2010), 13371359. 

12 
L. R. Ford Jr. and D. R. Fulkerson, Flows in Networks, Princeton Landmarks in Mathematics, Princeton University Press, Princeton, NJ, 1962. 

13 
A. Freno and E. Trentin, Hybrid Random Fields: A Scalable Approach to Structure and Parameter Learning in Probabilistic Graphical Models, Intelligent Systems Reference Library, Springer, 2011. 

14 
E. Giusti, Minimal Surfaces and Functions of Bounded Variation, Vol. 80 of Monographs in Mathematics, Birkhäuser Boston, 1984. 

15 
M. Gröschel, A. Keimer, G. Leugering and Z. Wang, Regularity theory and adjoint based optimality conditions for a nonlinear transport equation with nonlocal velocity, SIAM J. Control Optim., 52 (2014), 21412163. 

16 
M. Gugat, F. M. Hante, M. HirschDick and G. Leugering, Stationary states in gas networks, Networks and Heterogeneous Media, 10 (2015), 295320. 

17 
M. Gugat, M. Herty, A. Klar and G. Leugering, Optimal control for traffic flow networks, Journal of Optimization Theory and Applications, 126 (2005), 589616. 

18 
M. Gugat, M. Herty, A. Klar, G. Leugering and V. Schleper, Wellposedness of networked hyperbolic systems of balance laws, in Constrained optimization and optimal control for partial differential equations, vol. 160 of Internat. Ser. Numer. Math., Birkhäuser/Springer Basel AG, Basel, 2012, 123146. 

19 
J. L. Kennington, A survey of linear cost multicommodity network flows, Operations Res., 26 (1978), 209236. 

20 
M. La Marca, D. Armbruster, M. Herty and C. Ringhofer, Control of continuum models of production systems, IEEE Trans. Automat. Contr., 55 (2010), 25112526. 

21 
G. Leoni, A First Course in Sobolev Spaces, vol. 105 of Graduate Studies in Mathematics, American Mathematical Society, Providence, RI, 2009. 

22 
J. Simon, Compact sets in the space $L^p(0,T;B)$, Ann. Mat. Pura Appl. (4), 146 (1987), 6596 

23 
D. W. Stroock, Essentials of Integration Theory for Analysis, vol. 262, Springer, 2011. 

24 
W. W.Y. Wong, Compactness in $L^{2}$, 2013, Personal Communication., http://math.stackexchange.com/questions/329253/compactnessinl2. 

25 
J. J. Yeh, Lectures On Real Analysis, World Scientific Publishing Co. Inc., River Edge, NJ, 2000. 

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