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March  2019, 24(3): 1309-1340. doi: 10.3934/dcdsb.2019018

Partial differential inclusions of transport type with state constraints

Applied Mathematics, RheinMain University of Applied Sciences, Wiesbaden Rüsselsheim, Germany

Received  December 2017 Revised  May 2018 Published  January 2019

The focus is on the existence of weak solutions to the quasilinear first-order partial differential inclusion
$\partial_t \; f \:\: ∈ \:\: - {\rm{div}}_{\mathbf{x}} \big( \mathcal{G}(t, f) \; f \big) + \mathcal{U}(t, f) · f + \mathcal{W}(t, f)$
with values in
$L^p({{\mathbb{R}}^{N}})$
for
$p ∈ (1, ∞)$
. The solution is to satisfy state constraints in addition, i.e., all its values belong to a given set
$\mathcal{V} \subset L^p({{\mathbb{R}}^{N}})$
of constraints. We specify sufficient conditions such that every function in
$\mathcal{V}$
initializes at least one weak solution with all its values in
$\mathcal{V}$
(so-called weak invariance a.k.a. viability of
$\mathcal{V}$
). Due to the regularity assumptions about the set-valued coefficient mappings, these solutions prove to be renormalized (in the sense of Di Perna and Lions).
Citation: Thomas Lorenz. Partial differential inclusions of transport type with state constraints. Discrete & Continuous Dynamical Systems - B, 2019, 24 (3) : 1309-1340. doi: 10.3934/dcdsb.2019018
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References:
[1]

A. AggarwalR. M. Colombo and P. Goatin, Nonlocal systems of conservation laws in several space dimensions, SIAM J. Numer. Anal., 53 (2015), 963-983. doi: 10.1137/140975255.

[2]

H. W. Alt, Linear Functional Analysis. An Application-Oriented Introduction, London: Springer, 2016, Translated from the 6th German edition by Robert Nürnberg. doi: 10.1007/978-1-4471-7280-2.

[3]

L. Ambrosio, Transport equation and Cauchy problem for non-smooth vector fields, in Calculus of Variations and Nonlinear Partial Differential Equations (eds. B. Dacorogna and P. Marcellini), vol. 1927 of Lecture Notes in Math., Springer, Berlin, 2008, 1–41. doi: 10.1007/978-3-540-75914-0_1.

[4]

L. Ambrosio, Well posedness of ODE's and continuity equations with nonsmooth vector fields, and applications, Rev. Mat. Complut., 30 (2017), 427-450. doi: 10.1007/s13163-017-0244-3.

[5]

L. Ambrosio and G. Crippa, Existence, uniqueness, stability and differentiability properties of the flow associated to weakly differentiable vector fields, in Transport Equations and Multi-D Hyperbolic Conservation Laws (eds. F. Ancona, S. Bianchini, R. M. Colombo, C. De Lellis, A. Marson and A. Montanari), vol. 5 of Lect. Notes Unione Mat. Ital., Springer, Berlin, 2008, 3–57. doi: 10.1007/978-3-540-76781-7_1.

[6]

L. Ambrosio and P. Tilli, Topics on Analysis in Metric Spaces, vol. 25 of Oxford Lecture Series in Mathematics and its Applications, Oxford University Press, Oxford, 2004.

[7]

J.-P. Aubin and H. Frankowska, Feedback controls for uncertain systems, in Modeling, Estimation and Control of Systems with Uncertainty (Sopron, 1990) (eds. G. B. Di Masi, A. Gombani and A. B. Kurzhansky), vol. 10 of Progr. Systems Control Theory, Birkh¨auser Boston, Boston, MA, 1991, 1–21.

[8]

J.-P. Aubin, Viability Theory, Systems & Control: Foundations & Applications, Birkhäuser Boston, Inc., Boston, MA, 1991.

[9]

J.-P. Aubin, Mutational equations in metric spaces, Set-Valued Anal., 1 (1993), 3-46. doi: 10.1007/BF01039289.

[10]

J.-P. Aubin, Mutational and Morphological Analysis, Systems & Control: Foundations & Applications, Birkhäuser Boston Inc., Boston, MA, 1999, Tools for shape evolution and morphogenesis. doi: 10.1007/978-1-4612-1576-9.

[11]

J.-P. Aubin, A. M. Bayen and P. Saint-Pierre, Viability Theory, 2nd edition, Springer, Heidelberg, 2011, New directions. doi: 10.1007/978-3-642-16684-6.

[12]

J.-P. Aubin and A. Cellina, Differential Inclusions, vol. 264 of Grundlehren der mathematischen Wissenschaften, Springer-Verlag, Berlin, 1984, Set-valued maps and viability theory. doi: 10.1007/978-3-642-69512-4.

[13]

J.-P. Aubin and A. Désilles, Traffic Networks as Information Systems, Mathematical Engineering, Springer-Verlag, Berlin Heidelberg, 2017, A viability approach. doi: 10.1007/978-3-642-54771-3.

[14]

J.-P. Aubin and H. Frankowska, Set-valued Analysis, vol. 2 of Systems & Control: Foundations & Applications, Birkhäuser Boston Inc., Boston, MA, 1990.

[15]

D. Averna, Lusin type theorems for multifunctions, Scorza Dragoni's property and Carathéodory selections, Boll. Unione Mat. Ital., VII. Ser., A, 8 (1994), 193-202.

[16]

V. Barbu, Nonlinear Differential Equations of Monotone Types in Banach Spaces, Springer Monographs in Mathematics, Springer, New York, 2010. doi: 10.1007/978-1-4419-5542-5.

[17]

V. I. Bogachev, Measure Theory. Vol. I, II, Springer-Verlag, Berlin, 2007. doi: 10.1007/978-3-540-34514-5.

[18]

R. BorscheR. M. ColomboM. Garavello and A. Meurer, Differential equations modeling crowd interactions, J. Nonlinear Sci., 25 (2015), 827-859. doi: 10.1007/s00332-015-9242-0.

[19]

D. Bothe, Multivalued differential equations on graphs, Nonlinear Anal., 18 (1992), 245-252. doi: 10.1016/0362-546X(92)90062-J.

[20]

D. Bothe, Flow invariance for perturbed nonlinear evolution equations, Abstr. Appl. Anal., 1 (1996), 417-433. doi: 10.1155/S1085337596000231.

[21]

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

[22]

O. Cârjǎ and M. D. P. Monteiro Marques, Weak tangency, weak invariance, and Carathéodory mappings, J. Dynam. Control Systems, 8 (2002), 445-461. doi: 10.1023/A:1020765401015.

[23]

O. Cârjǎ, M. Necula and I. I. Vrabie, Viability, Invariance and Applications, vol. 207 of North-Holland Mathematics Studies, Elsevier Science B.V., Amsterdam, 2007.

[24]

C. CastaingM. Moussaoui and A. Syam, Multivalued differential equations on closed convex sets in Banach spaces, Set-Valued Anal., 1 (1993), 329-353. doi: 10.1007/BF01027824.

[25]

C. Castaing and M. Valadier, Convex Analysis and Measurable Multifunctions, vol. 580 of Lecture Notes in Mathematics, Springer-Verlag, Berlin-New York, 1977.

[26]

F. H. Clarke, Y. S. Ledyaev, R. J. Stern and P. R. Wolenski, Nonsmooth Analysis and Control Theory, vol. 178 of Graduate Texts in Mathematics, Springer-Verlag, New York, 1998.

[27]

R. M. Colombo and G. Guerra, Nonlocal sources in hyperbolic balance laws with applications, in Hyperbolic Problems: Theory, Numerics, Applications (eds. S. Benzoni-Gavage and D. Serre), Springer, Berlin, 2008,577–584. doi: 10.1007/978-3-540-75712-2_56.

[28]

R. M. ColomboA. Corli and M. D. Rosini, Non local balance laws in traffic models and crystal growth, ZAMM Z. Angew. Math. Mech., 87 (2007), 449-461. doi: 10.1002/zamm.200710327.

[29]

R. M. Colombo, M. Garavello and M. Lécureux-Mercier, A class of nonlocal models for pedestrian traffic, Math. Models Methods Appl. Sci., 22 (2012), 1150023, 34pp. doi: 10.1142/S0218202511500230.

[30]

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