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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:
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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. Google Scholar

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[3]

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

[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. Google Scholar

[11]

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[12]

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[13]

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[14]

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[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. Google Scholar

[17]

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

[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. Google Scholar

[19]

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

[20]

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

[21]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[25]

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[27]

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