Partial differential inclusions of transport type with state constraints

Thomas Lorenz

doi:10.3934/dcdsb.2019018

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2019, Volume 24, Issue 3: 1309-1340. Doi: 10.3934/dcdsb.2019018

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Partial differential inclusions of transport type with state constraints

Thomas Lorenz

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

Received: November 30, 2017

Revised: April 30, 2018

Early access: January 2019

Published: January 2019

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Abstract

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).

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Mathematics Subject Classification: Primary: 35F25, 35R70; Secondary: 35B30, 35D30, 47J35.

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