Mathematical Models for Erosion and the Optimal Transportation of Sediment

Birnir, Björn; Rowlett, Julie

doi:10.1515/ijnsns-2013-0048

Cited by 8 publications

(4 citation statements)

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“…, P-a.e. ω ∈ Ω, (5) where ∆ u : Ω → R, with ∆ u (ω) := ||u(ω) − (u(ω))|| 2 L 2 (S) . Hereby it is actually possible to explicitly determine c 1 ≥ 0.…”

Section: The Main Resultsmentioning

confidence: 99%

“…Mazón, J. Rossi and J. Toledo in [2]. In addition, the importance of the PDE (2) to the evolution of fluvial landscapes has been discussed by B. Birnir and J. Rowlett in [5]. Moreover, the asymptotic properties of the solution of (2) have been studied in [7].…”

Section: Introductionmentioning

confidence: 99%

“…t ∈ (0, ∞) admits a unique strong solution and to determine asymptotic properties of this solution. and J. Rowlett in [5]. Moreover, the asymptotic properties of the solution of (2) have been studied in [7].…”

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confidence: 99%

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A randomized weighted $${\varvec{p}}$$-Laplacian evolution equation with Neumann boundary conditions

Nerlich

2018

Nonlinear Differ. Equ. Appl.

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The purpose of this paper is to show that the randomized weighted p-Laplacian evolution equation given byfor P-a.e. ω ∈ Ω and a.e. t ∈ (0, ∞) admits a unique strong solution and to determine asymptotic properties of this solution. and J. Rowlett in [5]. Moreover, the asymptotic properties of the solution of (2) have been studied in [7]. 2The purpose of this paper is to study a random version of the PDE (2); more precisely: The weight function γ : S → (0, ∞) occurring in (2) is replaced by a (vector-valued) random variable g : Ω → L 1 (S).Hereby (Ω, F , P) denotes a complete probability space, S ⊆ R n , where n ∈ N \ {1}, is a sufficiently regular set, η is the unit outer normal on ∂S and p ∈ (1, ∞) \ {2}.The weight function g being random of course implies that the solution has to be random as well. Consequently, it is appropriate to assume that the initial value is no longer deterministic but also a random quantity. Therefore it is natural to consider the randomized PDE (1) as the PDE corresponding to (2) for a random weight function.From an applied point of view, the weight function γ in (2) models a stationary water depth which occurs due to rain, on the landscape v. The motivation for considering the randomized PDE (3) is that this water depth might be not precisely known, which makes it reasonable to view it as a random quantity.Hereby, the assumptions made on g(ω) for a given ω ∈ Ω are, due to technical reasons, actually stronger than those made on γ in [2].It seems important to point out that this paper is not concerned with any kind of stochastic differential equation. The noise occurring in this paper's setting does not come from integrating with respect to Brownian motions or other stochastic processes, but originates from the random weight function g and the random initial value u.for any α, t ∈ (0, ∞). Hereby c 3 , c 4 ≥ 0 are again constants which can be determined explicitly.One should note that if n = 2, which is from an applied point of view the interesting case, one can apply either (6) or (8), resp. (9), given that the initial is sufficiently integrable. This paper is structured as follows: Section 3 contains all assumptions, notations and basic definitions which are needed throughout this paper. Section 4 (Section 5, resp.) deals with the existence and uniqueness of mild (strong, resp.) solutions of (4). The assertions (5)-(7) are established in Section 6.Finally, Section 7 deals with the tail function bounds (8) and (9). Moreover, this article contains two appendices. The first answers some technical measurability questions which occur while defining A and the second one provides some delicate results about the deterministic counterpart of A. Those are mostly needed to prove the existence and uniqueness of mild solutions of (4). Assumptions, notations and preliminary resultsSome notational preliminaries are in order: For any m-dimensional Borel measurable set B, where m ∈ N, B(B) denotes the Borel σ-algebra on this set. Moreover, for any measure µ : B(B) → [0, ∞] and any q ∈ [1, ∞], L q (B, µ;...

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“…, P-a.e. ω ∈ Ω, (5) where ∆ u : Ω → R, with ∆ u (ω) := ||u(ω) − (u(ω))|| 2 L 2 (S) . Hereby it is actually possible to explicitly determine c 1 ≥ 0.…”

Section: The Main Resultsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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A randomized weighted $${\varvec{p}}$$-Laplacian evolution equation with Neumann boundary conditions

Nerlich

2018

Nonlinear Differ. Equ. Appl.

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“…There is, however, no physical basis for predicting this 'optimal' river configuration; one can only assert that the observed state of a river is optimal. Recent developments in the mathematical theory of ramified optimal transport, which seeks solutions that minimize transportation cost 150 , may eventually yield a more formal treatment for routing of water and sediment by rivers 151 -and, consequently, their associated hydraulic geometry.…”

Section: Alternatives To the 1 + ε Modelmentioning

confidence: 99%

The importance of threshold in alluvial river channel geometry and dynamics

Phillips¹,

Masteller²,

Slater³

et al. 2021

Preprint

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Many cities and settlements are organized around alluvial rivers, which are self-formed channels composed of gravel, sand and mud. Much of the time alluvial river channels are oversized, in that they could accommodate greater water flow; yet during extreme storms they are woefully undersized, and potentially catastrophic flooding can occur. Considering widely varying hydroclimates, sediment supply, geologic constraints and varying vegetation, it is not altogether obvious that rivers should achieve an average channel geometry that is pattern stable -let alone predictable from theory. Yet, natural rivers follow remarkably consistent hydraulic-geometry scaling relations, that are reproduced in laboratory experiments. Starting with the constraint that channel formation requires that fluid stress exceeds the threshold for sediment entrainment, we review the explanatory power of threshold channel models. Moreover we explore how deviations from threshold channel theory relate to higher-order dynamics of fluid and sediment transport -essentially perturbations to the threshold state -clarifying misconceptions regarding model applicability. Finally, we demonstrate the utility of the threshold channel framework for understanding channel patterns and responses to variations in external forcing such as hydroclimate and land use. Accurate field determination of the entrainment threshold itself is a notorious problem, and emerges as a central challenge in further development and application of threshold channel theory.

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Asymptotic results for solutions of a weighted $${\varvec{p}}$$-Laplacian evolution equation with Neumann boundary conditions

Nerlich

2017

Nonlinear Differ. Equ. Appl.

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Mathematical Models for Erosion and the Optimal Transportation of Sediment

Cited by 8 publications

References 50 publications

A randomized weighted $${\varvec{p}}$$-Laplacian evolution equation with Neumann boundary conditions

A randomized weighted $${\varvec{p}}$$-Laplacian evolution equation with Neumann boundary conditions

The importance of threshold in alluvial river channel geometry and dynamics

Asymptotic results for solutions of a weighted $${\varvec{p}}$$-Laplacian evolution equation with Neumann boundary conditions

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