Stagnation, Creation, Breaking

Piotr B. Mucha

arXiv:1402.2450·math.AP·February 12, 2014·1 cites

Stagnation, Creation, Breaking

Piotr B. Mucha

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TL;DR

This paper investigates the behavior of solutions to a one-dimensional PDE with monotone flux function, focusing on the formation, stability, and breaking of flat regions called facets.

Contribution

It provides examples illustrating the creation, stability, and breaking of facets in solutions to a monotone PDE, enhancing understanding of their dynamics.

Findings

01

Facets can form and persist in solutions with monotone flux.

02

Strong stability of facets under certain conditions is demonstrated.

03

Facets can also break, indicating complex solution behaviors.

Abstract

This note deals with the mono-dimensional equation: $\d_{t} u - \d_{x} L (u_{x}) = f$ with $L (\cdot)$ merely monotone. The goal is to examine the features of facets -- flat regions of graphs of solutions appearing as $L (\cdot)$ suffers jumps. We concentrate on examples explaining strong stability, creation and even breaking of facets.

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Taxonomy

TopicsNonlinear Partial Differential Equations · Stability and Controllability of Differential Equations · Nonlinear Differential Equations Analysis