$ BV $ solution for a non-linear Hamilton-Jacobi system

Ahmad El Hajj; Hassan Ibrahim; Vivian Rizik

doi:10.3934/dcds.2020405

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2021, Volume 41, Issue 7: 3273-3293. Doi: 10.3934/dcds.2020405

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$ BV $ solution for a non-linear Hamilton-Jacobi system

1.
Université de Technologie de Compiègne, LMAC, 60205 Compiègne Cedex, France
2.
Université Libanaise, EDST, Hadath, Beyrouth, Liban

^* Corresponding author: Ahmad El Hajj
^* Corresponding author: Ahmad El Hajj

Received: January 31, 2020

Revised: October 31, 2020

Early access: December 2020

Published: July 2021

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Abstract

In this work, we are dealing with a non-linear eikonal system in one dimensional space that describes the evolution of interfaces moving with non-signed strongly coupled velocities. For such kind of systems, previous results on the existence and uniqueness are available for quasi-monotone systems and other special systems in Lipschitz continuous space. It is worth mentioning that our system includes, in particular, the case of non-decreasing solution where some existence and uniqueness results arose for strictly hyperbolic systems with a small total variation. In the present paper, we consider initial data with unnecessarily small $ BV $ seminorm, and we use some $ BV $ bounds to prove a global-in-time existence result of this system in the framework of discontinuous viscosity solution.

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Mathematics Subject Classification: 58F15, 58F1735A01, 74G25, 35F20, 35F21, 35L40, 35Q35, 35D40.

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