Abstract
In this paper, we study a non-local coupled system that arises in the theory of dislocations densities dynamics. Based on a new gradient entropy estimate in \(L \log L\) space, we prove the global existence of a continuous solution. The approach is made by adding a viscosity term and passing to the limit for vanishing viscosity. A comparison principle with respect to time is used for proving uniqueness of the solution for the local problem.
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Communicated by Arash Yavari.
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El Hajj, A., Oussaily, A. Existence and Uniqueness of Continuous Solution for a Non-local Coupled System Modeling the Dynamics of Dislocation Densities. J Nonlinear Sci 31, 20 (2021). https://doi.org/10.1007/s00332-021-09676-7
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DOI: https://doi.org/10.1007/s00332-021-09676-7
Keywords
- Hamilton–Jacobi system
- Non-local eikonal system
- Non-local transport system
- Entropy estimate
- Viscosity solution
- Dislocation dynamics