Existence and Uniqueness of Continuous Solution for a Non-local Coupled System Modeling the Dynamics of Dislocation Densities

Hajj, Ahmad El; Oussaily, Aya

doi:10.1007/s00332-021-09676-7

Cited by 5 publications

(4 citation statements)

References 22 publications

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“…In a similar framework, in El Hajj [4], the existence and uniqueness of strong solution 1,2 loc ([0, ] × R) was proved. Recently, a result of global existence and uniqueness was proved in El Hajj and Oussaily [7] for continuous solutions satisfying some gradient entropy estimates. We refer the readers to El Hajj et al [8] for the discontinuous solutions of this system.…”

Section: Recall Of Useful Resultsmentioning

confidence: 99%

Convergence of an implicit scheme for diagonal non-conservative hyperbolic systems

2021

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In this paper, we consider diagonal non-conservative hyperbolic systems in one space dimension with monotone and large Lipschitz continuous data. Under a certain nonnegativity condition on the Jacobian matrix of the velocity of the system, global existence and uniqueness results of a Lipschitz solution for this system, which is not necessarily strictly hyperbolic, was already proven. We propose a natural implicit scheme satisfiying a similar Lipschitz estimate at the discrete level. This property allows us to prove the convergence of the scheme without assuming it strictly hyperbolic.

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Section: Recall Of Useful Resultsmentioning

confidence: 99%

Convergence of an implicit scheme for diagonal non-conservative hyperbolic systems

2021

Self Cite

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“…First, motivated by dislocation dynamics, we can point out the result done by El Hajj and Boudjerada in [6] wherein the global existence of discontinuous viscosity BV solutions was proved for scalar one dimensional non-linear and non-local eikonal equations, including in particular the case d = 1 in system (1.1). Always within the framework of dislocation dynamics, let us mention that, it was proven in El Hajj et al [9,11,16] some existence and uniqueness results of continuous viscosity solution for a particular quasi-monotone (2 × 2) system which corresponds to the case where (λ i ,j )i,j=1,2 = 1 −1 −1 1 . The proof in [11,16] was based on the work of Ishii, Koike [18,17] wherein, a general theory concerning the existence and the uniqueness of viscosity solutions for quasi-monotone Hamilton-Jacobi systems, has been developed.…”

Section: 3mentioning

confidence: 99%

“…Always within the framework of dislocation dynamics, let us mention that, it was proven in El Hajj et al [9,11,16] some existence and uniqueness results of continuous viscosity solution for a particular quasi-monotone (2 × 2) system which corresponds to the case where (λ i ,j )i,j=1,2 = 1 −1 −1 1 . The proof in [11,16] was based on the work of Ishii, Koike [18,17] wherein, a general theory concerning the existence and the uniqueness of viscosity solutions for quasi-monotone Hamilton-Jacobi systems, has been developed. However, in [9], the author proved a L 2 energy estimate in order to get the existence and the uniqueness.…”

Section: 3mentioning

confidence: 99%

Continuous solution for a non-linear eikonal system

Hajj¹,

Oussaily²

2021

CPAA

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<p style='text-indent:20px;'>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. We prove a global existence result in the framework of continuous viscosity solution. The approach is made by adding a viscosity term and passing to the limit for vanishing viscosity, relying on a new gradient entropy and <inline-formula><tex-math id="M1">\begin{document}$ BV $\end{document}</tex-math></inline-formula> estimates. A uniqueness result is also proved through a comparison principle property.</p>

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“…Also in the framework of viscosity solutions, the global existence of a discontinuous solution was proven in El Hajj et al [30] for BV initial data. Further more, in El Hajj, Oussaily [32], the global existence and uniqueness of a continuous viscosity solution has been presented, based on an entropy estimate and under a control on the gradient of the solution.…”

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confidence: 99%

Entropy solutions to a non-conservative and non-strictly hyperbolic diagonal system inspired by dislocation dynamics

Zohbi

Junca

2023

J. Evol. Equ.

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In this work, we study the existence of solutions to a 2 × 2 non-conservative and non-strictly hyperbolic system in one space dimension related to the dynamics of dislocation densities in crystallography, propagating in two opposite directions. For such systems, existence results are mainly established in the sense of viscosity solutions for Hamilton-Jacobi equations. We study this problem for large initial data using the notions of the theory of conservation laws by constructing entropy solutions through the means of an adapted Godunov scheme, where the associated Riemann problem enjoys new features, more elementary waves than usual, and loss of uniqueness in many cases. The existence is obtained in spaces of functions with bounded fractional total variation BV s , for all 0 < s ≤ 1, such that BV 1 = BV .

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Existence and Uniqueness of Continuous Solution for a Non-local Coupled System Modeling the Dynamics of Dislocation Densities

Cited by 5 publications

References 22 publications

Convergence of an implicit scheme for diagonal non-conservative hyperbolic systems

Convergence of an implicit scheme for diagonal non-conservative hyperbolic systems

Continuous solution for a non-linear eikonal system

Entropy solutions to a non-conservative and non-strictly hyperbolic diagonal system inspired by dislocation dynamics

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