Global existence results for eikonal equation with BV initial data

Abstract

In this paper, we study a local and a non-local eikonal equations in one dimensional space describing the evolution of interfaces moving with non-signed velocity. For these equations, the global existence and uniqueness are available only of Lipschitz continuous viscosity solutions in some particular cases. In the present paper, we are interested in the study of the global in time existence of these equations, considering BV initial data. Based on a fundamental uniform BV estimate and the finite speed propagation property of these equations, we show, in a particular setting, global existence results of discontinuous viscosity solutions of this problem. An interesting application of these results is shown in the case of dislocation dynamics.Mathematics Subject Classification. 35A01, 74G25, 35F20, 35F21, 70H20, 35Q74.

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

“…This result has been extended in El Hajj et al [14] to a more general nonlinear and non-local (2 × 2) system where the authors were able to rely on the fact that this (2 × 2) system has a particular local structure, making it possible to verify some quasi-monotonicity properties. This allowed to attain the same estimates obtained in [8] and thus to prove the global existence of discontinuous solutions of the considered particular (2 × 2) system, inspired by the work of Ishii et al [18,19]. Also, thanks to those quasi-monotonicity properties and relying on the work of Ishii et al [18,19], an existence and uniqueness result of a Lipschitz viscosity solution was proved by El Hajj and Forcadel in [13] for the same (2×2) system.…”

“…where u i and u i are, respectively, the upper and lower relaxed semi-limits defined in (7) and (8).…”

“…Also, thanks to those quasi-monotonicity properties and relying on the work of Ishii et al [18,19], an existence and uniqueness result of a Lipschitz viscosity solution was proved by El Hajj and Forcadel in [13] for the same (2×2) system. In our paper, we will generalize the result established in [8] to the case of non-linear and strongly coupled (d × d) eikonal system (1) but without the quasi-monotonicity condition and with very low regularity on initial data and on the velocity. The idea is based on regularizing the initial data and the velocity in order to preserve the a priori estimates obtained previously in [8] for the scalar equation (independent of u), particularly the BV estimate.…”

“…In our paper, we will generalize the result established in [8] to the case of non-linear and strongly coupled (d × d) eikonal system (1) but without the quasi-monotonicity condition and with very low regularity on initial data and on the velocity. The idea is based on regularizing the initial data and the velocity in order to preserve the a priori estimates obtained previously in [8] for the scalar equation (independent of u), particularly the BV estimate. Then, after freezing the vector field u = (u i ) 1≤i≤d that appears in the velocity term λ i inside an appropriate Banach space, we will show, using a fixed point theorem and the BV estimate, the global-in-time existence result of the regularized problem.…”

“…Let us mention some of the known results. First, motivated by dislocation dynamics, we can point out the result done by El Hajj and Boudjerada in [8] who were able to prove the global existence of discontinuous viscosity BV solutions for scalar one dimensional non-linear and non-local eikonal equations, including in particular the case d = 1 in system (1), where the velocity is independent of the solution. This result has been extended in El Hajj et al [14] to a more general nonlinear and non-local (2 × 2) system where the authors were able to rely on the fact that this (2 × 2) system has a particular local structure, making it possible to verify some quasi-monotonicity properties.…”