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July  2021, 41(7): 3273-3293. doi: 10.3934/dcds.2020405

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

Ahmad El Hajj 1,, , Hassan Ibrahim 2, and  Vivian Rizik 2,

1. 

Université de Technologie de Compiègne, LMAC, 60205 Compiègne Cedex, France
2. 

Université Libanaise, EDST, Hadath, Beyrouth, Liban

* Corresponding author: Ahmad El Hajj

Received  February 2020 Revised  November 2020 Published  July 2021 Early access  December 2020

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.

Keywords: Hamilton-Jacobi system, non-linear eikonal system, non-linear hyperbolic system, BV estimate, discontinuous viscosity solution.
Mathematics Subject Classification: 58F15, 58F1735A01, 74G25, 35F20, 35F21, 35L40, 35Q35, 35D40.
Citation: Ahmad El Hajj, Hassan Ibrahim, Vivian Rizik. $ BV $ solution for a non-linear Hamilton-Jacobi system. Discrete & Continuous Dynamical Systems, 2021, 41 (7) : 3273-3293. doi: 10.3934/dcds.2020405
References:
[1]

L. Ambrosio, N. Fusco and D. Pallara, Functions of Bounded Variations and Free Discontinuity Problems, Oxford Mathematical Monographs, The Clarendon Press Oxford University Press, New York, 2000.  Google Scholar

[2]

G. Barles, Discontinuous viscosity solutions of first-order Hamilton-Jacobi equations: A guided visit, Nonlinear Anal., 20 (1993), 1123-1134.  doi: 10.1016/0362-546X(93)90098-D.  Google Scholar

[3]

G. Barles, Solutions de Viscosité Des Équations de Hamilton-Jacobi, vol. 17 of Mathématiques et Applications (Berlin), Springer-Verlag, Paris, 1994.  Google Scholar

[4]

G. Barles and B. Perthame, Exit time problems in optimal control and vanishing viscosity method, SIAM J. Control Optim., 26 (1988), 1133-1148.  doi: 10.1137/0326063.  Google Scholar

[5]

G. Barles and B. Perthame, Comparison principle for Dirichlet-type Hamilton-Jacobi equations and singular perturbations of degenerated elliptic equations, Appl. Math. Optim., 21 (1990), 21-44.  doi: 10.1007/BF01445155.  Google Scholar

[6]

G. Barles, H. M. Soner and P. E. Souganidis, Front propagation and phase field theory, SIAM J. Control Optim, 31 (1993), 439-496.  doi: 10.1137/0331021.  Google Scholar

[7]

S. Bianchini and A. Bressan, Vanishing viscosity solutions of nonlinear hyperbolic systems, Ann. Math., 161 (2005), 223-342.  doi: 10.4007/annals.2005.161.223.  Google Scholar

[8]

R. Boudjerada and A. El Hajj, Global existence results for eikonal equation with $BV$ initial data, Nonlinear Differ. Equ. Appl., 22 (2015), 947-978.  doi: 10.1007/s00030-015-0310-9.  Google Scholar

[9]

M. G. Crandall, H. Ishii and P.-L. Lions, User's guide to viscosity solutions of second order partial differential equations, Bull. Amer. Math. Soc. (N.S.), 27 (1992), 1-67.  doi: 10.1090/S0273-0979-1992-00266-5.  Google Scholar

[10]

M. G. Crandall and P.-L. Lions, Viscosity solutions of Hamilton-Jacobi equations, Trans. Amer. Math. Soc., 277 (1983), 1-42.  doi: 10.1090/S0002-9947-1983-0690039-8.  Google Scholar

[11]

R. J. DiPerna, Convergence of approximate solutions to conservation laws, Arch. Ration. Mech. Anal., 82 (1983), 27-70.  doi: 10.1007/BF00251724.  Google Scholar

[12]

R. J. DiPerna, Compensated compactness and general systems of conservation laws, Trans. Amer. Math. Soc., 292 (1985), 383-420.  doi: 10.1090/S0002-9947-1985-0808729-4.  Google Scholar

[13]

A. El Hajj and N. Forcadel, A convergent scheme for a non-local coupled system modelling dislocation densities dynamics, Math. Comp., 77 (2008), 789-812.  doi: 10.1090/S0025-5718-07-02038-8.  Google Scholar

[14]

A. El Hajj, H. Ibrahim and V. Rizik, Global $BV$ solution for a non-local coupled system modeling the dynamics of dislocation densities, J. Differential Equations, 264 (2018), 1750-1785.  doi: 10.1016/j.jde.2017.10.004.  Google Scholar

[15]

A. El Hajj and R. Monneau, Uniqueness results for diagonal hyperbolic systems with large and monotone data, J. Hyper. Differ. Equ., 10 (2013), 461-494.  doi: 10.1142/S0219891613500161.  Google Scholar

[16]

A. El Hajj and R. Monneau, Global continuous solutions for diagonal hyperbolic systems with large and monotone data, J. Hyper. Differ. Equ., 7 (2010), 139-164.  doi: 10.1142/S0219891610002050.  Google Scholar

[17]

J. Glimm, Solutions in the large for nonlinear hyperbolic systems of equations, Commun. Pure Appl. Math., 18 (1965), 697-715.  doi: 10.1002/cpa.3160180408.  Google Scholar

[18]

H. Ishii, Perron's method for monotone systems of second-order elliptic partial differential equations, Differential Integral Equations, 5 (1992), 1-24.   Google Scholar

[19]

H. Ishii and S. Koike, Viscosity solution for monotone systems of second-order elliptic PDEs, Comm. Partial Differential Equations, 16 (1991), 1095-1128.  doi: 10.1080/03605309108820791.  Google Scholar

[20]

P. D. Lax, Hyperbolic systems of conservation laws and the mathematical theory of shock waves, CBMS Regional Conference Series in Mathematics, Vol. 11 (SIAM, Philadelphia, 1973).  Google Scholar

[21]

P. LeFloch, Entropy weak solutions to nonlinear hyperbolic systems under nonconservative form, Commun. Partial Differential Equations, 13 (1988), 669-727.  doi: 10.1080/03605308808820557.  Google Scholar

[22]

P. LeFloch and T.-P. Liu, Existence theory for nonlinear hyperbolic systems in nonconservative form, Forum Math., 5 (1993), 261-280.  doi: 10.1515/form.1993.5.261.  Google Scholar

[23]

O. Ley, Lower-bound gradient estimates for first-order Hamilton-Jacobi equations and applications to the regularity of propagating fronts, Adv. Differential Equations, 6 (2001), 547-576.   Google Scholar

[24]

J. Simon, Compacts sets in the space $L^p(0; T; B)$, Ann. Mat. Pura. Appl., 146 (1987), 65-96.  doi: 10.1007/BF01762360.  Google Scholar

show all references

References:
[1]

L. Ambrosio, N. Fusco and D. Pallara, Functions of Bounded Variations and Free Discontinuity Problems, Oxford Mathematical Monographs, The Clarendon Press Oxford University Press, New York, 2000.  Google Scholar

[2]

G. Barles, Discontinuous viscosity solutions of first-order Hamilton-Jacobi equations: A guided visit, Nonlinear Anal., 20 (1993), 1123-1134.  doi: 10.1016/0362-546X(93)90098-D.  Google Scholar

[3]

G. Barles, Solutions de Viscosité Des Équations de Hamilton-Jacobi, vol. 17 of Mathématiques et Applications (Berlin), Springer-Verlag, Paris, 1994.  Google Scholar

[4]

G. Barles and B. Perthame, Exit time problems in optimal control and vanishing viscosity method, SIAM J. Control Optim., 26 (1988), 1133-1148.  doi: 10.1137/0326063.  Google Scholar

[5]

G. Barles and B. Perthame, Comparison principle for Dirichlet-type Hamilton-Jacobi equations and singular perturbations of degenerated elliptic equations, Appl. Math. Optim., 21 (1990), 21-44.  doi: 10.1007/BF01445155.  Google Scholar

[6]

G. Barles, H. M. Soner and P. E. Souganidis, Front propagation and phase field theory, SIAM J. Control Optim, 31 (1993), 439-496.  doi: 10.1137/0331021.  Google Scholar

[7]

S. Bianchini and A. Bressan, Vanishing viscosity solutions of nonlinear hyperbolic systems, Ann. Math., 161 (2005), 223-342.  doi: 10.4007/annals.2005.161.223.  Google Scholar

[8]

R. Boudjerada and A. El Hajj, Global existence results for eikonal equation with $BV$ initial data, Nonlinear Differ. Equ. Appl., 22 (2015), 947-978.  doi: 10.1007/s00030-015-0310-9.  Google Scholar

[9]

M. G. Crandall, H. Ishii and P.-L. Lions, User's guide to viscosity solutions of second order partial differential equations, Bull. Amer. Math. Soc. (N.S.), 27 (1992), 1-67.  doi: 10.1090/S0273-0979-1992-00266-5.  Google Scholar

[10]

M. G. Crandall and P.-L. Lions, Viscosity solutions of Hamilton-Jacobi equations, Trans. Amer. Math. Soc., 277 (1983), 1-42.  doi: 10.1090/S0002-9947-1983-0690039-8.  Google Scholar

[11]

R. J. DiPerna, Convergence of approximate solutions to conservation laws, Arch. Ration. Mech. Anal., 82 (1983), 27-70.  doi: 10.1007/BF00251724.  Google Scholar

[12]

R. J. DiPerna, Compensated compactness and general systems of conservation laws, Trans. Amer. Math. Soc., 292 (1985), 383-420.  doi: 10.1090/S0002-9947-1985-0808729-4.  Google Scholar

[13]

A. El Hajj and N. Forcadel, A convergent scheme for a non-local coupled system modelling dislocation densities dynamics, Math. Comp., 77 (2008), 789-812.  doi: 10.1090/S0025-5718-07-02038-8.  Google Scholar

[14]

A. El Hajj, H. Ibrahim and V. Rizik, Global $BV$ solution for a non-local coupled system modeling the dynamics of dislocation densities, J. Differential Equations, 264 (2018), 1750-1785.  doi: 10.1016/j.jde.2017.10.004.  Google Scholar

[15]

A. El Hajj and R. Monneau, Uniqueness results for diagonal hyperbolic systems with large and monotone data, J. Hyper. Differ. Equ., 10 (2013), 461-494.  doi: 10.1142/S0219891613500161.  Google Scholar

[16]

A. El Hajj and R. Monneau, Global continuous solutions for diagonal hyperbolic systems with large and monotone data, J. Hyper. Differ. Equ., 7 (2010), 139-164.  doi: 10.1142/S0219891610002050.  Google Scholar

[17]

J. Glimm, Solutions in the large for nonlinear hyperbolic systems of equations, Commun. Pure Appl. Math., 18 (1965), 697-715.  doi: 10.1002/cpa.3160180408.  Google Scholar

[18]

H. Ishii, Perron's method for monotone systems of second-order elliptic partial differential equations, Differential Integral Equations, 5 (1992), 1-24.   Google Scholar

[19]

H. Ishii and S. Koike, Viscosity solution for monotone systems of second-order elliptic PDEs, Comm. Partial Differential Equations, 16 (1991), 1095-1128.  doi: 10.1080/03605309108820791.  Google Scholar

[20]

P. D. Lax, Hyperbolic systems of conservation laws and the mathematical theory of shock waves, CBMS Regional Conference Series in Mathematics, Vol. 11 (SIAM, Philadelphia, 1973).  Google Scholar

[21]

P. LeFloch, Entropy weak solutions to nonlinear hyperbolic systems under nonconservative form, Commun. Partial Differential Equations, 13 (1988), 669-727.  doi: 10.1080/03605308808820557.  Google Scholar

[22]

P. LeFloch and T.-P. Liu, Existence theory for nonlinear hyperbolic systems in nonconservative form, Forum Math., 5 (1993), 261-280.  doi: 10.1515/form.1993.5.261.  Google Scholar

[23]

O. Ley, Lower-bound gradient estimates for first-order Hamilton-Jacobi equations and applications to the regularity of propagating fronts, Adv. Differential Equations, 6 (2001), 547-576.   Google Scholar

[24]

J. Simon, Compacts sets in the space $L^p(0; T; B)$, Ann. Mat. Pura. Appl., 146 (1987), 65-96.  doi: 10.1007/BF01762360.  Google Scholar

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