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A constructive approach to robust chaos using invariant manifolds and expanding cones
$ 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 |
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.
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. |
[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. |
[3] |
G. Barles, Solutions de Viscosité Des Équations de Hamilton-Jacobi, vol. 17 of Mathématiques et Applications (Berlin), Springer-Verlag, Paris, 1994. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[11] |
R. J. DiPerna,
Convergence of approximate solutions to conservation laws, Arch. Ration. Mech. Anal., 82 (1983), 27-70.
doi: 10.1007/BF00251724. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[18] |
H. Ishii,
Perron's method for monotone systems of second-order elliptic partial differential equations, Differential Integral Equations, 5 (1992), 1-24.
|
[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. |
[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). |
[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. |
[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. |
[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.
|
[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. |
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. |
[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. |
[3] |
G. Barles, Solutions de Viscosité Des Équations de Hamilton-Jacobi, vol. 17 of Mathématiques et Applications (Berlin), Springer-Verlag, Paris, 1994. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[11] |
R. J. DiPerna,
Convergence of approximate solutions to conservation laws, Arch. Ration. Mech. Anal., 82 (1983), 27-70.
doi: 10.1007/BF00251724. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[18] |
H. Ishii,
Perron's method for monotone systems of second-order elliptic partial differential equations, Differential Integral Equations, 5 (1992), 1-24.
|
[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. |
[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). |
[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. |
[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. |
[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.
|
[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. |
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