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November  2021, 20(11): 3795-3823. doi: 10.3934/cpaa.2021131

Continuous solution for a non-linear eikonal system

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

* Corresponding author

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

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 $ BV $ estimates. A uniqueness result is also proved through a comparison principle property.

Citation: Ahmad El Hajj, Aya Oussaily. Continuous solution for a non-linear eikonal system. Communications on Pure and Applied Analysis, 2021, 20 (11) : 3795-3823. doi: 10.3934/cpaa.2021131
References:
[1] R. A. Adams, Sobolev Spaces, Academic Press, New York, London, 1975. 
[2]

M. Al Zohbi, A. El Hajj and M. Jazar, Global existence to a diagonal hyperbolic system for any bv initial data, to appear in Nonlinearity, 2021.

[3]

M. Bardi and I. Capuzzo-Dolcetta, Optimal control and viscosity solutions of Hamilton-Jacobi-Bellman equations, in Systems & Control: Foundations & Applications, Birkhäuser Boston, Inc., Boston, MA, 1997. With appendices by Maurizio Falcone and Pierpaolo Soravia. doi: 10.1007/978-0-8176-4755-1.

[4]

(Berlin) G. Barles, Solutions de viscosité des équations de Hamilton-Jacobi, in Mathématiques & Applications, Springer-Verlag, Paris, 1994.

[5]

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.

[6]

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.

[7]

M. CannoneA. El HajjR. Monneau and F. Ribaud, Global existence for a system of non-linear and non-local transport equations describing the dynamics of dislocation densities, Arch. Ration. Mech. Anal., 196 (2010), 71-96.  doi: 10.1007/s00205-009-0235-8.

[8]

M. G. CrandallH. 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.

[9]

A. El Hajj, Well-posedness theory for a nonconservative Burgers-type system arising in dislocation dynamics, SIAM J. Math. Anal., 39 (2007), 965-986.  doi: 10.1137/060672170.

[10]

A. El Hajj, Global solution for a non-local eikonal equation modelling dislocation dynamics, Nonlinear Anal., 168 (2018), 154-175.  doi: 10.1016/j.na.2017.11.012.

[11]

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

[12]

A. El HajjH. Ibrahim and V. Rizik, Global BV solution for a non-local coupled system modeling the dynamics of dislocation densities, J. Differ. Equ., 264 (2018), 1750-1785.  doi: 10.1016/j.jde.2017.10.004.

[13]

A. El HajjH. Ibrahim and V. Rizik, BV solution for a non-linear hamilton-jacobi system, Discret. Contin. Dyn. Syst. Series A, 41 (2021), 3273-3293. 

[14]

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

[15]

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

[16]

A. El Hajj and A. Oussaily, Existence and uniqueness of continuous solution for a non-local coupled system modeling the dynamics of dislocation densities, J. Nonlinear Sci., 31 (2021), 1-41.  doi: 10.1007/s00332-021-09676-7.

[17]

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

[18]

H. Ishii and S. Koike, Viscosity solutions for monotone systems of second-order elliptic PDEs, Commun. Partial Differ. Equ., 16 (1991), 1095–1128. doi: 10.1080/03605309108820791.

[19]

H. Ishii and S. Koike, Viscosity solutions of a system of nonlinear second-order elliptic PDEs arising in switching games, Funkcial. Ekvac., 34 (1991), 143-155. 

[20]

O. A. Ladyzhenskaia, V. A. Solonnikov and N. N. Ural'Tseva, Linear and Quasi-Linear Equations of Parabolic Type, American Mathematical Soc., 1988.

[21]

P. D. Lax, Hyperbolic Systems of Conservation Laws and the Mathematical Theory of Shock Waves, Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1973. Conference Board of the Mathematical Sciences Regional Conference Series in Applied Mathematics, No. 11.

[22]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.

[23]

R. Redheffer and W. Walter, The total variation of solutions of parabolic differential equations and a maximum principle in unbounded domains, Math. Ann., 209 (1974), 57–67. doi: 10.1007/BF01432886.

[24]

J. Simon, Compact sets in the space $L^p$, Ann. Mat. Pura Appl., 146 (1987), 65-96.  doi: 10.1007/BF01762360.

show all references

References:
[1] R. A. Adams, Sobolev Spaces, Academic Press, New York, London, 1975. 
[2]

M. Al Zohbi, A. El Hajj and M. Jazar, Global existence to a diagonal hyperbolic system for any bv initial data, to appear in Nonlinearity, 2021.

[3]

M. Bardi and I. Capuzzo-Dolcetta, Optimal control and viscosity solutions of Hamilton-Jacobi-Bellman equations, in Systems & Control: Foundations & Applications, Birkhäuser Boston, Inc., Boston, MA, 1997. With appendices by Maurizio Falcone and Pierpaolo Soravia. doi: 10.1007/978-0-8176-4755-1.

[4]

(Berlin) G. Barles, Solutions de viscosité des équations de Hamilton-Jacobi, in Mathématiques & Applications, Springer-Verlag, Paris, 1994.

[5]

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.

[6]

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.

[7]

M. CannoneA. El HajjR. Monneau and F. Ribaud, Global existence for a system of non-linear and non-local transport equations describing the dynamics of dislocation densities, Arch. Ration. Mech. Anal., 196 (2010), 71-96.  doi: 10.1007/s00205-009-0235-8.

[8]

M. G. CrandallH. 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.

[9]

A. El Hajj, Well-posedness theory for a nonconservative Burgers-type system arising in dislocation dynamics, SIAM J. Math. Anal., 39 (2007), 965-986.  doi: 10.1137/060672170.

[10]

A. El Hajj, Global solution for a non-local eikonal equation modelling dislocation dynamics, Nonlinear Anal., 168 (2018), 154-175.  doi: 10.1016/j.na.2017.11.012.

[11]

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

[12]

A. El HajjH. Ibrahim and V. Rizik, Global BV solution for a non-local coupled system modeling the dynamics of dislocation densities, J. Differ. Equ., 264 (2018), 1750-1785.  doi: 10.1016/j.jde.2017.10.004.

[13]

A. El HajjH. Ibrahim and V. Rizik, BV solution for a non-linear hamilton-jacobi system, Discret. Contin. Dyn. Syst. Series A, 41 (2021), 3273-3293. 

[14]

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

[15]

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

[16]

A. El Hajj and A. Oussaily, Existence and uniqueness of continuous solution for a non-local coupled system modeling the dynamics of dislocation densities, J. Nonlinear Sci., 31 (2021), 1-41.  doi: 10.1007/s00332-021-09676-7.

[17]

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

[18]

H. Ishii and S. Koike, Viscosity solutions for monotone systems of second-order elliptic PDEs, Commun. Partial Differ. Equ., 16 (1991), 1095–1128. doi: 10.1080/03605309108820791.

[19]

H. Ishii and S. Koike, Viscosity solutions of a system of nonlinear second-order elliptic PDEs arising in switching games, Funkcial. Ekvac., 34 (1991), 143-155. 

[20]

O. A. Ladyzhenskaia, V. A. Solonnikov and N. N. Ural'Tseva, Linear and Quasi-Linear Equations of Parabolic Type, American Mathematical Soc., 1988.

[21]

P. D. Lax, Hyperbolic Systems of Conservation Laws and the Mathematical Theory of Shock Waves, Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1973. Conference Board of the Mathematical Sciences Regional Conference Series in Applied Mathematics, No. 11.

[22]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.

[23]

R. Redheffer and W. Walter, The total variation of solutions of parabolic differential equations and a maximum principle in unbounded domains, Math. Ann., 209 (1974), 57–67. doi: 10.1007/BF01432886.

[24]

J. Simon, Compact sets in the space $L^p$, Ann. Mat. Pura Appl., 146 (1987), 65-96.  doi: 10.1007/BF01762360.

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