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doi: 10.3934/cpaa.2021131
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## 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 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 &amp; Applied Analysis, doi: 10.3934/cpaa.2021131
##### References:
 [1] R. A. Adams, Sobolev Spaces, Academic Press, New York, London, 1975.   Google Scholar [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. Google Scholar [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.  Google Scholar [4] (Berlin) G. Barles, Solutions de viscosité des équations de Hamilton-Jacobi, in Mathématiques & Applications, Springer-Verlag, Paris, 1994. Google Scholar [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.  Google Scholar [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.  Google Scholar [7] M. Cannone, A. El Hajj, R. 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.  Google Scholar [8] 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 [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [12] A. El Hajj, H. 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.  Google Scholar [13] A. El Hajj, H. Ibrahim and V. Rizik, BV solution for a non-linear hamilton-jacobi system, Discret. Contin. Dyn. Syst. Series A, 41 (2021), 3273-3293.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [17] H. Ishii, Perron's method for monotone systems of second-order elliptic partial differential equations, Differ. Integral Equ., 5 (1992), 1-24.   Google Scholar [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.  Google Scholar [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.   Google Scholar [20] O. A. Ladyzhenskaia, V. A. Solonnikov and N. N. Ural'Tseva, Linear and Quasi-Linear Equations of Parabolic Type, American Mathematical Soc., 1988.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [24] J. Simon, Compact sets in the space $L^p$, Ann. Mat. Pura Appl., 146 (1987), 65-96.  doi: 10.1007/BF01762360.  Google Scholar

show all references

##### References:
 [1] R. A. Adams, Sobolev Spaces, Academic Press, New York, London, 1975.   Google Scholar [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. Google Scholar [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.  Google Scholar [4] (Berlin) G. Barles, Solutions de viscosité des équations de Hamilton-Jacobi, in Mathématiques & Applications, Springer-Verlag, Paris, 1994. Google Scholar [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.  Google Scholar [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.  Google Scholar [7] M. Cannone, A. El Hajj, R. 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.  Google Scholar [8] 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 [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [12] A. El Hajj, H. 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.  Google Scholar [13] A. El Hajj, H. Ibrahim and V. Rizik, BV solution for a non-linear hamilton-jacobi system, Discret. Contin. Dyn. Syst. Series A, 41 (2021), 3273-3293.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [17] H. Ishii, Perron's method for monotone systems of second-order elliptic partial differential equations, Differ. Integral Equ., 5 (1992), 1-24.   Google Scholar [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.  Google Scholar [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.   Google Scholar [20] O. A. Ladyzhenskaia, V. A. Solonnikov and N. N. Ural'Tseva, Linear and Quasi-Linear Equations of Parabolic Type, American Mathematical Soc., 1988.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [24] J. Simon, Compact sets in the space $L^p$, Ann. Mat. Pura Appl., 146 (1987), 65-96.  doi: 10.1007/BF01762360.  Google Scholar
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