# American Institute of Mathematical Sciences

September  2015, 35(9): 4225-4239. doi: 10.3934/dcds.2015.35.4225

## Global propagation of singularities for time dependent Hamilton-Jacobi equations

 1 Dipartimento di Matematica, Università degli Studi di Roma "Tor Vergata", Via della Ricerca Scientifica, 00133 Roma 2 CNRS, IMJ-PRG, UMR 7586, Sorbonne Universités, UPMC Univ Paris Diderot, Sorbonne Paris Cité, Case 247, 4 Place Jussieu, 75252 Paris, France 3 Dipartimento di Matematica, Università di Roma, Via della Ricerca Scientifica 1, 00133 Roma

Received  August 2014 Revised  October 2014 Published  April 2015

We investigate the properties of the set of singularities of semiconcave solutions of Hamilton-Jacobi equations of the form $$\label{abstract:EQ} u_t(t,x)+H(\nabla u(t,x))=0, \qquad\text{a.e. }(t,x)\in (0,+\infty)\times\Omega\subset\mathbb{R}^{n+1}\,.$$ It is well known that the singularities of such solutions propagate locally along generalized characteristics. Special generalized characteristics, satisfying an energy condition, can be constructed, under some assumptions on the structure of the Hamiltonian $H$. In this paper, we provide estimates of the dissipative behavior of the energy along such curves. As an application, we prove that the singularities of any viscosity solution of (1) cannot vanish in a finite time.
Citation: Piermarco Cannarsa, Marco Mazzola, Carlo Sinestrari. Global propagation of singularities for time dependent Hamilton-Jacobi equations. Discrete and Continuous Dynamical Systems, 2015, 35 (9) : 4225-4239. doi: 10.3934/dcds.2015.35.4225
##### References:
 [1] P. Albano, Propagation of singularities for solutions of Hamilton-Jacobi equations, J. Math. Anal. Appl., 411 (2014), 684-687. doi: 10.1016/j.jmaa.2013.10.015. [2] P. Albano and P. Cannarsa, Propagation of singularities for concave solutions of Hamilton-Jacobi equations, in EQUADIFF 99 Proceedings of the International Conference on Differential Equations (eds. D. Fiedler, K. Gröger and J. Sprekels), World Scientific, Singapore, (2000), 583-588. [3] P. Albano and P. Cannarsa, Propagation of singularities for solutions of nonlinear first order partial differential equations, Arch. Ration. Mech. Anal., 162 (2002), 1-23. doi: 10.1007/s002050100176. [4] P. Albano, P. Cannarsa, K. T. Nguyen and C. Sinestrari, Singular gradient flow of the distance fundtion and homotopy equivalence, Math. Ann., 356 (2013), 23-43. doi: 10.1007/s00208-012-0835-8. [5] M. Bardi and I. Capuzzo Dolcetta, Optimal Control and Viscosity Solutions of Hamilton-Jacobi Equations, Birkhäuser, Boston, 1997. doi: 10.1007/978-0-8176-4755-1. [6] P. Cannarsa, A. Mennucci and C. Sinestrari, Regularity results for solutions of a class of Hamilton-Jacobi equations, Arch. Rational Mech. Anal., 140 (1997), 197-223. doi: 10.1007/s002050050064. [7] P. Cannarsa and C. Sinestrari, Semiconcave Functions, Hamilton-Jacobi Equations and Optimal Control, Birkhäuser, Boston, 2004. [8] P. Cannarsa and Y. Yu, Singular dynamics for semiconcave functions, J. Eur. Math. Soc. (JEMS), 11 (2009), 999-1024. doi: 10.4171/JEMS/173. [9] 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. [10] M. G. Crandall, L. C. Evans and P. L. Lions, Some properties of viscosity solutions of Hamilton-Jacobi equations, Trans. Amer. Math. Soc., 282 (1984), 487-502. doi: 10.1090/S0002-9947-1984-0732102-X. [11] C. Dafermos, Generalized characteristics and the structure of solutions of hyperbolic conservation laws, Indiana Univ. Math. J., 26 (1977), 1097-1119. doi: 10.1512/iumj.1977.26.26088. [12] A. Douglis, The continuous dependence of generalized solutions of non-linear partial differential equations upon initial data, Comm. Pure Appl. Math., 14 (1961), 267-284. doi: 10.1002/cpa.3160140307. [13] W. H. Fleming and H. M. Soner, Controlled Markov Processes and Viscosity Solutions, Springer Verlag, Berlin, 1993. [14] S. N. Kruzhkov, Generalized solutions of the Hamilton-Jacobi equations of the eikonal type I, Math. USSR Sb., 27 (1975), 406-445. [15] N. V. Krylov, Nonlinear Elliptic and Parabolic Equations of the Second Order. Translated from the Russian by P. L. Buzytsky, Mathematics and its Applications (Soviet Series), 7. D. Reidel Publishing Co., Dordrecht, 1987. doi: 10.1007/978-94-010-9557-0. [16] P. L. Lions, Generalized Solutions of Hamilton-Jacobi Equations, Pitman, London, 1982. [17] T. Strömberg, Propagation of singularities along broken characteristics, Nonlinear Anal., 85 (2013), 93-109. doi: 10.1016/j.na.2013.02.024. [18] T. Strömberg and F. Ahmadzadeh, Excess action and broken characteristics for Hamilton-Jacobi equations, Nonlinear Anal., 110 (2014), 113-129. doi: 10.1016/j.na.2014.08.001. [19] C. Villani, Optimal Transport, Old and New, Springer, Berlin - Heidelberg, 2009. doi: 10.1007/978-3-540-71050-9. [20] Y. Yu, A simple proof of the propagation of singularities for solutions of Hamilton-Jacobi equations, Ann. Sc. Norm. Super. Pisa Cl. Sci., 5 (2006), 439-444.

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##### References:
 [1] P. Albano, Propagation of singularities for solutions of Hamilton-Jacobi equations, J. Math. Anal. Appl., 411 (2014), 684-687. doi: 10.1016/j.jmaa.2013.10.015. [2] P. Albano and P. Cannarsa, Propagation of singularities for concave solutions of Hamilton-Jacobi equations, in EQUADIFF 99 Proceedings of the International Conference on Differential Equations (eds. D. Fiedler, K. Gröger and J. Sprekels), World Scientific, Singapore, (2000), 583-588. [3] P. Albano and P. Cannarsa, Propagation of singularities for solutions of nonlinear first order partial differential equations, Arch. Ration. Mech. Anal., 162 (2002), 1-23. doi: 10.1007/s002050100176. [4] P. Albano, P. Cannarsa, K. T. Nguyen and C. Sinestrari, Singular gradient flow of the distance fundtion and homotopy equivalence, Math. Ann., 356 (2013), 23-43. doi: 10.1007/s00208-012-0835-8. [5] M. Bardi and I. Capuzzo Dolcetta, Optimal Control and Viscosity Solutions of Hamilton-Jacobi Equations, Birkhäuser, Boston, 1997. doi: 10.1007/978-0-8176-4755-1. [6] P. Cannarsa, A. Mennucci and C. Sinestrari, Regularity results for solutions of a class of Hamilton-Jacobi equations, Arch. Rational Mech. Anal., 140 (1997), 197-223. doi: 10.1007/s002050050064. [7] P. Cannarsa and C. Sinestrari, Semiconcave Functions, Hamilton-Jacobi Equations and Optimal Control, Birkhäuser, Boston, 2004. [8] P. Cannarsa and Y. Yu, Singular dynamics for semiconcave functions, J. Eur. Math. Soc. (JEMS), 11 (2009), 999-1024. doi: 10.4171/JEMS/173. [9] 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. [10] M. G. Crandall, L. C. Evans and P. L. Lions, Some properties of viscosity solutions of Hamilton-Jacobi equations, Trans. Amer. Math. Soc., 282 (1984), 487-502. doi: 10.1090/S0002-9947-1984-0732102-X. [11] C. Dafermos, Generalized characteristics and the structure of solutions of hyperbolic conservation laws, Indiana Univ. Math. J., 26 (1977), 1097-1119. doi: 10.1512/iumj.1977.26.26088. [12] A. Douglis, The continuous dependence of generalized solutions of non-linear partial differential equations upon initial data, Comm. Pure Appl. Math., 14 (1961), 267-284. doi: 10.1002/cpa.3160140307. [13] W. H. Fleming and H. M. Soner, Controlled Markov Processes and Viscosity Solutions, Springer Verlag, Berlin, 1993. [14] S. N. Kruzhkov, Generalized solutions of the Hamilton-Jacobi equations of the eikonal type I, Math. USSR Sb., 27 (1975), 406-445. [15] N. V. Krylov, Nonlinear Elliptic and Parabolic Equations of the Second Order. Translated from the Russian by P. L. Buzytsky, Mathematics and its Applications (Soviet Series), 7. D. Reidel Publishing Co., Dordrecht, 1987. doi: 10.1007/978-94-010-9557-0. [16] P. L. Lions, Generalized Solutions of Hamilton-Jacobi Equations, Pitman, London, 1982. [17] T. Strömberg, Propagation of singularities along broken characteristics, Nonlinear Anal., 85 (2013), 93-109. doi: 10.1016/j.na.2013.02.024. [18] T. Strömberg and F. Ahmadzadeh, Excess action and broken characteristics for Hamilton-Jacobi equations, Nonlinear Anal., 110 (2014), 113-129. doi: 10.1016/j.na.2014.08.001. [19] C. Villani, Optimal Transport, Old and New, Springer, Berlin - Heidelberg, 2009. doi: 10.1007/978-3-540-71050-9. [20] Y. Yu, A simple proof of the propagation of singularities for solutions of Hamilton-Jacobi equations, Ann. Sc. Norm. Super. Pisa Cl. Sci., 5 (2006), 439-444.
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