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Effective junction conditions for degenerate parabolic equations

  • Cyril ImbertEmail author
  • Vinh Duc Nguyen
Article
  • 131 Downloads

Abstract

We are interested in the study of parabolic equations on a multi-dimensional junction, i.e. the union of a finite number of copies of a half-hyperplane of dimension \(d+1\) whose boundaries are identified. The common boundary is referred to as the junction hyperplane. The parabolic equations on the half-hyperplanes are in non-divergence form, fully non-linear and possibly degenerate, and they do degenerate and are quasi-convex along the junction hyperplane. More precisely, along the junction hyperplane the nonlinearities do not depend on second order derivatives and their sublevel sets with respect to the gradient variable are convex. The parabolic equations are supplemented with a non-linear boundary condition of Neumann type, referred to as a generalized junction condition, which is compatible with the maximum principle. Our main result asserts that imposing a generalized junction condition in a weak sense reduces to imposing an effective one in a strong sense. This result extends the one obtained by Imbert and Monneau for Hamilton–Jacobi equations on networks and multi-dimensional junctions. We give two applications of this result. On the one hand, we give the first complete answer to an open question about these equations: we prove in the two-domain case that the vanishing viscosity limit associated with quasi-convex Hamilton–Jacobi equations coincides with the maximal Ishii solution identified by Barles et al. (ESAIM Control Optim Calc Var 19(3):710–739, 2013). On the other hand, we give a short and simple PDE proof of a large deviation result of Boué et al. (Probab Theory Relat Fields 116:125–149, 2000).

Mathematics Subject Classification

49L25 35K65 35R02 

Notes

Acknowledgements

The authors thank Russell Schwab for fruitful discussions in early stage of this work and for suggesting to address the large deviation problem. They also thank Guy Barles for stimulating discussions about the vanishing viscosity limit. The authors are also indebted to one of the referees who read very carefully seven successive versions of this work and made valuable recommandations about both presentation and proofs.

References

  1. 1.
    Achdou, Y., Camilli, F., Cutrì, A., Tchou, N.: Hamilton–Jacobi equations constrained on networks. NoDEA Nonlinear Differ. Equ. Appl. 20(3), 413–445 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    Andreianov, B., Sbihi, K.: Strong boundary traces and well-posedness for scalar conservation laws with dissipative boundary conditions. In: Benzoni-Gavage, S., Serre, D. (eds.) Hyperbolic Problems: Theory, Numerics, Applications, pp. 937–945. Springer, Berlin (2008)Google Scholar
  3. 3.
    Andreianov, B., Sbihi, K.: Scalar conservation laws with nonlinear boundary conditions. C. R. Math. Acad. Sci. Paris 345(8), 431–434 (2007)CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    Andreianov, B., Sbihi, K.: Well-posedness of general boundary-value problems for scalar conservation laws. Trans. Am. Math. Soc. 367(6), 3763–3806 (2015)CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    Barles, G., Briani, A., Chasseigne, E.: A Bellman approach for two-domains optimal control problems in \({\mathbb{R}}^N\). ESAIM Control Optim. Calc. Var. 19(3), 710–739 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    Barles, G., Briani, A., Chasseigne, E.: A Bellman approach for regional optimal control problems in \({\mathbb{R}}^N\). SIAM J. Control Optim. 52(3), 1712–1744 (2014)CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    Barles, G., Briani, A., Chasseigne, E., Imbert, C.: Flux-limited and classical viscosity solutions for regional control problems. Preprint HAL 01392414 (2016)Google Scholar
  8. 8.
    Boué, M., Dupuis, P., Ellis, R.S.: Large deviations for small noise diffusions with discontinuous statistics. Probab. Theory Relat. Fields 116(1), 125–149 (2000)CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Camilli, F., Marchi, C., Schieborn, D.: The vanishing viscosity limit for Hamilton–Jacobi equations on networks. J. Differ. Equ. 254(10), 4122–4143 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    Clarke, F.H., Ledyaev, Y.S., Stern, R.J., Wolenski, P.R.: Nonsmooth Analysis and Control Theory, Volume 178 of Graduate Texts in Mathematics. Springer, New York (1998)Google Scholar
  11. 11.
    Crandall, M.G., Evans, L.C., Lions, P.-L.: Some properties of viscosity solutions of Hamilton–Jacobi equations. Trans. Am. Math. Soc. 282(2), 487–502 (1984)CrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    Crandall, M.G., Ishii, H., Lions, P.-L.: User’s guide to viscosity solutions of second order partial differential equations. Bull. Am. Math. Soc. (N.S.) 27(1), 1–67 (1992)CrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    Crandall, M.G., Lions, P.-L.: Viscosity solutions of Hamilton–Jacobi equations. Trans. Am. Math. Soc. 277(1), 1–42 (1983)CrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    Elliott, C.M., Giga, Y., Goto, S.: Dynamic boundary conditions for Hamilton–Jacobi equations. SIAM J. Math. Anal. 34(4), 861–881 (2003)CrossRefzbMATHMathSciNetGoogle Scholar
  15. 15.
    Freidlin, M.I., Wentzell, A.D.: Diffusion processes on an open book and the averaging principle. Stoch. Process. Appl. 113(1), 101–126 (2004)CrossRefzbMATHMathSciNetGoogle Scholar
  16. 16.
    Guerand, J.: Effective nonlinear boundary conditions for 1D nonconvex Hamilton–Jacobi equations. Preprint HAL 01372892 (2016)Google Scholar
  17. 17.
    Imbert, C., Monneau, R.: Quasi-convex Hamilton–Jacobi equations posed on junctions: the multi-dimensional case. Second version (2016)Google Scholar
  18. 18.
    Imbert, C., Monneau, R.: Flux-limited solutions for quasi-convex Hamilton–Jacobi equations on networks. Annales Scientifiques de l’École Normale Supérieure 50(2), 357–448 (2017)CrossRefzbMATHMathSciNetGoogle Scholar
  19. 19.
    Imbert, C., Monneau, R., Zidani, H.: A Hamilton–Jacobi approach to junction problems and application to traffic flows. ESAIM Control Optim. Calc. Var. 19(1), 129–166 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  20. 20.
    Ishii, H.: Perron’s method for Hamilton–Jacobi equations. Duke Math. J. 55(2), 369–384 (1987)CrossRefzbMATHMathSciNetGoogle Scholar
  21. 21.
    Fijavž, M.K., Mugnolo, D., Sikolya, E.: Variational and semigroup methods for waves and diffusion in networks. Appl. Math. Optim. 55(2), 219–240 (2007)CrossRefzbMATHMathSciNetGoogle Scholar
  22. 22.
    Lions, P.-L.: Lectures at Collège de France (2015–2016)Google Scholar
  23. 23.
    Lions, P.-L., Souganidis, P.: Viscosity solutions for junctions: well posedness and stability. First version. arXiv:1608.03682 (2016)
  24. 24.
    Monneau, R.: Personnal communicationGoogle Scholar
  25. 25.
    Oudet, S.: Hamilton–Jacobi equations for optimal control on multidimensional junctions. Preprint arXiv:1412.2679 (2014)
  26. 26.
    Pokornyi, Y.V., Borovskikh, A.V.: Differential equations on networks (geometric graphs). J. Math. Sci. (N. Y.) 119(6), 691–718 (2004)CrossRefzbMATHMathSciNetGoogle Scholar
  27. 27.
    Schieborn, D.: Viscosity solutions of Hamilton–Jacobi equations of Eikonal type on ramified spaces. Ph.D. thesis, Tübingen (2006)Google Scholar
  28. 28.
    Schieborn, D., Camilli, F.: Viscosity solutions of Eikonal equations on topological networks. Calc. Var. Partial Differ. Equ. 46(3–4), 671–686 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  29. 29.
    von Below, J.: Classical solvability of linear parabolic equations on networks. J. Differ. Equ. 72(2), 316–337 (1988)CrossRefzbMATHMathSciNetGoogle Scholar
  30. 30.
    von Below, J.: A maximum principle for semilinear parabolic network equations. In: Goldstein, J.A., Kappel, F., Schappacher, W. (eds.) Differential Equations with Applications in Biology, Physics, and Engineering (Leibnitz, 1989), Volume 133 of Lecture Notes in Pure and Applied Mathematics, pp. 37–45. Marcel Dekker, New York (1991)Google Scholar
  31. 31.
    von Below, J.: An existence result for semilinear parabolic network equations with dynamical node conditions. In: Bandle, C., Bemelmans, J., Chipot, M., Grüter, M., Saint Jean Paulin, J. (eds.) Progress in Partial Differential Equations: Elliptic and Parabolic Problems (Pont-à-Mousson, 1991), Volume 266 of Pitman Research Notes in Mathematics Series, pp. 274–283. Longman Science and Technology, Harlow (1992)Google Scholar
  32. 32.
    von Below, J., Nicaise, S.: Dynamical interface transition in ramified media with diffusion. Commun. Partial Differ. Equ. 21(1–2), 255–279 (1996)CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany 2017

Authors and Affiliations

  1. 1.CNRS & Department of Mathematics and ApplicationsÉcole Normale Supérieure (Paris)ParisFrance
  2. 2.CERMICS, École Nationale des Ponts et ChausséesUniversité Paris-EstMarne-La-Vallee Cedex 2France

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