AIMS Mathematics, 2018, 3(1): 96-130. doi: 10.3934/Math.2018.1.96.

Research article

Export file:

Format

  • RIS(for EndNote,Reference Manager,ProCite)
  • BibTex
  • Text

Content

  • Citation Only
  • Citation and Abstract

Minimization solutions to conservation laws with non-smooth and non-strictly convex flux

University of Pittsburgh, Mathematics Department, 301 Thackeray Hall, Pittsburgh PA 15260

Conservation laws are usually studied in the context of suffcient regularity conditionsimposed on the flux function, usually C2 and uniform convexity. Some results are proven with theaid of variational methods and a unique minimizer such as Hopf-Lax and Lax-Oleinik. We show thatmany of these classical results can be extended to a flux function that is not necessarily smooth oruniformly or strictly convex. Although uniqueness a.e. of the minimizer will generally no longerhold, by considering the greatest (or supremum, where applicable) of all possible minimizers, wecan successfully extend the results. One specific nonlinear case is that of a piecewise linear fluxfunction, for which we prove existence and uniqueness results. We also approximate it by a smoothed,superlinearized version parameterized by ε and consider the characterization of the minimizers for thesmooth version and limiting behavior as ε ↓ 0 to that of the sharp, polygonal problem. In proving akey result for the solution in terms of the value of the initial condition, we provide a stepping stone toanalyzing the system under stochastic processes, which will be explored further in a future paper.
  Figure/Table
  Supplementary
  Article Metrics

Keywords conservation laws; Hopf-Lax-Oleinik; shocks; variational problems in differential equations; minimization

Citation: Carey Caginalp. Minimization solutions to conservation laws with non-smooth and non-strictly convex flux. AIMS Mathematics, 2018, 3(1): 96-130. doi: 10.3934/Math.2018.1.96

References

  • 1. D. Applebaum, Levy Processes and Stochastic Calculus, Cambridge Studies in Advanced Mathematics, 2 Eds, Cambridge: Cambridge University Press, 2009.
  • 2. J. Bertoin, Levy Processes, Cambridge: Cambridge University Press, 1996.
  • 3. Y. Brienier and E. Grenier, Sticky particles and scalar conservation laws, SIAM J. Numer. Anal., 35 (1998), 2317–2328.
  • 4. M. Chabanol and J. Duchon, Markovian solutions of inviscid Burgers equation, J. Stat. Phys., 114 (2004), 525–534.
  • 5. M. Crandall, L. Evans and P. Lions, Some properties of viscosity solutions of Hamilton-Jacobi equations, Trans. Amer. Math. Soc., 282 (1984), 487–502.
  • 6. C. Dafermos, Polygonal Approximations of Solutions of the Initial Value Problem for a Conservation Law, J. Math. Anal. Appl., 38 (1972), 33–41.
  • 7. C. Dafermos, Hyberbolic Conservation Laws in Continuum Physics, 3 Eds, New York: Springer, 2010.
  • 8. E.Weinan, G. Rykov and G. Sinai, Generalized variational principles, global weak solutions and behavior with random initial data for systems of conservation laws arising in adhesion particle dynamics, Commun. Math. Phys., 177 (1996), 349–380.
  • 9. C. Evans, Partial Differential Equations, 2 Eds, New York: Springer, 2010.
  • 10. L. Frachebourg and P. Martin, Exact statistical properties of the Burgers equation, J. Fluid. Mech., 417 (2000), 323–349.
  • 11. D. G. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, New York: Springer, 1977.
  • 12. P. Groeneboom, Brownian motion with a parabolic drift and Airy functions, Probab. Theory Rel., 81 (1989), 79–109.
  • 13. H. Holden and N. Risebro, Front Tracking for Hyperbolic Conservation Laws, New York: Springer, 2015.
  • 14. E. Hopf, The partial differential equation ut + uux = μuxx , Comm. Pure Appl. Math., 3 (1950), 201–230.
  • 15. D. Kaspar and F. Rezakhanlou, Scalar conservation laws with monotone pure-jump Markov initial conditions, Probab. Theory Rel., 165 (2016), 867–899.
  • 16. K. Karlsen and N. Risebro, A note on front tracking and equivalence between viscosity solutions of Hamilton-Jacobi equations and entropy solutions of scalar conservation law, J. Nonlin. Anal., 50 (2002), 455–469.
  • 17. P. Lax, Hyperbolic systems of conservation laws II, Comm. Pure Appl. Math., 10 (1957), 537–566.
  • 18. P. Lax, Hyperbolic systems of conservation laws and the mathematical theory of shock waves, CBMS-NSF Regional Conference Series in Applied Mathematics, 1973.
  • 19. G. Menon, Complete integrability of shock clustering and Burgers turbulence, Arch. Ration. Mech. An., 203 (2012), 853–882.
  • 20. G. Menon and R. Pego, Universality classes in Burgers turbulence, Commun. Math. Phys., 273 (2007), 177–202.
  • 21. G. Menon and R. Srinivasan, Kinetic theory and Lax equations for shock clustering and Burgers turbulence, J. Stat. Phys., 140 (2010), 1195–1223.
  • 22. H. L. Royden and P. Fitzpatrick, Real Analysis, 4 Eds, Boston: Prentice Hall, 2010.
  • 23. W. Rudin, Real and Complex Analysis, 3 Eds, Boston: McGraw-Hill, 1987.
  • 24. Z. Schuss, Theory and Applications of Stochastic Processes, New York: Springer, 2010.
  • 25. A. Vol'pert, Spaces BV and quasilinear equations, Mat. Sb. (N.S.), 73 (1967), 255–302.

 

Reader Comments

your name: *   your email: *  

© 2018 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution Licese (http://creativecommons.org/licenses/by/4.0)

Download full text in PDF

Export Citation

Copyright © AIMS Press All Rights Reserved