AIMS Mathematics, 2018, 3(1): 148-182. doi: 10.3934/Math.2018.1.148

Research article

Export file:


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


  • Citation Only
  • Citation and Abstract

A minimization approach to conservation laws with random initialconditions and non-smooth, non-strictly convex flux

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

We obtain solutions to conservation laws under any random initial conditionsthat are described by Gaussian stochastic processes (in some cases discretized). We analyze thegeneralization of Burgers' equation for a smooth flux function $H\left( p\right)=\left\vert p\right\vert ^{j}$ for $j\geq2$ under randominitial data. We then consider a piecewise linear, non-smooth and non-convexflux function paired with general discretized Gaussian stochastic process initial data. Bypartitioning the real line into a finite number of points, we obtain an exactexpression for the solution of this problem. From this we can also find exactand approximate formulae for the density of shocks in the solution profile ata given time $t$ and spatial coordinate $x$. We discuss the simplification ofthese results in specific cases, including Brownian motion and Brownianbridge, for which the inverse covariance matrix and corresponding eigenvaluespectrum have some special properties. We calculate the transition probabilities between various cases and examine the variance of the solution $w\left(x,t\right)$ in both $x$ and $t$. We also describe how results may be obtained for a non-discretized version of a Gaussian stochastic process bytaking the continuum limit as the partition becomes more fine.
  Article Metrics


1. D. Applebaum, Levy Processes and Stochastic Calculus, 2 Eds, Cambridge: Cambridge University Press, 2009.

2. J. Bertoin, Levy Processes, Cambridge: Cambridge University Press, 1996.

3. P. Billingsley, Probability and Measure, New York: Wiley, 2012.

4. Y. Brienier, E. Grenier, Sticky particles and scalar conservation laws, SIAM J. Numer. Anal., 35 (1998), 2317–2328.

5. C. Caginalp, Minimization Solutions to Conservation Laws with Non-Smooth and Non-Strictly Convex Flux, AIMS Mathematics, 3 (2018), 96–130.

6. M. Chabanol, J. Duchon, Markovian solutions of inviscid Burgers equation, J. Stat. Phys., 114 (2004), 525–534.

7. M. Crandall, L. Evans, P. Lions, Some properties of viscosity solutions of Hamilton-Jacobi equations, Trans. Amer. Math. Soc., 282 (1984), 487–502.

8. C. Dafermos, Polygonal approximations of solutions of the initial value problem for a conservation law, J. Math. Anal. Appl., 38 (1972), 33–41.

9. C. Dafermos, Hyberbolic Conservation Laws in Continuum Physics, 3 Eds, New York: Springer, 2010.

10. E. Weinan, G. Rykov, 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.

11. C. Evans, Partial Differential Equations, 2 Eds, New York: Springer, 2010.

12. L. Frachebourg, P. Martin, Exact statistical properties of the Burgers equation, J. Fluid Mech., 417 (2000), 323–349.

13. D. G. Gilbarg & N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, New York: Springer, 1977.

14. P. Groeneboom, Brownian motion with a parabolic drift and Airy functions, Probab. Theory Rel., 81 (1989), 79–109.

15. M. Hairer, J. Maas, H. Weber, Approximating rough stochastic PDEs, Comm. Pure Appl. Math., 67 (2013), 776–870.

16. H. Holden, N. Risebro, Front Tracking for Hyperbolic Conservation Laws, New York: Springer, 2015.

17. E. Hopf, The partial differential equation ut + uux = μuxx, Comm. Pure Appl. Math., 3 (1950), 201–230.

18. D. Kaspar, F. Rezakhanlou, Scalar conservation laws with monotone pure-jump Markov initial conditions, Probab. Theory Relat. Fields, 165 (2016), 867–899.

19. A. Kaufman, H. Lim, J. Glimm, Conservative front tracking: the algorithm, the rationale and the API, Bulletin of the Institute of Mathematics, 11 (2016), 115–130.

20. D. Khoshnevisan, Z. Shi, Chung's Law for Integrated Brownian Motion, T. Am. Math. Soc., 350 (1998), 4253–4264.

21. P. Lax, Hyperbolic systems of conservation laws II, Comm. Pure Appl. Math., 10 (1957), 537–566.

22. P. Lax, Hyperbolic systems of conservation laws and the mathematical theory of shock waves, Society for Industrial and Applied Mathematics, Conference Board of the Mathematical Sciences Regional Conference Series in Applied Mathematics, 1973.

23. G. Menon, Complete integrability of shock clustering and Burgers turbulence, Arch. Ration. Mech. An., 203 (2012), 853–882.

24. G. Menon, R. Pego, Universality classes in Burgers turbulence, Comm. Math. Phys., 273 (2007), 177–202.

25. G. Menon, R. Srinivasan, Kinetic theory and Lax equations for shock clustering and Burgers turbulence, J. Stat. Phys., 140 (2010), 1195–1223.

26. M. Pinsky, S. Karlin, An Introduction to Stochastic Modeling, Burlington: Elsevier, 2011.

27. S. Ross, Introduction to Proabbility Models, 10 Eds, Burlington: Elsevier, 2010.

28. H. Royden, P. Fitzpatrick, Real Analysis, 4 Eds, Boston: Prentice Hall, 2010.

29. Z. Schuss, Theory and Applications of Stochastic Processes: An Analytical Approach, New York: Springer, 2010.

30. S. Shankar, Burgers Equation in 1D and 2D, 2012. Available from:

31. M. Slemrod, (2013) Admissibility of the weak solutions for the compressible Euler equations, $n\geq2$, Philos. T. R. Soc. A, 371 (2013), pp20120351.

32. D. She, R. Kaufman, H. Lim, et al. Handbook of Numerical Analysis, Elsevier 17, 383–402.

33. A. Vol'pert, Spaces BV and quasilinear equations, Mat. Sb. (N.S.), 73 (1967), 255–302.

© 2018 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution Licese (

Download full text in PDF

Export Citation

Article outline

Show full outline
Copyright © AIMS Press All Rights Reserved