Research article

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

  • Received: 19 February 2018 Accepted: 11 March 2018 Published: 19 March 2018
  • We obtain solutions to conservation laws under any random initial conditions that are described by Gaussian stochastic processes (in some cases discretized). We analyze the generalization of Burgers' equation for a smooth flux function $H\left( p\right) = \left\vert p\right\vert .{j}$ for $j\geq2$ under random initial data. We then consider a piecewise linear, non-smooth and non-convex flux function paired with general discretized Gaussian stochastic process initial data. By partitioning the real line into a finite number of points, we obtain an exact expression for the solution of this problem. From this we can also find exact and approximate formulae for the density of shocks in the solution profile at a given time $t$ and spatial coordinate $x$. We discuss the simplification of these results in specific cases, including Brownian motion and Brownian bridge, for which the inverse covariance matrix and corresponding eigenvalue spectrum 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 by taking the continuum limit as the partition becomes more fine.

    Citation: Carey Caginalp. A minimization approach to conservation laws with random initialconditions and non-smooth, non-strictly convex flux[J]. AIMS Mathematics, 2018, 3(1): 148-182. doi: 10.3934/Math.2018.1.148

    Related Papers:

  • We obtain solutions to conservation laws under any random initial conditions that are described by Gaussian stochastic processes (in some cases discretized). We analyze the generalization of Burgers' equation for a smooth flux function $H\left( p\right) = \left\vert p\right\vert .{j}$ for $j\geq2$ under random initial data. We then consider a piecewise linear, non-smooth and non-convex flux function paired with general discretized Gaussian stochastic process initial data. By partitioning the real line into a finite number of points, we obtain an exact expression for the solution of this problem. From this we can also find exact and approximate formulae for the density of shocks in the solution profile at a given time $t$ and spatial coordinate $x$. We discuss the simplification of these results in specific cases, including Brownian motion and Brownian bridge, for which the inverse covariance matrix and corresponding eigenvalue spectrum 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 by taking the continuum limit as the partition becomes more fine.


    加载中
    [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: https://www.mathworks.com/matlabcentral/fileexchange/38087-burgers-equation-in-1d-and-2d?focused=5246985&tab=function
    [31] M. Slemrod, (2013) Admissibility of the weak solutions for the compressible Euler equations, n≥2, 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.
  • Reader Comments
  • © 2018 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Metrics

Article views(1842) PDF downloads(804) Cited by(0)

Article outline

Figures and Tables

Figures(6)

Other Articles By Authors

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog