# American Institute of Mathematical Sciences

August 2018, 23(6): 2043-2069. doi: 10.3934/dcdsb.2018225

## A survey of results on conservation laws with deterministic and random initial data

 301 Thackeray Hall, University of Pittsburgh, Pittsburgh PA 15213, USA

Received  March 2018 Revised  March 2018 Published  July 2018

Fund Project: The author thanks Professors Menon and Dafermos and Dr. Kaspar for valuable discussions. This work was partially supported by NSF grants DMS 1411278 and DMS 1148284 as well as the NSF Graduate Research Fellowship

This expository paper examines key results on the dynamics of nonlinear conservation laws with random initial data and applies some theorems to physically important situations. Conservation laws with some nonlinearity, e.g. Burgers' equation, exhibit discontinuous behavior, or shocks, even for smooth initial data. The introduction of randomness in any of several forms into the initial condition renders the analysis extremely complex. Standard methods for tracking a multitude of shock collisions are difficult to implement, suggesting other methods may be needed. We review several perspectives into obtaining the statistics of resulting states and shocks. We present a spectrum of results from a number of works, both deterministic and random. Some of the deep theorems are applied to important discrete examples where the results can be understood in a clearer, more physical context.

Citation: Carey Caginalp. A survey of results on conservation laws with deterministic and random initial data. Discrete & Continuous Dynamical Systems - B, 2018, 23 (6) : 2043-2069. doi: 10.3934/dcdsb.2018225
##### References:
 [1] D. Applebaum, Levy Processes and Stochastic Calculus, 2nd ed., Cambridge Studies in Advanced Mathematics, vol. 116. Cambridge University Press, Cambridge, 2009. doi: 10.1017/CBO9780511809781. [2] M. Avallaneda and W. E, Statistical properties of shocks in Burgers turbulence, Comm. Math Phys., 172 (1995), 13-38. doi: 10.1007/BF02104509. [3] M. Bardi and C. Evans, On Hopf's formulas for solutions of Hamilton-Jacobi equations, Nonlinear Anal., 8 (1984), 1373-1381. doi: 10.1016/0362-546X(84)90020-8. [4] J. Bertoin Levy Processes, Cambridge University Press, Cambridge, 1996. [5] J. Bertoin, The inviscid Burgers equation with Brownian initial velocity, Comm. Math Phys., 193 (1998), 397-406. doi: 10.1007/s002200050334. [6] H. Brezis, Functional Analysis, Sobolev Spaces, and Partial Differential Equations, Springer, New York, 2011. [7] Y. Brienier and E. Grenier, Sticky particles and scalar conservation laws, SIAM J. Numer. Anal., 35 (1998), 2317-2328. doi: 10.1137/S0036142997317353. [8] M. Chabanol and J. Duchon, Markovian solutions of inviscid Burgers equation, J. Stat. Phys., 114 (2004), 525-534. doi: 10.1023/B:JOSS.0000003120.32992.a9. [9] A. Chertock, A. Kurganov and Y. Rykov, A new sticky particle method for pressureless gas dynamics, SIAM J. on Num. Anal., 45 (2007), 2408-2441. doi: 10.1137/050644124. [10] C. Dafermos, Hyperbolic Conservation Laws in Continuum Physics, 3rd edition, Springer, New York, 2010. doi: 10.1007/978-3-642-04048-1. [11] A. Dermoune, Probabilistic interpretation of sticky particle model, Ann. of Prob., 27 (1999), 1357-1367. doi: 10.1214/aop/1022677451. [12] A. Dermoune, Probabilistic interpretation for system of conservation law arising in adhesion particle dynamics, Comp. Rend. de l'Académie des Sciences-Series I-Math, 326 (1998), 595-599. doi: 10.1016/S0764-4442(98)85013-1. [13] C. Evans, Partial Differential Equations, 2nd ed., Springer, New York, 1998. doi: 10.1090/gsm/019. [14] J. Feng and D. Nualart, Stochastic scalar conservation laws, J. Func. Anal, 255 (2008), 313-373. doi: 10.1016/j.jfa.2008.02.004. [15] L. Frachebourg and P. Martin, Ballistic aggregation: A solvable model of irreversible many partical dynamics, Phys. A: Stat. Mech. and Appl., 279 (2000), 69-99. doi: 10.1016/S0378-4371(99)00585-3. [16] L. Frachebourg and P. Martin, Exact statistical properties of the Burgers equation, J Fluid Mech, 417 (2000), 323-349. doi: 10.1017/S0022112000001142. [17] B. Gess and P. Souganidis, Long-Time Behavior, Invariant Measures, and Regularizing Effects for Stochastic Scalar Conservation Laws, Comm. on Pure and Appl. Math., 70 (2017), 1562-1597. doi: 10.1002/cpa.21646. [18] P. Grassia, Dissipation, fluctuations, and conservation laws, American J. of Phys., 69 (2001), 113-119. doi: 10.1119/1.1289211. [19] P. Groeneboom, Brownian motion with a parabolic drift and Airy functions, Probab. Theory Relat. Fields, 81 (1989), 79-109. doi: 10.1007/BF00343738. [20] H. Holden and N. Risebro, Front Tracking for Hyperbolic Conservation Laws, Springer, New York, 2015. doi: 10.1007/978-3-662-47507-2. [21] E. Hopf, The partial differential equation ut+uux = μuxx, Comm. Pure Appl. math., 3 (1950), 201-230. doi: 10.1002/cpa.3160030302. [22] F. Huang and Z. Wang, Well posedness for pressureless flow, Commun. Math Phys., 222 (2001), 117-146. doi: 10.1007/s002200100506. [23] D. Kaspar and F. Rezakhanlou, Scalar conservation laws with monotone pure-jump Markov initial conditions, Probab. Theory Relat. Fields, 165 (2016), 867-899. doi: 10.1007/s00440-015-0648-2. [24] S. Kida, Asymptotic properties of Burgers turbulence, J. Fluid Mech., 93 (1979), 337-377. doi: 10.1017/S0022112079001932. [25] L. Krapivsky and E. Ben-Naim, Aggregation with multiple conservation laws, Phys. Rev. E, 53 (1996), 291. doi: 10.1103/PhysRevE.53.291. [26] P. Lax, Hyperbolic systems of conservation laws, Comm. Pure Appl. Math., 10 (1957), 537-566. doi: 10.1002/cpa.3160100406. [27] P. Lax, Hyperbolic Systems of Conservation Laws and the Mathematical Theory of Shock Waves, SIAM, Philadelphia, Pa., Conference Board of the Mathematical Sciences Regional Conference Series in Applied Mathematics, 1973. [28] G. Menon and R. Srinivasan, Kinetic theory and Lax equations for shock clustering and Burgers turbulence, J. Stat. Phys., 140 (2010), 1195-1223. doi: 10.1007/s10955-010-0028-3. [29] G. Menon and R. Pego, Universality classes in Burgers turbulence, Comm. Math. Phys., 273 (2007), 177-202. doi: 10.1007/s00220-007-0251-1. [30] G. Menon, Complete integrability of shock clustering and Burgers turbulence, Archive for Rational Mechanics and Analysis, 203 (2012), 853-882. doi: 10.1007/s00205-011-0461-8. [31] S. Mishra, C. Schwab and J. Šukys, Multi-level Monte Carlo finite volume methods for nonlinear systems of conservation laws in multi-dimensions, J. Comp. Phys., 231 (2012), 3365-3388. doi: 10.1016/j.jcp.2012.01.011. [32] T. Nguyen and A. Tudorascu, Pressureless Euler/Euler-Poisson systems via adhesion dynamics and scalar conservation laws, SIAM J. on Math. Anal., 40 (2008), 754-775. doi: 10.1137/070704459. [33] M. Rost, L. Laurson, M. Dubé and M. Alava, Fluctuations in fluid invasion into disordered media, Phys. rev. letters, 98 (2007), 054502. doi: 10.1103/PhysRevLett.98.054502. [34] H. Royden and P. Fitzpatrick, Real Analysis, 4th ed, Prentice Hall, Boston, 2010. [35] W. E, 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. doi: 10.1007/BF02101897. [36] B. Saussereau and I. Stoica, Scalar conservation laws with fractional stochastic forcing: Existence, uniqueness and invariant measure, Stoch. Proc. and their Appl., 122 (2012), 1456-1486. doi: 10.1016/j.spa.2012.01.005. [37] Z. Schuss, Theory and Applications of Stochastic Processes, An Analytical Approach, Springer, New York, 2010. doi: 10.1007/978-1-4419-1605-1. [38] C. Schwab and S. Tokareva, High order approximation of probabilistic shock profiles in hyperbolic conservation laws with uncertain initial data, ESAIM: Math. Modelling and Num. Anal., 47 (2013), 807-835. doi: 10.1051/m2an/2012060. [39] H. Spohn, Large Scale Dynamics of Interacting Particles, Springer, New York, 2012. doi: 10.1007/978-3-642-84371-6. [40] H. Spohn, Kinetic equations from Hamiltonian dynamics: Markovian limits, Rev. of Mod. Phys., 3 (1980), 569-615. doi: 10.1103/RevModPhys.52.569. [41] P. Valageas, Ballistic aggregation for one-sided Brownian initial velocity, Physica A, 388 (2009), 1031-1045. doi: 10.1016/j.physa.2008.12.033. [42] A. Vol'pert, Spaces BV and quasilinear equations, Mat. Sb. (N.S.), 73 (1967), 255-302. doi: 10.1070/SM1967v002n02ABEH002340.

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##### References:
 [1] D. Applebaum, Levy Processes and Stochastic Calculus, 2nd ed., Cambridge Studies in Advanced Mathematics, vol. 116. Cambridge University Press, Cambridge, 2009. doi: 10.1017/CBO9780511809781. [2] M. Avallaneda and W. E, Statistical properties of shocks in Burgers turbulence, Comm. Math Phys., 172 (1995), 13-38. doi: 10.1007/BF02104509. [3] M. Bardi and C. Evans, On Hopf's formulas for solutions of Hamilton-Jacobi equations, Nonlinear Anal., 8 (1984), 1373-1381. doi: 10.1016/0362-546X(84)90020-8. [4] J. Bertoin Levy Processes, Cambridge University Press, Cambridge, 1996. [5] J. Bertoin, The inviscid Burgers equation with Brownian initial velocity, Comm. Math Phys., 193 (1998), 397-406. doi: 10.1007/s002200050334. [6] H. Brezis, Functional Analysis, Sobolev Spaces, and Partial Differential Equations, Springer, New York, 2011. [7] Y. Brienier and E. Grenier, Sticky particles and scalar conservation laws, SIAM J. Numer. Anal., 35 (1998), 2317-2328. doi: 10.1137/S0036142997317353. [8] M. Chabanol and J. Duchon, Markovian solutions of inviscid Burgers equation, J. Stat. Phys., 114 (2004), 525-534. doi: 10.1023/B:JOSS.0000003120.32992.a9. [9] A. Chertock, A. Kurganov and Y. Rykov, A new sticky particle method for pressureless gas dynamics, SIAM J. on Num. Anal., 45 (2007), 2408-2441. doi: 10.1137/050644124. [10] C. Dafermos, Hyperbolic Conservation Laws in Continuum Physics, 3rd edition, Springer, New York, 2010. doi: 10.1007/978-3-642-04048-1. [11] A. Dermoune, Probabilistic interpretation of sticky particle model, Ann. of Prob., 27 (1999), 1357-1367. doi: 10.1214/aop/1022677451. [12] A. Dermoune, Probabilistic interpretation for system of conservation law arising in adhesion particle dynamics, Comp. Rend. de l'Académie des Sciences-Series I-Math, 326 (1998), 595-599. doi: 10.1016/S0764-4442(98)85013-1. [13] C. Evans, Partial Differential Equations, 2nd ed., Springer, New York, 1998. doi: 10.1090/gsm/019. [14] J. Feng and D. Nualart, Stochastic scalar conservation laws, J. Func. Anal, 255 (2008), 313-373. doi: 10.1016/j.jfa.2008.02.004. [15] L. Frachebourg and P. Martin, Ballistic aggregation: A solvable model of irreversible many partical dynamics, Phys. A: Stat. Mech. and Appl., 279 (2000), 69-99. doi: 10.1016/S0378-4371(99)00585-3. [16] L. Frachebourg and P. Martin, Exact statistical properties of the Burgers equation, J Fluid Mech, 417 (2000), 323-349. doi: 10.1017/S0022112000001142. [17] B. Gess and P. Souganidis, Long-Time Behavior, Invariant Measures, and Regularizing Effects for Stochastic Scalar Conservation Laws, Comm. on Pure and Appl. Math., 70 (2017), 1562-1597. doi: 10.1002/cpa.21646. [18] P. Grassia, Dissipation, fluctuations, and conservation laws, American J. of Phys., 69 (2001), 113-119. doi: 10.1119/1.1289211. [19] P. Groeneboom, Brownian motion with a parabolic drift and Airy functions, Probab. Theory Relat. Fields, 81 (1989), 79-109. doi: 10.1007/BF00343738. [20] H. Holden and N. Risebro, Front Tracking for Hyperbolic Conservation Laws, Springer, New York, 2015. doi: 10.1007/978-3-662-47507-2. [21] E. Hopf, The partial differential equation ut+uux = μuxx, Comm. Pure Appl. math., 3 (1950), 201-230. doi: 10.1002/cpa.3160030302. [22] F. Huang and Z. Wang, Well posedness for pressureless flow, Commun. Math Phys., 222 (2001), 117-146. doi: 10.1007/s002200100506. [23] D. Kaspar and F. Rezakhanlou, Scalar conservation laws with monotone pure-jump Markov initial conditions, Probab. Theory Relat. Fields, 165 (2016), 867-899. doi: 10.1007/s00440-015-0648-2. [24] S. Kida, Asymptotic properties of Burgers turbulence, J. Fluid Mech., 93 (1979), 337-377. doi: 10.1017/S0022112079001932. [25] L. Krapivsky and E. Ben-Naim, Aggregation with multiple conservation laws, Phys. Rev. E, 53 (1996), 291. doi: 10.1103/PhysRevE.53.291. [26] P. Lax, Hyperbolic systems of conservation laws, Comm. Pure Appl. Math., 10 (1957), 537-566. doi: 10.1002/cpa.3160100406. [27] P. Lax, Hyperbolic Systems of Conservation Laws and the Mathematical Theory of Shock Waves, SIAM, Philadelphia, Pa., Conference Board of the Mathematical Sciences Regional Conference Series in Applied Mathematics, 1973. [28] G. Menon and R. Srinivasan, Kinetic theory and Lax equations for shock clustering and Burgers turbulence, J. Stat. Phys., 140 (2010), 1195-1223. doi: 10.1007/s10955-010-0028-3. [29] G. Menon and R. Pego, Universality classes in Burgers turbulence, Comm. Math. Phys., 273 (2007), 177-202. doi: 10.1007/s00220-007-0251-1. [30] G. Menon, Complete integrability of shock clustering and Burgers turbulence, Archive for Rational Mechanics and Analysis, 203 (2012), 853-882. doi: 10.1007/s00205-011-0461-8. [31] S. Mishra, C. Schwab and J. Šukys, Multi-level Monte Carlo finite volume methods for nonlinear systems of conservation laws in multi-dimensions, J. Comp. Phys., 231 (2012), 3365-3388. doi: 10.1016/j.jcp.2012.01.011. [32] T. Nguyen and A. Tudorascu, Pressureless Euler/Euler-Poisson systems via adhesion dynamics and scalar conservation laws, SIAM J. on Math. Anal., 40 (2008), 754-775. doi: 10.1137/070704459. [33] M. Rost, L. Laurson, M. Dubé and M. Alava, Fluctuations in fluid invasion into disordered media, Phys. rev. letters, 98 (2007), 054502. doi: 10.1103/PhysRevLett.98.054502. [34] H. Royden and P. Fitzpatrick, Real Analysis, 4th ed, Prentice Hall, Boston, 2010. [35] W. E, 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. doi: 10.1007/BF02101897. [36] B. Saussereau and I. Stoica, Scalar conservation laws with fractional stochastic forcing: Existence, uniqueness and invariant measure, Stoch. Proc. and their Appl., 122 (2012), 1456-1486. doi: 10.1016/j.spa.2012.01.005. [37] Z. Schuss, Theory and Applications of Stochastic Processes, An Analytical Approach, Springer, New York, 2010. doi: 10.1007/978-1-4419-1605-1. [38] C. Schwab and S. Tokareva, High order approximation of probabilistic shock profiles in hyperbolic conservation laws with uncertain initial data, ESAIM: Math. Modelling and Num. Anal., 47 (2013), 807-835. doi: 10.1051/m2an/2012060. [39] H. Spohn, Large Scale Dynamics of Interacting Particles, Springer, New York, 2012. doi: 10.1007/978-3-642-84371-6. [40] H. Spohn, Kinetic equations from Hamiltonian dynamics: Markovian limits, Rev. of Mod. Phys., 3 (1980), 569-615. doi: 10.1103/RevModPhys.52.569. [41] P. Valageas, Ballistic aggregation for one-sided Brownian initial velocity, Physica A, 388 (2009), 1031-1045. doi: 10.1016/j.physa.2008.12.033. [42] A. Vol'pert, Spaces BV and quasilinear equations, Mat. Sb. (N.S.), 73 (1967), 255-302. doi: 10.1070/SM1967v002n02ABEH002340.
(a) By taking a cross-section in time, one can obtain a cumulative distribution function of the mass as a function of position; (b) Illustration of the potential as a function of mass; (c) Illustration of the flux function of mass
(a) Cumulative distribution function of mass as a function of position; (b) Construction of $\Psi\left( 0, x\right)$
(a) Representation of mass in cumulative distribution form up to a point $x$ in the $xt$ plane; (b)-(d) Plot of the expression $\Phi^{0}\left( m\right) +tA\left( m\right)$ for times $t = 0, 1, 2$ respectively in solid lines; following the dashed lines forms the convex hull, yielding the Legendre transform $\Phi_{n}\left( t, m\right) .$ Note that for (b) and (c), the expression and its convex hull are identical, and in (d) there is a distinction, with the convex hull indicated by the dashed blue line
Evolution of the discrete example and mapping back using the flow map. Highlighted in blue (long-short dash lines) are intervals unchanged under the flow map. In red (long dashed line) are intervals for which the flow map inverse is undefined. The points in green correspond to single points for which an entire interval is mapped back onto, which occurs in notably many cases. For example, $\varphi_{t_{2}^{\ast}}^{-1}\left( I_{2}^{1\ast}\right) = \left\{ 0\right\}$ and $\varphi_{t_{1}^{\ast}}^{-1}\left( \left\{ -2\right\} \right) = \left\{ \emptyset\right\}$
Graphs of (a) $\int_{0+0}^{y-0}tu_{0}\left( \eta\right) dm_{0}\left( \eta\right)$ (with $t = 1$), (b) $\int_{0+0}^{y-0}\left( \eta-x\right) dm_{0}\left( \eta\right)$, (c) $F\left( y;0, 1\right)$
For a shock at a point $x^{\ast}$, we slide the parabola $\left( x-x^{\ast}\right) ^{2}/2$ down until we have (at least) two contact points with the Brownian path, but in such a way that the parabola does not cross the Brownian path. If there are more than two, we consider only the first and last contact points. These points are given by $\left( \xi_{-}, \left( \xi _{-}-x^{\ast}\right) ^{2}/2\right)$ and $\left( \xi_{+}, \left( \xi _{+}-x^{\ast}\right) ^{2}/2\right)$. The shock is then described by the parameters $\mu = \xi_{+}-\xi_{-}$ and $\nu = x^{\ast}-\xi_{-}.$ This figure is based off Figure 1, [16]
Illustration of the flux function as described above
(a)-(b) Construction of the test functions $\varphi_{k}\left( u\right)$ and $\psi_{k}\left( u\right)$; (c) Illustration of a shock, with positive contribution from $\partial_{t}p_{1}\left( x, t;u_{l}\right)$ (upward arrow, blue), and negative contribution from $p_{2}\left( x, x+, t;u_{l}, u_{r}\right)$ (right arrow, red)
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