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

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

Carey Caginalp

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

Received  March 2018 Revised  March 2018 Published  August 2018 Early access  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.

Keywords: Partial differential equations, randomness, stochastics, Euler equations.
Mathematics Subject Classification: Primary: 35F50, 35F55; Secondary: 35L65.
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.  Google Scholar

[2]

M. Avallaneda and W. E, Statistical properties of shocks in Burgers turbulence, Comm. Math Phys., 172 (1995), 13-38.  doi: 10.1007/BF02104509.  Google Scholar

[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.  Google Scholar

[4]

J. Bertoin Levy Processes, Cambridge University Press, Cambridge, 1996.  Google Scholar

[5]

J. Bertoin, The inviscid Burgers equation with Brownian initial velocity, Comm. Math Phys., 193 (1998), 397-406.  doi: 10.1007/s002200050334.  Google Scholar

[6]

H. Brezis, Functional Analysis, Sobolev Spaces, and Partial Differential Equations, Springer, New York, 2011.  Google Scholar

[7]

Y. Brienier and E. Grenier, Sticky particles and scalar conservation laws, SIAM J. Numer. Anal., 35 (1998), 2317-2328.  doi: 10.1137/S0036142997317353.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

C. Dafermos, Hyperbolic Conservation Laws in Continuum Physics, 3rd edition, Springer, New York, 2010. doi: 10.1007/978-3-642-04048-1.  Google Scholar

[11]

A. Dermoune, Probabilistic interpretation of sticky particle model, Ann. of Prob., 27 (1999), 1357-1367.  doi: 10.1214/aop/1022677451.  Google Scholar

[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.  Google Scholar

[13]

C. Evans, Partial Differential Equations, 2nd ed., Springer, New York, 1998. doi: 10.1090/gsm/019.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

L. Frachebourg and P. Martin, Exact statistical properties of the Burgers equation, J Fluid Mech, 417 (2000), 323-349.  doi: 10.1017/S0022112000001142.  Google Scholar

[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.  Google Scholar

[18]

P. Grassia, Dissipation, fluctuations, and conservation laws, American J. of Phys., 69 (2001), 113-119.  doi: 10.1119/1.1289211.  Google Scholar

[19]

P. Groeneboom, Brownian motion with a parabolic drift and Airy functions, Probab. Theory Relat. Fields, 81 (1989), 79-109.  doi: 10.1007/BF00343738.  Google Scholar

[20]

H. Holden and N. Risebro, Front Tracking for Hyperbolic Conservation Laws, Springer, New York, 2015. doi: 10.1007/978-3-662-47507-2.  Google Scholar

[21]

E. Hopf, The partial differential equation ut+uux = μuxx, Comm. Pure Appl. math., 3 (1950), 201-230.  doi: 10.1002/cpa.3160030302.  Google Scholar

[22]

F. Huang and Z. Wang, Well posedness for pressureless flow, Commun. Math Phys., 222 (2001), 117-146.  doi: 10.1007/s002200100506.  Google Scholar

[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.  Google Scholar

[24]

S. Kida, Asymptotic properties of Burgers turbulence, J. Fluid Mech., 93 (1979), 337-377.  doi: 10.1017/S0022112079001932.  Google Scholar

[25]

L. Krapivsky and E. Ben-Naim, Aggregation with multiple conservation laws, Phys. Rev. E, 53 (1996), 291. doi: 10.1103/PhysRevE.53.291.  Google Scholar

[26]

P. Lax, Hyperbolic systems of conservation laws, Comm. Pure Appl. Math., 10 (1957), 537-566.  doi: 10.1002/cpa.3160100406.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[34]

H. Royden and P. Fitzpatrick, Real Analysis, 4th ed, Prentice Hall, Boston, 2010. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[37]

Z. Schuss, Theory and Applications of Stochastic Processes, An Analytical Approach, Springer, New York, 2010. doi: 10.1007/978-1-4419-1605-1.  Google Scholar

[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.  Google Scholar

[39]

H. Spohn, Large Scale Dynamics of Interacting Particles, Springer, New York, 2012. doi: 10.1007/978-3-642-84371-6.  Google Scholar

[40]

H. Spohn, Kinetic equations from Hamiltonian dynamics: Markovian limits, Rev. of Mod. Phys., 3 (1980), 569-615.  doi: 10.1103/RevModPhys.52.569.  Google Scholar

[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.  Google Scholar

[42]

A. Vol'pert, Spaces BV and quasilinear equations, Mat. Sb. (N.S.), 73 (1967), 255-302.  doi: 10.1070/SM1967v002n02ABEH002340.  Google Scholar

show all references

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.  Google Scholar

[2]

M. Avallaneda and W. E, Statistical properties of shocks in Burgers turbulence, Comm. Math Phys., 172 (1995), 13-38.  doi: 10.1007/BF02104509.  Google Scholar

[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.  Google Scholar

[4]

J. Bertoin Levy Processes, Cambridge University Press, Cambridge, 1996.  Google Scholar

[5]

J. Bertoin, The inviscid Burgers equation with Brownian initial velocity, Comm. Math Phys., 193 (1998), 397-406.  doi: 10.1007/s002200050334.  Google Scholar

[6]

H. Brezis, Functional Analysis, Sobolev Spaces, and Partial Differential Equations, Springer, New York, 2011.  Google Scholar

[7]

Y. Brienier and E. Grenier, Sticky particles and scalar conservation laws, SIAM J. Numer. Anal., 35 (1998), 2317-2328.  doi: 10.1137/S0036142997317353.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

C. Dafermos, Hyperbolic Conservation Laws in Continuum Physics, 3rd edition, Springer, New York, 2010. doi: 10.1007/978-3-642-04048-1.  Google Scholar

[11]

A. Dermoune, Probabilistic interpretation of sticky particle model, Ann. of Prob., 27 (1999), 1357-1367.  doi: 10.1214/aop/1022677451.  Google Scholar

[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.  Google Scholar

[13]

C. Evans, Partial Differential Equations, 2nd ed., Springer, New York, 1998. doi: 10.1090/gsm/019.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

L. Frachebourg and P. Martin, Exact statistical properties of the Burgers equation, J Fluid Mech, 417 (2000), 323-349.  doi: 10.1017/S0022112000001142.  Google Scholar

[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.  Google Scholar

[18]

P. Grassia, Dissipation, fluctuations, and conservation laws, American J. of Phys., 69 (2001), 113-119.  doi: 10.1119/1.1289211.  Google Scholar

[19]

P. Groeneboom, Brownian motion with a parabolic drift and Airy functions, Probab. Theory Relat. Fields, 81 (1989), 79-109.  doi: 10.1007/BF00343738.  Google Scholar

[20]

H. Holden and N. Risebro, Front Tracking for Hyperbolic Conservation Laws, Springer, New York, 2015. doi: 10.1007/978-3-662-47507-2.  Google Scholar

[21]

E. Hopf, The partial differential equation ut+uux = μuxx, Comm. Pure Appl. math., 3 (1950), 201-230.  doi: 10.1002/cpa.3160030302.  Google Scholar

[22]

F. Huang and Z. Wang, Well posedness for pressureless flow, Commun. Math Phys., 222 (2001), 117-146.  doi: 10.1007/s002200100506.  Google Scholar

[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.  Google Scholar

[24]

S. Kida, Asymptotic properties of Burgers turbulence, J. Fluid Mech., 93 (1979), 337-377.  doi: 10.1017/S0022112079001932.  Google Scholar

[25]

L. Krapivsky and E. Ben-Naim, Aggregation with multiple conservation laws, Phys. Rev. E, 53 (1996), 291. doi: 10.1103/PhysRevE.53.291.  Google Scholar

[26]

P. Lax, Hyperbolic systems of conservation laws, Comm. Pure Appl. Math., 10 (1957), 537-566.  doi: 10.1002/cpa.3160100406.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[34]

H. Royden and P. Fitzpatrick, Real Analysis, 4th ed, Prentice Hall, Boston, 2010. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[37]

Z. Schuss, Theory and Applications of Stochastic Processes, An Analytical Approach, Springer, New York, 2010. doi: 10.1007/978-1-4419-1605-1.  Google Scholar

[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.  Google Scholar

[39]

H. Spohn, Large Scale Dynamics of Interacting Particles, Springer, New York, 2012. doi: 10.1007/978-3-642-84371-6.  Google Scholar

[40]

H. Spohn, Kinetic equations from Hamiltonian dynamics: Markovian limits, Rev. of Mod. Phys., 3 (1980), 569-615.  doi: 10.1103/RevModPhys.52.569.  Google Scholar

[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.  Google Scholar

[42]

A. Vol'pert, Spaces BV and quasilinear equations, Mat. Sb. (N.S.), 73 (1967), 255-302.  doi: 10.1070/SM1967v002n02ABEH002340.  Google Scholar

Figure 1.  (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
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Figure 2.  (a) Cumulative distribution function of mass as a function of position; (b) Construction of $\Psi\left( 0, x\right) $
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Figure 3.  (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
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Figure 4.  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\} $
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Figure 5.  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) $
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Figure 1, [16]">Figure 6.  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]
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Figure 7.  Illustration of the flux function as described above
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Figure 8.  (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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