American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Constructing set-valued fundamental diagrams from Jamiton solutions in second order traffic models
  • NHM Home
  • This Issue
  • Next Article
    Viability approach to Hamilton-Jacobi-Moskowitz problem involving variable regulation parameters
September  2013, 8(3): 727-744. doi: 10.3934/nhm.2013.8.727

Explicit construction of solutions to the Burgers equation with discontinuous initial-boundary conditions

Anya Désilles 1, and  Hélène Frankowska 2,

1. 

ENSTA ParisTech, 828, Boulevard des Maréchaux, 91762 Palaiseau Cedex
2. 

CNRS and Institut de Mathématiques de Jussieu, 4 place Jussieu, Université Pierre et Marie Curie, case 247, 75252 Paris,

Received  April 2012 Revised  August 2013 Published  October 2013

A solution of the initial-boundary value problem on the strip $(0,\infty) \times [0,1]$ for scalar conservation laws with strictly convex flux can be obtained by considering gradients of the unique solution $V$ to an associated Hamilton-Jacobi equation (with appropriately defined initial and boundary conditions). It was shown in Frankowska (2010) that $V$ can be expressed as the minimum of three value functions arising in calculus of variations problems that, in turn, can be obtained from the Lax formulae. Moreover the traces of the gradients $V_x$ satisfy generalized boundary conditions (as in LeFloch (1988)). In this work we illustrate this approach in the case of the Burgers equation and provide numerical approximation of its solutions.
Keywords: Hamilton-Jacobi equation, Lax formulae, boundary conditions, Burgers equation, value function..
Mathematics Subject Classification: 35C99, 35D40, 35F31, 35L65, 49L9.
Citation: Anya Désilles, Hélène Frankowska. Explicit construction of solutions to the Burgers equation with discontinuous initial-boundary conditions. Networks & Heterogeneous Media, 2013, 8 (3) : 727-744. doi: 10.3934/nhm.2013.8.727
References:
[1]

D. Amadori, Initial-boundary value problems for nonlinear systems of conservation laws,, NoDEA Nonlinear Differential Equations and Applications, 4 (1997), 1.  doi: 10.1007/PL00001406.  Google Scholar

[2]

C. Bardos, A. Leroux and J. Nedelec, First order quasilinear equations with boundary conditions,, Commun. Partial Diff. Equat., 4 (1979), 1017.  doi: 10.1080/03605307908820117.  Google Scholar

[3]

A. Bressan, "Hyperbolic Systems of Conservation Laws: The One-dimensional Cauchy Problem,", Oxford Lecture Series in Mathematics and its Applications, (2000).   Google Scholar

[4]

M. G. Crandall, L. C. Evans and P.-L. Lions, Some properties of viscosity solutions of Hamilton-Jacobi equations,, Trans. AMS., 282 (1984), 487.  doi: 10.1090/S0002-9947-1984-0732102-X.  Google Scholar

[5]

L. C. Evans, "Partial Differential Equations,", Graduate Studies in Mathematics, (1998).   Google Scholar

[6]

H. Frankowska, On LeFloch solutions to initial-boundary value problem for scalar conservation laws,, Journal of Hyperbolic Differential Equations, 7 (2010), 503.  doi: 10.1142/S0219891610002219.  Google Scholar

[7]

H. Frankowska, Lower semicontinuous solutions to Hamilton-Jacobi-Bellman equations,, Proceedings of 30th CDC Conference, (1991), 11.   Google Scholar

[8]

H. Frankowska, Lower semicontinuous solutions of Hamilton-Jacobi-Bellman equation,, SIAM J. Control and Optimization, 31 (1993), 257.  doi: 10.1137/0331016.  Google Scholar

[9]

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

[10]

P. G. LeFloch, Explicit formula for scalar nonlinear conservation laws with boundary condition,, Math. Methods Appl. Sci., 10 (1988), 265.  doi: 10.1002/mma.1670100305.  Google Scholar

[11]

M. Lighthill and G. Whitham, On kinematic waves, II: A theory of traffic flow on long crowded roads,, Proceedings of the Royal Society of London Ser. A., 229 (1955), 317.  doi: 10.1098/rspa.1955.0089.  Google Scholar

[12]

P. Richards, Shock waves on the highway,, Operations Research, 4 (1956), 42.  doi: 10.1287/opre.4.1.42.  Google Scholar

[13]

I. Strub and A. Bayen, Weak formulation of boundary conditions for scalar conservation laws: An application to highway traffic modelling,, Int. J. Robust Nonlinear Control, 16 (2006), 733.  doi: 10.1002/rnc.1099.  Google Scholar

show all references

References:
[1]

D. Amadori, Initial-boundary value problems for nonlinear systems of conservation laws,, NoDEA Nonlinear Differential Equations and Applications, 4 (1997), 1.  doi: 10.1007/PL00001406.  Google Scholar

[2]

C. Bardos, A. Leroux and J. Nedelec, First order quasilinear equations with boundary conditions,, Commun. Partial Diff. Equat., 4 (1979), 1017.  doi: 10.1080/03605307908820117.  Google Scholar

[3]

A. Bressan, "Hyperbolic Systems of Conservation Laws: The One-dimensional Cauchy Problem,", Oxford Lecture Series in Mathematics and its Applications, (2000).   Google Scholar

[4]

M. G. Crandall, L. C. Evans and P.-L. Lions, Some properties of viscosity solutions of Hamilton-Jacobi equations,, Trans. AMS., 282 (1984), 487.  doi: 10.1090/S0002-9947-1984-0732102-X.  Google Scholar

[5]

L. C. Evans, "Partial Differential Equations,", Graduate Studies in Mathematics, (1998).   Google Scholar

[6]

H. Frankowska, On LeFloch solutions to initial-boundary value problem for scalar conservation laws,, Journal of Hyperbolic Differential Equations, 7 (2010), 503.  doi: 10.1142/S0219891610002219.  Google Scholar

[7]

H. Frankowska, Lower semicontinuous solutions to Hamilton-Jacobi-Bellman equations,, Proceedings of 30th CDC Conference, (1991), 11.   Google Scholar

[8]

H. Frankowska, Lower semicontinuous solutions of Hamilton-Jacobi-Bellman equation,, SIAM J. Control and Optimization, 31 (1993), 257.  doi: 10.1137/0331016.  Google Scholar

[9]

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

[10]

P. G. LeFloch, Explicit formula for scalar nonlinear conservation laws with boundary condition,, Math. Methods Appl. Sci., 10 (1988), 265.  doi: 10.1002/mma.1670100305.  Google Scholar

[11]

M. Lighthill and G. Whitham, On kinematic waves, II: A theory of traffic flow on long crowded roads,, Proceedings of the Royal Society of London Ser. A., 229 (1955), 317.  doi: 10.1098/rspa.1955.0089.  Google Scholar

[12]

P. Richards, Shock waves on the highway,, Operations Research, 4 (1956), 42.  doi: 10.1287/opre.4.1.42.  Google Scholar

[13]

I. Strub and A. Bayen, Weak formulation of boundary conditions for scalar conservation laws: An application to highway traffic modelling,, Int. J. Robust Nonlinear Control, 16 (2006), 733.  doi: 10.1002/rnc.1099.  Google Scholar

[1]

Alexander Quaas, Andrei Rodríguez. Analysis of the attainment of boundary conditions for a nonlocal diffusive Hamilton-Jacobi equation. Discrete & Continuous Dynamical Systems - A, 2018, 38 (10) : 5221-5243. doi: 10.3934/dcds.2018231
[2]

Joan-Andreu Lázaro-Camí, Juan-Pablo Ortega. The stochastic Hamilton-Jacobi equation. Journal of Geometric Mechanics, 2009, 1 (3) : 295-315. doi: 10.3934/jgm.2009.1.295
[3]

Yoshikazu Giga, Przemysław Górka, Piotr Rybka. Nonlocal spatially inhomogeneous Hamilton-Jacobi equation with unusual free boundary. Discrete & Continuous Dynamical Systems - A, 2010, 26 (2) : 493-519. doi: 10.3934/dcds.2010.26.493
[4]

Nicolas Forcadel, Mamdouh Zaydan. A comparison principle for Hamilton-Jacobi equation with moving in time boundary. Evolution Equations & Control Theory, 2019, 8 (3) : 543-565. doi: 10.3934/eect.2019026
[5]

Tomoki Ohsawa, Anthony M. Bloch. Nonholonomic Hamilton-Jacobi equation and integrability. Journal of Geometric Mechanics, 2009, 1 (4) : 461-481. doi: 10.3934/jgm.2009.1.461
[6]

Nalini Anantharaman, Renato Iturriaga, Pablo Padilla, Héctor Sánchez-Morgado. Physical solutions of the Hamilton-Jacobi equation. Discrete & Continuous Dynamical Systems - B, 2005, 5 (3) : 513-528. doi: 10.3934/dcdsb.2005.5.513
[7]

María Barbero-Liñán, Manuel de León, David Martín de Diego, Juan C. Marrero, Miguel C. Muñoz-Lecanda. Kinematic reduction and the Hamilton-Jacobi equation. Journal of Geometric Mechanics, 2012, 4 (3) : 207-237. doi: 10.3934/jgm.2012.4.207
[8]

Larry M. Bates, Francesco Fassò, Nicola Sansonetto. The Hamilton-Jacobi equation, integrability, and nonholonomic systems. Journal of Geometric Mechanics, 2014, 6 (4) : 441-449. doi: 10.3934/jgm.2014.6.441
[9]

Yuxiang Li. Stabilization towards the steady state for a viscous Hamilton-Jacobi equation. Communications on Pure & Applied Analysis, 2009, 8 (6) : 1917-1924. doi: 10.3934/cpaa.2009.8.1917
[10]

Renato Iturriaga, Héctor Sánchez-Morgado. Limit of the infinite horizon discounted Hamilton-Jacobi equation. Discrete & Continuous Dynamical Systems - B, 2011, 15 (3) : 623-635. doi: 10.3934/dcdsb.2011.15.623
[11]

Manuel de León, Juan Carlos Marrero, David Martín de Diego. Linear almost Poisson structures and Hamilton-Jacobi equation. Applications to nonholonomic mechanics. Journal of Geometric Mechanics, 2010, 2 (2) : 159-198. doi: 10.3934/jgm.2010.2.159
[12]

Daniele Castorina, Annalisa Cesaroni, Luca Rossi. On a parabolic Hamilton-Jacobi-Bellman equation degenerating at the boundary. Communications on Pure & Applied Analysis, 2016, 15 (4) : 1251-1263. doi: 10.3934/cpaa.2016.15.1251
[13]

Jean-Claude Zambrini. On the geometry of the Hamilton-Jacobi-Bellman equation. Journal of Geometric Mechanics, 2009, 1 (3) : 369-387. doi: 10.3934/jgm.2009.1.369
[14]

Olivier Bokanowski, Maurizio Falcone, Roberto Ferretti, Lars Grüne, Dante Kalise, Hasnaa Zidani. Value iteration convergence of $\epsilon$-monotone schemes for stationary Hamilton-Jacobi equations. Discrete & Continuous Dynamical Systems - A, 2015, 35 (9) : 4041-4070. doi: 10.3934/dcds.2015.35.4041
[15]

Claudio Marchi. On the convergence of singular perturbations of Hamilton-Jacobi equations. Communications on Pure & Applied Analysis, 2010, 9 (5) : 1363-1377. doi: 10.3934/cpaa.2010.9.1363
[16]

Isabeau Birindelli, J. Wigniolle. Homogenization of Hamilton-Jacobi equations in the Heisenberg group. Communications on Pure & Applied Analysis, 2003, 2 (4) : 461-479. doi: 10.3934/cpaa.2003.2.461
[17]

Manuel de León, David Martín de Diego, Miguel Vaquero. A Hamilton-Jacobi theory on Poisson manifolds. Journal of Geometric Mechanics, 2014, 6 (1) : 121-140. doi: 10.3934/jgm.2014.6.121
[18]

John D. Towers. The Lax-Friedrichs scheme for interaction between the inviscid Burgers equation and multiple particles. Networks & Heterogeneous Media, 2020, 15 (1) : 143-169. doi: 10.3934/nhm.2020007
[19]

Joachim von Below, Gaëlle Pincet Mailly, Jean-François Rault. Growth order and blow up points for the parabolic Burgers' equation under dynamical boundary conditions. Discrete & Continuous Dynamical Systems - S, 2013, 6 (3) : 825-836. doi: 10.3934/dcdss.2013.6.825
[20]

Laura Caravenna, Annalisa Cesaroni, Hung Vinh Tran. Preface: Recent developments related to conservation laws and Hamilton-Jacobi equations. Discrete & Continuous Dynamical Systems - S, 2018, 11 (5) : ⅰ-ⅲ. doi: 10.3934/dcdss.201805i

2018 Impact Factor: 0.871

Tools

Metrics

  • PDF downloads (14)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences