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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), 1-42. 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), 1017-1034. 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, 20. Oxford University Press, Oxford, 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), 487-502. doi: 10.1090/S0002-9947-1984-0732102-X.  Google Scholar

[5]

L. C. Evans, "Partial Differential Equations," Graduate Studies in Mathematics, 19. American Mathematical Society, Providence, RI, 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), 503-543. doi: 10.1142/S0219891610002219.  Google Scholar

[7]

H. Frankowska, Lower semicontinuous solutions to Hamilton-Jacobi-Bellman equations, Proceedings of 30th CDC Conference, IEEE, Brighton, December 11-13, (1991), 265-270. Google Scholar

[8]

H. Frankowska, Lower semicontinuous solutions of Hamilton-Jacobi-Bellman equation, SIAM J. Control and Optimization, 31 (1993), 257-272. doi: 10.1137/0331016.  Google Scholar

[9]

P. D. Lax, Hyperbolic systems of conservation laws II, Comm. Pure Appl. Math., 10 (1957), 537-566. 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), 265-287. 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), 317-345. doi: 10.1098/rspa.1955.0089.  Google Scholar

[12]

P. Richards, Shock waves on the highway, Operations Research, 4 (1956), 42-51. 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), 733-748. 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), 1-42. 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), 1017-1034. 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, 20. Oxford University Press, Oxford, 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), 487-502. doi: 10.1090/S0002-9947-1984-0732102-X.  Google Scholar

[5]

L. C. Evans, "Partial Differential Equations," Graduate Studies in Mathematics, 19. American Mathematical Society, Providence, RI, 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), 503-543. doi: 10.1142/S0219891610002219.  Google Scholar

[7]

H. Frankowska, Lower semicontinuous solutions to Hamilton-Jacobi-Bellman equations, Proceedings of 30th CDC Conference, IEEE, Brighton, December 11-13, (1991), 265-270. Google Scholar

[8]

H. Frankowska, Lower semicontinuous solutions of Hamilton-Jacobi-Bellman equation, SIAM J. Control and Optimization, 31 (1993), 257-272. doi: 10.1137/0331016.  Google Scholar

[9]

P. D. Lax, Hyperbolic systems of conservation laws II, Comm. Pure Appl. Math., 10 (1957), 537-566. 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), 265-287. 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), 317-345. doi: 10.1098/rspa.1955.0089.  Google Scholar

[12]

P. Richards, Shock waves on the highway, Operations Research, 4 (1956), 42-51. 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), 733-748. doi: 10.1002/rnc.1099.  Google Scholar

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