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

Anya Désilles; Hélène Frankowska; Anya Désilles; Hélène Frankowska

doi:10.3934/nhm.2013.8.727

Networks and Heterogeneous Media

2013, Volume 8, Issue 3: 727-744. doi: 10.3934/nhm.2013.8.727

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

Anya Désilles ^{1
,},
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: 01 April 2012 Revised: 01 August 2013
35C99, 35D40, 35F31, 35L65, 49L99.

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.
- Burgers equation,
- boundary conditions,
- Hamilton-Jacobi equation,
- Lax formulae,
- value function
Citation: Anya Désilles, Hélène Frankowska. Explicit construction of solutions to the Burgers equation with discontinuous initial-boundary conditions[J]. Networks and Heterogeneous Media, 2013, 8(3): 727-744. doi: 10.3934/nhm.2013.8.727

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Abstract

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.

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