Explicit construction of solutions to the Burgers equation with discontinuous initial-boundary conditions
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ENSTA ParisTech, 828, Boulevard des Maréchaux, 91762 Palaiseau Cedex
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CNRS and Institut de Mathématiques de Jussieu, 4 place Jussieu, Université Pierre et Marie Curie, case 247, 75252 Paris,
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Received:
01 April 2012
Revised:
01 August 2013
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35C99, 35D40, 35F31, 35L65, 49L99.
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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.
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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