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A comparison principle for Hamilton-Jacobi equation with moving in time boundary. (English) Zbl 1426.35078

Summary: In this paper we consider an Hamilton-Jacobi equation on a moving in time domain. The boundary is described by a \(C^1\) function. We show how we derive this equation from the work of J. P. Lebacque, J. B. Lesort and F. Giorgi [“Introducing buses into first-order macroscopic traffic flow models”, Transp. Res. Record: J. Transp. Res. Board 1644, No. 1, 70–79 (1998; doi:10.3141/1644-08)]. We only prove a comparison principle since the proof of other theoritical results can be found in [C. Imbert and R. Monneau, Ann. Sci. Éc. Norm. Supér. (4) 50, No. 2, 357–448 (2017; Zbl 1382.35075)]. At the end of the paper, we consider a short homogenization result in order to reinforce the traffic flow interpretation of the equation.

MSC:

35F21 Hamilton-Jacobi equations
35D40 Viscosity solutions to PDEs
90B20 Traffic problems in operations research
35B27 Homogenization in context of PDEs; PDEs in media with periodic structure
35F20 Nonlinear first-order PDEs
45K05 Integro-partial differential equations
35B51 Comparison principles in context of PDEs

Citations:

Zbl 1382.35075
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