分野別セミナー

The viscous Burgers equation is well known as a simple equation describing fluid phenomena, and it is also known as a mathematical model of traffic flow. As a model of traffic flow, we consider the one-dimensional Cauchy problem of the generalized viscous Burgers equations with a time delay and analyze a delay effect. Indeed, because the viscous Burgers equation is a parabolic partial differential equation, its solution has an infinite speed of propagation. In terms of traffic flow, this means that drivers and their vehicles are assumed to react instantly changing the density and its gradient. To improve this troubling feature, we modify the term without a time delay to the one with a time delay. In this talk, we show the existence of the global-in-time solution when the product of the size of the delay parameter and the one of the initial history is suitably small. Moreover, we also prove some theorems concerning the regularity of the global-in-time solution. This result is joint research with Takayuki Kubo of Ochanomizu University.

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