The initial-boundary-value problem for a viscous scalar conservation law in a bounded interval $I=(-\ell,\ell)$ is considered, with emphasis on the metastable dynamics, whereby the time-dependent solution develops internal transition layers that approach their steady state in an asymptotically exponentially long time interval as the viscosity coecient $\epsilon>0$ goes to zero. We describe such behavior by deriving an ODE for the position of the internal interface. The main tool of our analysis is the construction of a one-parameter family of approximate stationary solutions $\U^\epsilon(\cdot;\xi)\$ , parametrized by the location of the shock layer $\xi$, to be considered as an approximate invariant manifold for the problem. By using the properties of the linearized operator at $U^\epsilon$, we estimate the size of the layer location.

### Metastability for scalar conservation laws in a bounded domain

#### Abstract

The initial-boundary-value problem for a viscous scalar conservation law in a bounded interval $I=(-\ell,\ell)$ is considered, with emphasis on the metastable dynamics, whereby the time-dependent solution develops internal transition layers that approach their steady state in an asymptotically exponentially long time interval as the viscosity coecient $\epsilon>0$ goes to zero. We describe such behavior by deriving an ODE for the position of the internal interface. The main tool of our analysis is the construction of a one-parameter family of approximate stationary solutions $\U^\epsilon(\cdot;\xi)\$ , parametrized by the location of the shock layer $\xi$, to be considered as an approximate invariant manifold for the problem. By using the properties of the linearized operator at $U^\epsilon$, we estimate the size of the layer location.
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Actes congres SMAI 2013
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/10278/3699472