Metastability for scalar conservation laws in a bounded domain

Abstract

Abstract. The initial-boundary-value problem for a viscous scalar conservation law in a bounded interval I = (− , ) 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 coefficient ε > 0 goes to zero. We describe such behavior by deriving an ODE for the position ξ of the internal interface. The main tool of our analysis is the construction of a one-parameter family of approximate stationary solutions {U ε (·; ξ)} ξ∈I , parametrized by the location of the shock layer ξ, to be considered as an approximate invariant manifold for the problem. By using the properties of the linearized operator at U ε , we estimate the size of the layer location.

Slow Motion of Internal Shock Layers for the Jin–Xin System in One Space Dimension

“…This stabilization property has been proved for the first time in [17], using the theory of generalized characteristic, firstly introduced in [6]. In this framework, assumptions (1.3) on the flux function f are crucial (see [20,Theorem 6.1]). Hence, every entropy solution to the initial-boundary value problem…”

“…) is an approximate stationary solution to (2.1), in the sense that it satisfies the stationary equation up to an error that is small in ε. More precisely, following the idea firstly introduced in [20], we assume that there exist two families of smooth functions Ω ε 1 = Ω ε 1 (ξ) and Ω ε 2 = Ω ε 2 (ξ), uniformly convergent to zero as ε → 0, such that, for any ξ ∈ I, the following estimates hold…”

“…In order to determine the behavior of P ε 1 [W ε (·; ξ)] for small ε, we need an asymptotic description of the values k ± . Following the idea of [20], let us set k ± := ∓u ± (1 + h ± ) and ∆ ± := ∓ ξ. Relation (2.5) becomes…”

“…Mimicking the approach of [20] let us assume that, for any ξ, the linear operator L ε ξ has a sequence of eigenvalues λ ε k = λ ε k (ξ) with corresponding (right) eigenfunctions φ ε k = φ ε k (ξ, ·) (for more details see Section 3). Denoting by ψ ε k = ψ ε k (ξ, ·) the eigenfunctions of the corresponding adjoint operator L ε * ξ and setting Y k = Y k (ξ; t) := ψ ε k (·; ξ), Y (·, t) , we impose that the component Y 1 = ψ ε 1 (·; ξ), Y (·, t) is identically zero.…”

“…The main difference with respect to [20] is that here we deal with an hyperbolic system. Hence, since the linearized operator around the reference state {W ε } is not necessarily self-adjoint, we have to consider the chance of having complex eigenvalues, and the spectral analysis need much more care.…”