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The Semigroup Approach to Conservation Laws with Discontinuous Flux

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Hyperbolic Conservation Laws and Related Analysis with Applications

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 49))

Abstract

The model one-dimensional conservation law with discontinuous spatially heterogeneous flux is

$$\displaystyle{ u_{t} + \mathfrak{f}(x,u)_{x} = 0,\quad \mathfrak{f}(x,\cdot ) = {f}^{l}(x,\cdot )1\!\!1_{ x<0}\! + {f}^{r}(x,\cdot )1\!\!1_{ x>0}. }$$
(EvPb)

We prove well-posedness for the Cauchy problem for (EvPb) in the framework of solutions satisfying the so-called adapted entropy inequalities.Exploiting the notion of integral solution that comes from nonlinear semigroup theory, we propose a way to circumvent the use of strong interface traces for the evolution problem (EvPb) (in fact, proving the existence of such traces for the case of x-dependent f l,r would be a delicate technical issue). The difficulty is shifted to the study of the associated one-dimensional stationary problem \(u + \mathfrak{f}(x,u)_{x} = g\), where the existence of strong interface traces of entropy solutions is an easy fact. We give a direct proof of this, avoiding the subtle arguments of the kinetic formulation (Kwon YS, Vasseur A (2007) Arch Ration Mech Anal 185(3):495–513) and of the H-measure approach (Panov EY (2007) J Hyperbolic Differ Equ 4(4):729–770).

2010 Mathematics Subject Classification Primary: 35L65, 35L04; Secondary: 47H06, 47H20

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Notes

  1. 1.

    Actually, a non-degeneracy of f l,r on intervals is needed for existence of such traces, see assumption (H3). But if the degeneracy happens, one can reformulate (1) in terms of the traces of some “singular mapping functions” Vf l,r(u), see [9].

  2. 2.

    See [9] for more general assumptions that ensure L bounds, which have to be adapted to the inhomogeneous case.

  3. 3.

    For the multi-dimensional domains treated in the Appendix, one uses an analogous definition based upon a parametrizaton of a neighbourhood of \(\partial \Omega \) by \((\sigma,h) \in \partial \Omega \times (0, 1)\).

  4. 4.

    To be specific, the Bardos-LeRoux-Nédélec formulation with a strong boundary trace (cf. [30]) is used not in \(\Omega \) but in specially selected subdomains of \(\Omega \), so that the existence of strong boundary traces comes “for free”

  5. 5.

    Consider a conservation law of the form div(t,x)ϕ(t,x,u) = h(t,x) set up in a space-time domain Q. We say that the boundary ∂ Q is space-like if the map u↦ϕ(t,x,u) ⋅ n(t,x) is strictly decreasing for all points (t,x) of the boundary. In this case, the local change of variables w(t,x):= ϕ(t,x,u) ⋅ n(t,x) (the field of exterior unit normal vectors n(⋅) on ∂ Q should be lifted in a neighbourhood of ∂ Q) reduces the situation to a standard conservation law with the time direction given by the vector field n(⋅).

  6. 6.

    To justify this claim, the arguments are the same as for the time-continuity of entropy solutions. Indeed, we have ensured that the normal component of the flux is a strictly increasing function: this makes the normal direction to the boundary time-like. Let us stress that the existence of a strong trace for this case is considerably easier to justify than in the general case: as a matter of fact, it follows from a local application of entropy inequalities. We refer to [19] and to [8, Lemma A4] for the arguments that can be used in this context.

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Acknowledgements

The author thanks Kenneth H. Karlsen for turning his attention to the difficulty treated in the Appendix. The work on this paper was partially supported by the French ANR project CoToCoLa.

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Correspondence to Boris Andreianov .

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Andreianov, B. (2014). The Semigroup Approach to Conservation Laws with Discontinuous Flux. In: Chen, GQ., Holden, H., Karlsen, K. (eds) Hyperbolic Conservation Laws and Related Analysis with Applications. Springer Proceedings in Mathematics & Statistics, vol 49. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-39007-4_1

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