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Multi-dimensional Systems of Conservation Laws: An Introductory Lecture

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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

These notes are written after the crash course given at the ICMS conference on Hyperbolic conservation laws. We intend to review several aspects of the theory of the Cauchy problem and the Initial-boundary value problem (IBVP). On the one hand, we give a thorough account of the theory for linear, constant coefficient operators, following Gårding, Hersch, Kreiss and others. Hyperbolicity raises interesting questions in real algebraic geometry, a topic to which Petrowski’s school (in particular Oleĭnik) contributed. Next, we turn towards quasilinear systems and recall the interplay between entropies and symmetrizability. This leads us to the local existence of a classical solution. The global-in-time Cauchy problem necessitates weak solutions; these must be selected by admissibility criteria. We give a review of the various criteria that have been elaborated so far. Some of them lead us to the ‘viscous’ approximation of hyperbolic systems. We review the structural properties of these models, whose paradigm is the Navier-Stokes-Fourier (NSF) system of gas dynamics. This is more or less Kawashima’s theory, in the simplified description that we have given in recent papers. We end with results about singular limits, such as the convergence of NSF towards Euler-Fourier when Newtonian viscosity tends to zero, and the analysis of the principal sub-systems introduced by Boillat and Ruggeri. Despite the length of these notes, they contain only very few proofs. We focus instead on the concepts and the theorems of the theory.

2010 Mathematics Subject Classification Primary: 35L04, 35L60, 35L65, 35L67, 35K40, 35M30; Secondary: 76L05, 76N17

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Notes

  1. 1.

    Gårding’s original definition involves well-posedness in C ∞, a weaker notion than the one used here. In particular it is not tailored to apply Duhamel’s principle. Our stronger version of hyperbolicity used to be called strong hyperbolicity. It is more practical, at least when we have quasi-linear Cauchy problems in mind.

  2. 2.

    This is always true in the constant rank hyperbolic case.

  3. 3.

    A classical result for Lyapunov equations \(\Sigma X + {X}^{{\ast}}\Sigma = S\) where S ∈H n is given and \(\Sigma \in \mathbf{H}_{n}\) is the unknown.

  4. 4.

    In [1], we missed this easy argument.

  5. 5.

    But it does not tell us whether these flows solve the Euler equation, that is if the Euler system is a good physical model.

  6. 6.

    Such perturbative effects are treated in Sect. 5.

  7. 7.

    This is associated with the conservation of mass.

  8. 8.

    There is at least one notable exception, namely the Maxwell system governing the electro-magnetic field in the vacuum. If it was dissipative, our world would be completely dark after billions of years. There are also the Einstein equations of the gravitational field and more generally all models dealing with fundamental forces.

  9. 9.

    The big open problem!

  10. 10.

    With the exception of the unphysical case s = s(θ), which means that p = p(ρ) only.

  11. 11.

    Actually, only strict convexity with respect to w is needed here.

  12. 12.

    The upper-left block is the Schur complement of D ww 2η in D2η.

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Authors and Affiliations

  1. UMPA, UMR 5669 CNRS, École Normale Supérieure de Lyon, Lyon, France

    Denis Serre

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Correspondence to Denis Serre .

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Editors and Affiliations

  1. University of Oxford Mathematical Institute, Oxford, United Kingdom

    Gui-Qiang G. Chen

  2. Department of Mathematical Sciences, Norwegian University of Science and Technology, Trondheim, Norway

    Helge Holden

  3. University of Oslo Centre of Mathematics for Applications, Oslo, Norway

    Kenneth H. Karlsen

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Serre, D. (2014). Multi-dimensional Systems of Conservation Laws: An Introductory Lecture. 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_12

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