Abstract
The initial-value problem for a hyperbolic system of conservation laws will be formulated, and classical as well as weak solutions will be considered. Nonuniqueness of weak solutions will be demonstrated in the context of the simplest nonlinear scalar conservation law, the well-known Burgers equation. This raises the need to devise conditions that will weed out unstable, physically irrelevant, or otherwise undesirable solutions, hopefully singling out a unique admissible solution of the initial-value problem. Two admissibility criteria will be introduced in this chapter: The requirement that admissible solutions satisfy a designated entropy inequality; and the principle that the hyperbolic system should be viewed as the “vanishing viscosity” limit of a family of systems with diffusive terms. A preliminary investigation of the compatibility of the above two criteria will be conducted. The chapter will close with remarks on the interpretation of boundary conditions in the context of weak solutions.
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Bateman, H.: Some recent researches on the motion of fluids. Monthly Weather Review 43 (1915), 163–170.
Hopf, E.: The partial differential equation u, + uu,- = µu,,. Comm. Pure Appl. Math. 3 (1950), 201–230.
Stokes, G.G.: On a difficulty in the theory of sound. Philos. Magazine 33 (1848), 349–356.
Rayleigh, Lord (J.W. Strutt): The Theory of Sound, Vol I1. London: Macmillan, 1878.
Stokes, G.G.: On a difficulty in the theory of sound. Mathematical and Physical Papers, Vol II, pp. 51–55. Cambridge: Cambridge University Press, 1883.
Burton, C.V.: On plane and spherical sound-waves of finite amplitude. Philos. Magazine, ’ cr. 5, Cabannes, H.: (1893), 317–333.
Weber, H.: Die Partiellen Differential-Gleichungen der Mathematischen Physik, Zweiter Band, Vierte Auflage. Braunschweig: Friedrich Vieweg und Sohn, 1901.
Rayleigh, Lord (J.W. Strutt): Note on tidal bores. Proc. Royal Soc. London 81A (1908), 448–449.
Jouguet, E.: Sur la propagation des discontinuités dans les fluides. C. R. Acad. Sc1. Paris 132 (1901), 673–676.
Kruzkov, S.: First-order quasilinear equations with several space variables. Mat. Sbornik 123 (1970), 228–255. English translation: Math. USSR Sbomik 10 (1970), 217–273.
Lax, P.D.: Shock waves and entropy. Contributions to Functional Analysis pp. 603–634, ed. E.A. Zarantonello. New York: Academic Press, 1971.
Truesdell, C.A. and R.A. Toupin: The Classical Field Theories. Handbuch der Physik, Vol. III/1. Berlin: Springer, 1960.
Truesdell, C.A. and W. Noll: The Non-Linear Field Theories of - Mechanics. Handbuch der Physik, Vol. 111/3. Berlin: Springer, 1965.
Gurtin, M.E.: An Introduction to Continuum Mechanics. New York: Academic Press, 1981.
Silhavÿ, M.: The Mechanics and Thermodynamic’s of Continuous Media. Berlin: Springer, 1997.
Ciarlet, P.G.: Mathematical Elasticity. Amsterdam: North-Holland, 1988.
Hanyga, A.: Mathematical Theory of Non-Linear Elasticity. Warszawa: PWN, 1985.
Marsden, J.E. and T.J.R. Hughes: Mathematical Foundations of Elasticity. Englewood Cliffs: Prentice-Hall, 1983.
Wang, C.C. and C. Truesdell: Introduction to Rational Elasticity. Leyden: Noordhoff, 1973.
Antman, S.S.: Nonlinear Problems of Elasticity. New York: Springer, 1995.
Luo, T. and D. Serre: Linear stability of shock profiles for a rate-type viscoelastic system with relaxation. Quart. Appl. Math. 56 (1998), 569–586.
Amadori, D., Baiti, P., LeFloch, P.G. and B. Piccoli: Nonclassical shocks and the Cauchy problem for nonconvex conservation laws. J. Diff. Eqs. 151 (1999), 345–372.
Gisclon, M.: Etude des conditions aux limites pour un système strictement hyperbolique via l’approximation parabolique. J. Math. Pures Appl. 75 (1996), 485–508.
Gisclon, M. and D. Serre: Etude des conditions aux limites pour un système strictement hyperbolique via l’approximation parabolique. C. R. Acad. Sc1. Paris, Série I, 319 (1994), 377–382.
Grenier, E.: Boundary layers for viscous perturbations of noncharacteristic quasilinear hyperbolic problems. J. Diff. Eqs. 143 (1998), 110–146.
Hayes, B. and P.G. LeFloch: Nonclassical shocks and kinetic relations: Scalar conservaton laws. Arch. Rational Mech. Anal. 139 (1997), 1–56.
Otto, F.: Initial-boundary value problem for a scalar conservation law. C. R. Acad. Sc1. Paris, Série I, 322 (1996), 729–734.
Smoller, J.A., Temple, J.B. and Z.-P. Xin: Instability of rarefaction shocks in systems of conservation laws. Arch. Rational Mech. Anal. 112 (1990), 63–81.
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Dafermos, C.M. (2000). The Initial-Value Problem: Admissibility of Solutions. In: Hyperbolic Conservation Laws in Continuum Physics. Grundlehren der mathematischen Wissenschaften, vol 325. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-22019-1_4
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