Math. Nachr. 280, No. 8, 907 – 915 (2007) / DOI 10.1002/mana.200410523 Hugoniot–Maslov chains of a shock wave in conservation law with poly- nomial flow Panters Rodr´ ıguez Berm ´ udez ∗ 1 and Baldomero Vali˜ no Alonso ∗∗2 1 Instituto Nacional de Matem´ atica Pura e Aplicada, Estrada Dona Castorina, 110, CEP 22460-320 Rio de Janeiro, Brazil 2 Facultad de Matem´ atica y Computaci´ on, Universidad de La Habana, San L´ azaro y L, CP 10400, Cuba Received 15 December 2004, revised 3 November 2005, accepted 27 November 2005 Published online 7 May 2007 Key words Colombeau algebras of generalized functions, conservation laws, shock waves, Hugoniot– Maslov chains MSC (2000) Primary: 35L67; Secondary: 46F30 In this paper we give a theoretical foundation to the asymptotical development proposed by V. P.Maslov for shock type singular solutions of conservations laws, in the framework of Colombeau theory of generalized functions. Indeed, operating with Colombeau differential algebra of simplified generalized functions, we proof that Hugoniot–Maslov chains are necessary conditions for the existence of shock waves in conservation laws with polynomial flows. As a particular case, these equations include the Hugoniot–Maslov chains for shock waves in the Hopf equation. c  2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim 1 Introduction Shock waves under conservation laws are one of the types of propagating singular solutions that preserve their structure. As observed V. P. Maslov [10], the propagationof the shock waves can be described by an infinite system of ordinary differential equations known as Hugoniot–Maslov chain (see Dobrokhotov [7]). The holds of the chain is a necessary condition for the existence of the shock wave solution. Singular solutions of conservation laws, in particular, shock waves, are described by means of distributions like the Heaviside H and the Dirac δ functions, and others. So the consideration of such type of solutions requires of some kind of differential algebra to operate with them. Many ways to construction of differential algebras have been used to define the multiplication of distributions in a wider space than the space of distributions D ′ (Ω) after L. Schwartz’s proof of the imposibility of multi- plication of distributions (see Colombeau [4]). A new space of generalized functions G(Ω) was constructed by J. F. Colombeau in the early eighties [3], which contains the space of distributions D ′ (Ω). Many applications and further developments of Colombeau algebras have been done later (see, for instance, [1], [4], and others). For the problem considered in this paper, we will use a simplified algebra G s (Ω) of generalized functions, which is a subalgebra of G(Ω) but does not contain canonically the space of distributions. Nevertheless, the “classical” distributions H of Heaviside and δ of Dirac can be considered as “macroscopic aspects” of simplified gener- alized functions of the algebra G s (Ω), so its application will be enough for the study of shock waves in scalar conservations laws with polynomial flows. To justify the Hugoniot–Maslov chain corresponding to a shock wave in a conservation law with such a flow is the main purpose of this paper. We do that in the framework of Colombeau algebra G s (Ω). 2 Shock waves of scalar conservations laws with polynomial flows We begin with the formulation of the principal problem and the main result of the paper. ∗ Corresponding author: e-mail: panters@impa.br, Phone: +55 21 2529 5231, Fax: +55 21 2529 5075 ∗∗ e-mail: bval@matcom.uh.cu, Phone: +537 870 43 67, Fax: +537 873 63 10 c  2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim