Conservation laws describe the conservation of some basic physical quantities of a system and they arise in all branches of science and engineering. In this chapter, we study first-order, quasi-linear, partial differential equations which become conservation laws. We discuss the fundamental role of characteristics in the study of quasi-linear equations and then solve the nonlinear, initial-value problems with both continuous and discontinuous initial data. Special attention is given to discontinuous (or weak) solutions, development of shock waves, and breaking phenomena. As we have observed, quasi-linear equations arise from integral conservation laws which may be satisfied by functions which are not differentiable, and not even continuous, but simply bounded and measurable. These functions are called weak or generalized solutions, in contrast to classical solutions, which are smooth (differentiable) functions. It is shown that the integral conservation law can be used to derive the jump condition, which allows us to determine the speed of discontinuity or shock waves. Finally, a formal definition of a shock wave is given.


Shock Wave Classical Solution Wave Profile Discontinuous Solution Breaking Phenomenon 
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  1. Buckley, S.E. and Leverett, M.C. (1942). Mechanisms of fluid displacement in sands, Trans. AIME 146, 107–116. Google Scholar
  2. Lax, P.D. (1973). Hyperbolic Systems of Conservation Laws and the Mathematical Theory of Shock Waves, Society for Industrial and Applied Mathematics, Philadelphia. MATHGoogle Scholar

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© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of Texas, Pan AmericanEdinburgUSA

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