Conservation laws arise from the modeling of physical processes through the following three steps:
  1. 1)

    The appropriate physical balance laws are derived for m-physical quantities, u1,...,um with u = t(u1,...,Um) and u(x,t) defined for x = (x1,...,xN) ∈ RN (N = 1,2, or 3), t ≥ 0 and with the values u(x,t) lying in an open subset, G, of Rm, the state space. The state space G arises because physical quantities such as the density or total energy should always be positive; thus the values of u are often constrained to an open set G.

  2. 2)

    The flux functions appearing in these balance laws are through prescribed nonlinear functions, Fj(u), mapping G into Rm, j = 1,...,N while source terms are defined by S(u,x,t) with S a given smooth function of these arguments with values in Rm. In particular, the detailed microscopic effects of diffusion and dissipation are ignored.

  3. 3)

    A generalized version of the principle of virtual work is applied (see Antman [1]).



Nonlinear Wave Equation Entropy Condition Simple Wave Shallow Water Theory Nonlinear Wave Propagation 
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Copyright information

© Springer Science+Business Media New York 1984

Authors and Affiliations

  • A. Majda
    • 1
  1. 1.Department of MathematicsUniversity of CaliforniaBerkeleyUSA

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