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Abstract

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]).

     

Keywords

Nonlinear Wave Equation Entropy Condition Simple Wave Shallow Water Theory Nonlinear Wave Propagation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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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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