WIAS Preprint No. 438, (1998)

Initial and Boundary Value Problems of Hyperbolic Heat Conduction


  • Dreyer, Wolfgang
  • Kunik, Matthias

2010 Mathematics Subject Classification

  • 80-99 35L15 35L20 35L65 35L67


  • heat transfer, initial and boundary value problems for a hyperbolic system, shock waves, kinetic theory




This is a study on the initial and boundary value problem of a symmetric hyperbolic system which is related to the conduction of heat in solids at low temperatures. The nonlinear system consists of a conservation equation for the energy density e and a balance equation for the heat flux Qi, where e and Qi are the four basic fields of the theory. The initial and boundary value problem that uses exclusively prescribed boundary data for the energy density e is solved by a new kinetic approach. This method includes the formation of shock fronts and the broadening of heat pulses. These effects cannot be observed in the linearized theory, as it is described in [4]. The kinetic representations of the initial and boundary value problem reveal a peculiar phenomenon. To the solution there contribute integrals containing the initial fields e0 (x), Q0 (x) as well as integrals that need knowledge on energy and heat flux at a boundary. However, only one of these quantities can be controlled in an experiment. When this ambiguity is removed by a continuity condition, it will turn out that after some very short time energy and heat flux are related to each other according to the Rankine Hugoniot relation.

Appeared in

  • Initial and boundary value problems of hyperbolic heat conduction, Contin. Mech. Thermodyn., 11 (1999), No. 4, pp. 227-245

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