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Unique solvability of the periodic Cauchy problem for wave‐hierarchy problems with dissipation
Author(s) -
Göz Manfred F.
Publication year - 1994
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.1670171004
Subject(s) - mathematics , initial value problem , sobolev space , mathematical analysis , boundary value problem , dissipation , compressibility , periodic boundary conditions , bifurcation , hierarchy , nonlinear system , mechanics , economics , market economy , physics , quantum mechanics , thermodynamics
Wave‐hierarchy problems appear in a variety of applications such as traffic flows, roll waves down an open inclined channel and multiphase flows. Usually, these are described by the compressible Navier‐Stokes equations with specific non‐linearities; in a fluidized bed model they contain an additional pressure gradient term and are supplemented by an elliptic equation for this unknown pressure. These equations admit solutions periodic in space as well as in time, i.e. periodic travelling waves. Therefore, the corresponding initial value problem with periodic boundary conditions is solved locally in time in appropriate Sobolev spaces. Some remarks are made concerning global solutions, the occurrence of clusters or voids and the bifurcation of time periodic solutions, respectively.