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Mathematical analysis of a Saint-Venant model with variable temperature. (English) Zbl 1204.35134
A Saint-Venant system of partial differential equations with an energy equation and with the temperature depending on the transport coefficients is investigated. A symmetric conservative formulation of the system is derived. Then, the system is written in the form of a symmetric hyperbolic-parabolic composite system.
In the next section, the global existence around an equilibrium state and the asymptotic stability of the resulting system of partial differential equations are established.
In the last two sections, a three-dimensional model of a thin viscous sheet over a fluid substrate is studied. The authors consider the 3D system of partial differential equations governing a thin viscous layer of an incompressible fluid with two free boundaries, an upper fluid/gas boundary and a lower fluid/substrate boundary. By means of an asymptotic analysis, the Saint-Venant equations with an energy equation and with the temperature depending on the transport coefficients are derived from the considered 3D model.

MSC:
35Q35 PDEs in connection with fluid mechanics
76A20 Thin fluid films
76D27 Other free boundary flows; Hele-Shaw flows
35B35 Stability in context of PDEs
35B40 Asymptotic behavior of solutions to PDEs
76M45 Asymptotic methods, singular perturbations applied to problems in fluid mechanics
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