99-367 Irene M. Gamba, Ansgar Jungel

Positive solutions to singular second and third order differential equations for quantum fluids (231K, amstex) Sep 30, 99

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Abstract. A steady-state hydrodynamic model for quantum fluids is analyzed. The momentum equation can be written as a dispersive third-order equation for the particle density where viscous effects can be incorporated. The phenomena that admit positivity of the solutions are studied. The cases: dispersive or non-dispersive, viscous or non-viscous are thoroughly analyzed with respect to positivity and existence or non-existence of solutions. It is proven that in the dispersive, non-viscous model, a classical positive solution only exists for ``small'' data and no weak solution can exist for ``large'' data, whereas the dispersive, viscous problem admits a classical positive solution for all data. The viscous term is shown to correspond to hyper-viscosity or ultra-diffusion. The proofs are based on a reformulation of the equations as a singular elliptic second-order problem and on a variant of the Stampacchia truncation technique. The results are extended to general third-order equations in any space dimension.

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