Irene M. Gamba, Ansgar Jungel
Positive solutions to singular second and third order differential
equations for quantum fluids
(231K, amstex)
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.