2015
DOI: 10.1002/mma.3749
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Large data solutions to the viscous quantum hydrodynamic model with barrier potential

Abstract: We discuss analytically the stationary viscous quantum hydrodynamic model including a barrier potential, which is a nonlinear system of partial differential equations of mixed order in the sense of Douglis–Nirenberg. Combining a reformulation by means of an adjusted Fermi level, a variational functional, and a fixed point problem, we prove the existence of a weak solution. There are no assumptions on the size of the given data or their variation. We also provide various estimates of the solution that are indep… Show more

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Cited by 1 publication
(4 citation statements)
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“…The maximum cannot be attained on the left endpoint s 0 , by assumption. If the maximum is attained at the right endpoint, then (9) and the Lipschitz continuity of u 0 imply max J κ u 2 κ (x) ≤ u 2 0 (s 0 +) + Cκ 1/4 . And if the maximum is attained inside of J κ , then (40) and (36) yield…”
Section: The Results Then Ismentioning
confidence: 97%
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“…The maximum cannot be attained on the left endpoint s 0 , by assumption. If the maximum is attained at the right endpoint, then (9) and the Lipschitz continuity of u 0 imply max J κ u 2 κ (x) ≤ u 2 0 (s 0 +) + Cκ 1/4 . And if the maximum is attained inside of J κ , then (40) and (36) yield…”
Section: The Results Then Ismentioning
confidence: 97%
“…Although formulated for the isothermal case p(n) = (T 0 + μ)n, the results of [12] seem to generalize to the case of general pressure terms p(n). And we also mention [9], where it was shown (in the isothermal case) that solutions (n, V, J) ∈ W 2,2 (0, 1) × W 2,2 (0, 1) × W 1,2 (0, 1) to (2) for given (possibly large) Dirichlet boundary values for V and periodic boundary values for n do exist. The purpose of the present paper is to extend the solution theory of [9] towards an asymptotic expansion of the solution, for vanishing values of the quantum mechanical parameters ε and ν, focusing on the equilibrium case.…”
Section: Introductionmentioning
confidence: 97%
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