2009
DOI: 10.1007/s00605-009-0092-4
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On the blowing up of solutions to quantum hydrodynamic models on bounded domains

Abstract: Abstract. The blow-up in finite time for the solutions to the initial-boundary value problem associated to the multi-dimensional quantum hydrodynamic model in a bounded domain is proved. The model consists on conservation of mass equation and a momentum balance equation equivalent to a compressible Euler equations corrected by a dispersion term of the third order in the momentum balance. The proof is based on a-priori estimates for the energy functional for a new observable constructed with an auxiliary functi… Show more

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Cited by 19 publications
(10 citation statements)
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“…employing the bound (12). Hence, as we are working in finite dimensions, the operators are invertible with…”
Section: Linear Faedo-galerkin Approximationmentioning
confidence: 98%
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“…employing the bound (12). Hence, as we are working in finite dimensions, the operators are invertible with…”
Section: Linear Faedo-galerkin Approximationmentioning
confidence: 98%
“…Theorem 1 is proved in Section 4, whereas Section 5 is concerned with the proof of Theorem 2. We remark that the a priori estimates derived from the energy functional (5) and its corresponding energy production were already employed in [5,12,18]. …”
Section: The Condition H( Y)mentioning
confidence: 99%
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“…Formally, if we set C ≡ 0, λ = 0, take τ a = τ b = τ > 0, P λ a = A(ρ λ a ) γ , and P λ b = A(ρ λ b ) γ for the constants A > 0 and γ > 1 in system (1.1)-(1.3), then we have ρ λ a = ρ λ b . Thus, the bipolar QHD model (1.1)-(1.3) will become the QHD equations [3] (if ρ and v, the limits of ρ λ i and u λ i , i = a, b exist, respectively):…”
Section: Introductionmentioning
confidence: 99%
“…In[3], the quantum hydrodynamic equations (1.5)-(1.6) had been introduced to investigate the blow-up phenomena in semiconductors. Our results in this paper indicate that system (1.5)-(1.6) can be obtained from the bipolar QHD model (1.1)-(1.3) by taking the process of quasineutral limit.…”
mentioning
confidence: 99%