Genuine Hydrodynamic Analysis to the 1-D QHD System: Existence, Dispersion and Stability

Antonelli, Paolo; Marcati, Pierangelo; Zheng, Hao

doi:10.1007/s00220-021-03998-z

Cited by 14 publications

(16 citation statements)

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confidence: 59%

Remarks on the derivation of finite energy weak solutions to the QHD system

Antonelli

2021

Proc. Amer. Math. Soc.

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In this paper we consider the multi-dimensional Quantum Hydrodynamics (QHD) system, by adopting an intrinsically hydrodynamic approach. The present work continues the analysis initiated in [6] where the one dimensional case was studied. Here we extend the analysis to the multi-dimensional problem, in particular by considering two physically relevant classes of solutions. First of all we consider two-dimensional initial data endowed with point vortices; by assuming the continuity of the mass density and a quantization rule for the vorticity we are able to study the Cauchy problem and provide global finite energy weak solutions. The same result can be obtained also by considering spherically symmetric initial data in the multi-dimensional setting. For rough solutions with finite energy, we are able to provide suitable dispersive estimates, which also apply to a more general class of Euler-Korteweg equations.Moreover we are also able to show the sequential stability of weak solutions with positive density. Analogously to the one dimensional case this is achieved through the a priori bounds given by a new functional first introduced in [6].

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confidence: 59%

Remarks on the derivation of finite energy weak solutions to the QHD system

Antonelli

2021

Proc. Amer. Math. Soc.

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“…Indeed, if ψ is a solution to NLS, then the associated observables (ρ, J) defined by ρ = |ψ| 2 , J = Im( ψ∇ψ) formally solve the QHD system. This approach was made rigorous in [4,5,7], by exploiting a polar factorization method, see also Lemma 2.8 below for a similar result. More precisely, given a wave function ψ ∈ H 1 , it is possible to define the hydrodynamic state ( √ ρ, Λ) by √ ρ = |ψ|, Λ = Im( φ∇ψ), where ϕ is a polar factor of ψ, see (2.7) below for a rigorous definition.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…T W 1/2,6 are uniformly bounded, where we set σ := min{σ, 7 6 } > 0; (2) there exits δ > 0 such that χ α ψ (n)…”

Section: Proof Of the Main Resultsmentioning

confidence: 99%

Existence and Stability of almost finite energy weak solutions to the Quantum Euler-Maxwell system

Antonelli¹,

Marcati²,

Scandone³

2021

Preprint

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We prove the existence of global in time, finite energy, weak solutions to a quantum magneto-hydrodynamic system (QMHD), modeling a charged quantum fluid interacting with it self-generated electromagnetic field, with the additional presence of a pure-power, quantum statistical pressure which comes from the electron degeneracy, due to Heisenberg's uncertainty principle and Pauli's exclusion principle. No smallness and/or higher regularity is necessary. The QMHD system formally is strictly connected with the wave function dynamics, given by the related nonlinear Maxwell-Schrödinger system, via the so called Madelung transformations. To make rigorous this connection we actually need to be able to approximate non smooth solutions. Unfortunately the nonlinear Maxwell-Schrödinger system lacks of a well-posedness theory in the energy space, suitable to approximate the QMHD nonlinear terms, in particular the Lorentz force. To circumvent these difficulties here we perform a derivation by using the integral formulation for weak solutions. More precisely, we derive suitable a priori smoothing estimates for the Maxwell-Schrödinger system, so we can rigorously justify the derivation for a class of "almost" finite energy initial data. In the same regime of regularity we also prove a local smoothing effect, which guarantees the weak-stability of both the hydrodynamic variables and the Lorentz force associated to the electromagnetic field.

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“…The case of quantum hydrodynamics corresponds to K (ρ) = κ/ρ and, in this case, the (EK)-system is formally equivalent, via Madelung transform, to a semilinear Schrödinder equation on R d . Exploiting this fact, global in time weak solutions have been obtained by Antonelli-Marcati [4,5] also allowing ρ(t, x) to become zero (see also the recent paper [6]).…”

Section: Introductionmentioning

confidence: 99%