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

Abstract

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].

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

“…As a consequence (see e.g. the proof of [2, Proposition 4.3]), one can deduce the existence of a global, weak M (1,1) -solution (ψ, A) to the system (1.10) with initial data (ψ 0 , A 0 , A 1 ), such that for every T > 0…”

“…This approach was already exploited in [1] to provide an alternative way of deriving the QHD system from the NLS equation and it is inspired by an argument by Ozawa [24]. The main idea in [24] is to prove the well known conservation in time of the physical quantities associated to the NLS equation, without using an approximation procedure.…”

“…We start by considering the case s = 0. Let h be a smooth, real valued, increasing function, such h ′ (t) = 1 for |t| ≥ 1 2 , h ′ (t) = 0 for |t| ≥ 1, and sup t∈R |h(t)| ≤ 1. For every α ∈ Z 3 and spatial direction j ∈ {1, 2, 3}, we set h α,j (x) := h(x j − α j ), and with a slight abuse of notation we write h ′ α,j := ∂ j h α,j , and analogously for higher derivatives.…”

“…as an identity in H −1 (R 3 ) for every t ∈ [0, T ], where we set (4.2) 1) ψ. Finally, we say that a solution is global if it is defined on [0, T ] × R 3 fo every T > 0.…”