Ground States for Mass Critical Two Coupled Semi-Relativistic Hartree Equations with Attractive Interactions

We prove the existence and nonexistence of $L^{2}(\mathbb R^3)$-normalized solutions of two coupled semi-relativistic Hartree equations, which arisen from the studies of boson stars and multi-component Bose–Einstein condensates. Under certain condition on the strength of intra-specie and inter-specie interactions, by proving some delicate energy estimates, we give a precise description on the concentration behavior of ground state solutions of the system. Furthermore, an optimal blowing up rate for the ground state solutions of the system is also proved.

Full Analytical Modeling Of Intrawell Chemical Tracer Concentration For Robust Production Allocation In Challenging Environments

Conventional downhole dynamic characterization is based on data from standard production logging tool (PLT) strings. Such method is not a feasible option in long horizontal drains, deep water scenarios, subsea clusters, pump-assisted wells and in presence of asphaltenes/solids deposition, mainly due to high costs and risk of tools stuck. In this respect, intrawell chemical tracers (ICT) can represent a valid and unobtrusive monitoring alternative. This paper deals with a new production allocation interpretation model of tracer concentration behavior that can overcome the limitation of standard PLT analyses in challenging environments. ICT are installed along the well completion and are characterized by a unique oil and/or water tracer signature at each selected production interval. Tracer concentration is obtained by dedicated analyses performed for each fluid sample taken at surface during transient production. Next, tracer concentration behavior over time is interpreted, for each producing interval, by means of an ad-hoc one-dimensional partial differential equation model with proper initial and boundary conditions, which describes tracer dispersion and advection profiles in such transient conditions. The full time-dependent analytical solutions are then utilized to obtain the final production allocation. The methodology has been developed and validated using data from a dozen of tracer campaigns. The approach is here presented through a selected case study, where a parallel acquisition of standard PLT and ICT data has been carried out in an offshore well. The aim was to understand if ICT could be used in substitution of the more impacting PLT for the future development wells in the field. At target, the well completion consists of a perforated production liner with tubing. The latter, which is slotted in front of the perforations, includes oil and water tracer systems. The straightforward PLT interpretation shows a clear dynamic well behavior with an oil production profile in line with the expectations from petrophysical information. Then, after a short shut-in period, the ICT-based production allocation has been performed in transient conditions with a very good match with the available outcomes from PLT: in fact, the maximum observed difference in the relative production rates is 5%. In addition, the full analytical solution of the ICT model has been fundamental to completely characterize some complex tracer concentration behaviors over time, corresponding to non-simultaneous activation of the different producing intervals. Given the consistency of the independent PLT and ICT interpretations, the monitoring campaign for the following years has been planned based on ICT only, with consequent impact on risk and cost mitigations. Although the added value of ICT is relatively well known, the successful description of the tracer signals through the full mathematical model is a novel topic and it can open the way for even more advanced applications.

Blow-Up Phenomena and Asymptotic Profiles Passing from H 1-Critical to Super-Critical Quasilinear Schrödinger Equations

We study the asymptotic profile, as ℏ → 0 {\hbar\rightarrow 0} , of positive solutions to - ℏ 2 ⁢ Δ ⁢ u + V ⁢ ( x ) ⁢ u - ℏ 2 + γ ⁢ u ⁢ Δ ⁢ u 2 = K ⁢ ( x ) ⁢ | u | p - 2 ⁢ u , x ∈ ℝ N , -\hbar^{2}\Delta u+V(x)u-\hbar^{2+\gamma}u\Delta u^{2}=K(x)\lvert u\rvert^{p-2% }u,\quad x\in\mathbb{R}^{N}, where γ ⩾ 0 {\gamma\geqslant 0} is a parameter with relevant physical interpretations, V and K are given potentials and the dimension N is greater than or equal to 5, as we look for finite L 2 {L^{2}} -energy solutions. We investigate the concentrating behavior of solutions when γ > 0 {\gamma>0} and, differently from the case γ = 0 {\gamma=0} where the leading potential is V, the concentration is here localized by the source potential K. Moreover, surprisingly for γ > 0 {\gamma>0} we find a different concentration behavior of solutions in the case p = 2 ⁢ N N - 2 {p=\frac{2N}{N-2}} and when 2 ⁢ N N - 2 < p < 4 ⁢ N N - 2 {\frac{2N}{N-2}<p<\frac{4N}{N-2}} . This phenomenon does not occur when γ = 0 {\gamma=0} .

Existence and Concentration Behavior of Solutions of the Critical Schrödinger–Poisson Equation in R3

In this paper, we study the singularly perturbed problem for the Schrödinger–Poisson equation with critical growth. When the perturbed coefficient is small, we establish the relationship between the number of solutions and the profiles of the coefficients. Furthermore, without any restriction on the perturbed coefficient, we obtain a different concentration phenomenon. Besides, we obtain an existence result.