On properties of solutions to the α-harmonic equation

2019 ◽  
Vol 65 (12) ◽  
pp. 1981-1997
Author(s):  
Peijin Li ◽  
Antti Rasila ◽  
Zhi-Gang Wang
Keyword(s):  
Harmonic Equation

Classification of nonnegative solutions to a bi-harmonic equation with Hartree type nonlinearity

2018 ◽  
Vol 149 (04) ◽  
pp. 979-994 ◽  
Author(s):  
Daomin Cao ◽  
Wei Dai
Keyword(s):  
Critical Case ◽  
Classical Solutions ◽  
Sobolev Inequalities ◽  
Extremal Functions ◽  
Method Of Moving Planes ◽  
Nonnegative Solutions ◽  
Moving Planes ◽  
Image Position ◽  
Harmonic Equation

AbstractIn this paper, we are concerned with the following bi-harmonic equation with Hartree type nonlinearity $$\Delta ^2u = \left( {\displaystyle{1 \over { \vert x \vert ^8}}* \vert u \vert ^2} \right)u^\gamma ,\quad x\in {\open R}^d,$$where 0 < γ ⩽ 1 and d ⩾ 9. By applying the method of moving planes, we prove that nonnegative classical solutions u to (𝒫γ) are radially symmetric about some point x0 ∈ ℝd and derive the explicit form for u in the Ḣ2 critical case γ = 1. We also prove the non-existence of nontrivial nonnegative classical solutions in the subcritical cases 0 < γ < 1. As a consequence, we also derive the best constants and extremal functions in the corresponding Hardy-Littlewood-Sobolev inequalities.

scholarly journals Investigation of vibration characteristics for bladed-disks with dry friction nonlinearity

2018 ◽  
Vol 207 ◽  
pp. 04009
Author(s):  
Tianyuan Liu ◽  
Yonghui Xie ◽  
Di Zhang
Keyword(s):  
Dry Friction ◽  
Normal Load ◽  
Frequency Domain Analysis ◽  
Friction Model ◽  
Load Level ◽  
Vibration Characteristics ◽  
Lumped Parameter Model ◽  
Qualitative Assessment ◽  
Bladed Disk ◽  
Harmonic Equation

This paper focuses on the vibration characteristics of the bladed-disk subjected to the dry friction damping under periodic excitation. Firstly, the multi-harmonic equation basing on the frequency-domain analysis is established to predicate the steady response of the bladed-disk. Then, the algorithm to solve the nonlinear multi-harmonic balance equation is given step by step. In the numerical simulation, a simplified lumped parameter model of a turbine bladed-disk as well as the elastic Coulomb friction model between the root and disk contacting surfaces are applied. The normal load level of the friction interfaces, which are the focal points of the bladed-disk design, is analysed for the nonlinear vibration characteristics of the blades. The results show that vibration response of bladed-disk is affected significantly by normal load, and there exists an optimal value of the normal load under the operating condition of the turbomachinery, which can provide a qualitative assessment for the design practice of friction dampers.

Temporal patterns of CO 2 and CH 4 in a rural area in northern Spain described by a harmonic equation over 2010–2016

2017 ◽  
Vol 593-594 ◽  
pp. 1-9 ◽  
Author(s):  
Beatriz Fernández-Duque ◽  
Isidro A. Pérez ◽  
M. Luisa Sánchez ◽  
M. Ángeles García ◽  
Nuria Pardo
Keyword(s):  
Rural Area ◽  
Temporal Patterns ◽  
Northern Spain ◽  
Harmonic Equation

The effect of a semi-contaminated free surface on mass transport

1974 ◽  
Vol 75 (2) ◽  
pp. 283-294 ◽  
Author(s):  
D. Porter ◽  
B. D. Dore
Keyword(s):  
Free Surface ◽  
Velocity Field ◽  
Mass Transport ◽  
Mixed Boundary ◽  
Surface Film ◽  
Vertical Displacement ◽  
Mass Transport Velocity ◽  
Harmonic Equation ◽  
The Waves

AbstractThe mass transport velocity field is determined for surface waves which propagate from a region with a clean free surface into a region beneath an inextensible surface film. The waves are assumed to be incident normally on the edge of the film. Determination of this velocity field requires the investigation of a mixed boundary value problem for the bi-harmonic equation, the solution of which is obtained using the Wiener–Hopf technique. Streamlines for the mean motion of the fluid particles are thus obtained. It is found that considerable vertical displacement of fluid is possible due to the presence of the surface film.

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