Contraction for large perturbations of traveling waves in a hyperbolic–parabolic system arising from a chemotaxis model

2020 ◽  
Vol 30 (02) ◽  
pp. 387-437 ◽  
Author(s):  
Kyudong Choi ◽  
Moon-Jin Kang ◽  
Young-Sam Kwon ◽  
Alexis F. Vasseur
Keyword(s):  
Shock Wave ◽  
Parabolic System ◽  
Traveling Waves ◽  
Tumor Angiogenesis ◽  
Initial Perturbation ◽  
Shock Strength ◽  
Chemotaxis Model ◽  
Contraction Property ◽  
Singular Sensitivity ◽  
Large Initial Perturbation

We consider a hyperbolic–parabolic system arising from a chemotaxis model in tumor angiogenesis, which is described by a Keller–Segel equation with singular sensitivity. It is known to allow viscous shocks (so-called traveling waves). We introduce a relative entropy of the system, which can capture how close a solution at a given time is to a given shock wave in almost [Formula: see text]-sense. When the shock strength is small enough, we show the functional is non-increasing in time for any large initial perturbation. The contraction property holds independently of the strength of the diffusion.

scholarly journals Global Existence and Convergence Rates for the Strong Solutions inH2to the 3D Chemotaxis Model

2013 ◽  
Vol 2013 ◽  
pp. 1-9 ◽  
Author(s):  
Weijun Xie ◽  
Yinghui Zhang ◽  
Yuandong Xiao ◽  
Wei Wei
Keyword(s):  
Spectral Analysis ◽  
Global Existence ◽  
Parabolic System ◽  
Convergence Rates ◽  
Initial Perturbation ◽  
Strong Solutions ◽  
Energy Methods ◽  
Chemotaxis Model ◽  
Constant State ◽  
Optimal Convergence Rates

We are concerned with a 3D chemotaxis model arising from biology, which is a coupled hyperbolic-parabolic system. We prove the global existence of a strong solution whenH2-norm of the initial perturbation around a constant state is sufficiently small. Moreover, if additionally,L1-norm of the initial perturbation is bounded; the optimal convergence rates are also obtained for such a solution. The proofs are obtained by combining spectral analysis with energy methods.

Mechanics of the influx phase in the jet regurgitant mode of a powered resonance tube

2019 ◽  
Vol 18 (2-3) ◽  
pp. 279-298 ◽  
Author(s):  
Bhavraj Thethy ◽  
David Tairych ◽  
Daniel Edgington-Mitchell
Keyword(s):  
Shock Wave ◽  
High Speed ◽  
Fluid Velocity ◽  
Velocity Variation ◽  
Shock Strength ◽  
Velocity Change ◽  
Resonator Length ◽  
Time Resolved ◽  
Reflected Waves ◽  
Shock Position

Time-resolved visualisation of shock wave motion within a powered resonant tube (PRT) is presented for the regurgitant mode of operation. Shock position and velocity are measured as functions of both time and space from ultra-high-speed schlieren visualisations. The shock wave velocity is seen to vary across the resonator length for both the incident and reflected waves. Three mechanisms are explored as explanations for the variation in velocity: change in local fluid velocity, variation in shock strength and variations in local temperature. For the incident wave, local fluid velocity and shock strength are extracted from the data and both are demonstrated to contribute to the observed variation, with a non-trivial remainder likely explained by variation in temperature.

Shockwave and Noise Abatement of Transonic Fans

2004 ◽  
Author(s):  
Liping Xu
Keyword(s):  
Shock Wave ◽  
Noise Reduction ◽  
Three Dimensional ◽  
Leading Edge ◽  
Aerodynamic Performance ◽  
Test Results ◽  
Shock Strength ◽  
Noise Levels ◽  
Noise Abatement ◽  
Blade Passage

The aerodynamic sources of the forward tone noise of transonic civil fans are analysed. The leading edge shockwave near the rotor tip section is identified as the main source of tone noise. By comparing the tone noise levels of the same fan operating at two different working lines, numerical calculations show that on the lower working line, the main passage shockwave is swallowed and locked into the blade passage, and the fan blades act as a shield to prevent the strong passage shock wave from propagating upstream. The calculations show that, by running the fan at a lower working line, up to 6 db abatement in the blade passing frequency (BPF) tone can be achieved through shielding the shockwave. With three dimensional CFD it is possible to design swept rotors which have desired shockwave structures near the tip region. Fan rotors with different swept leading edges have been designed to study this effect and comparisons in aerodynamics performances as well as the tone noise levels are made. It is predicted that in a swept rotor the leading edge shock strength can be further weakened and up to 5db further reduction in tone noise is possible. With a more secure shockwave shielding, a forward swept rotor has the combination of better aerodynamic performance and better noise abatement feature. The design and test results of a three dimensional fan rotor LNR2, featuring localised forward swept rotor are presented. Rig test results show that although the noise reduction through shock shielding has been demonstrated, the aerodynamics and noise are complicated by the problems specific to such localised forward swept fan.

scholarly journals Nonlinear Instability for a Volume-Filling Chemotaxis Model with Logistic Growth

2014 ◽  
Vol 2014 ◽  
pp. 1-11 ◽  
Author(s):  
Haiyan Gao ◽  
Shengmao Fu
Keyword(s):  
Nonlinear Evolution ◽  
Initial Perturbation ◽  
Neumann Boundary ◽  
Logistic Growth ◽  
Nonlinear Instability ◽  
Chemotaxis Model ◽  
Linear Dynamics ◽  
Volume Filling ◽  
Neumann Boundary Value Problem ◽  
Time Period

This paper deals with a Neumann boundary value problem for a volume-filling chemotaxis model with logistic growth in ad-dimensional boxTd=(0,π)d  (d=1,2,3). It is proved that given any general perturbation of magnitudeδ, its nonlinear evolution is dominated by the corresponding linear dynamics along a finite number of fixed fastest growing modes, over a time period of the orderln⁡(1/δ). Each initial perturbation certainly can behave drastically different from another, which gives rise to the richness of patterns.

Simulation of Shock Wave Assisted Free and Shape Forming of Metallic Plates in a Shock Tube

2015 ◽  
Vol 813-814 ◽  
pp. 586-591 ◽  
Author(s):  
Kottakota Kalasagarreddi ◽  
Prem Sai Koppuravuri Sobhan ◽  
Vinay Kumar Gundu ◽  
S.R. Nagaraja
Keyword(s):  
Shock Wave ◽  
Shock Tube ◽  
Metal Forming ◽  
Von Mises Stress ◽  
Stress And Strain ◽  
Shock Strength ◽  
Engineering Problems ◽  
Experimental Values ◽  
Von Mises ◽  
Metallic Plates

Due to their complexity, certain engineering problems like finding shock strength, Mach number etc. and the interaction of shock wave with a structure in free and restricted metal forming techniques cannot be achieved in a single experimentation, these can be obtained only through a number of trials and that leads to increase in cost and time. In such cases both cost and time can be reduced by adopting numerical simulations. In this projectcommercial software ANSYS is used to simulate the propagation shock wave through a shock tube, free and shape forming of metallic plates subjected to this shock wave. Shock Mach numbers up to 2.12 have been generated by varying the driver to driven pressure ratios. Thin copper plates of diameter 60mm and thickness of 0.5mm and 0.3mm are subjected to shock wave loadingin order to form into dies.These dies,madeof structural steel are modelled with pre-defined shapes. The plate peakoverpressures ranging from 9 to 20bar have been generated.The midpoint deflection, Von Mises stress and strain are calculated for free forming copper plates. The simulated results are compared with the experimental values available in literature. The simulated results match well with the experimental values.

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