The Cauchy problem for an Oldroyd-B model in three dimensions

2019 ◽  
Vol 30 (01) ◽  
pp. 139-179
Author(s):  
Wenjun Wang ◽  
Huanyao Wen
Keyword(s):  
Cauchy Problem ◽  
Strong Solutions ◽  
Large Data ◽  
Finite Energy ◽  
Momentum Equation ◽  
Three Dimensions ◽  
Navier Stokes ◽  
Frequency Decomposition ◽  
Extra Stress Tensor ◽  
The Cauchy Problem

We consider an Oldroyd-B model which is derived in Ref. 4 [J. W. Barrett, Y. Lu and E. Süli, Existence of large-data finite-energy global weak solutions to a compressible Oldroyd-B model, Commun. Math. Sci. 15 (2017) 1265–1323] via micro–macro-analysis of the compressible Navier–Stokes–Fokker–Planck system. The global well posedness of strong solutions as well as the associated time-decay estimates in Sobolev spaces for the Cauchy problem are established near an equilibrium state. The terms related to [Formula: see text], in the equation for the extra stress tensor and in the momentum equation, lead to new technical difficulties, such as deducing [Formula: see text]-norm dissipative estimates for the polymer number density and its spatial derivatives. One of the main objectives of this paper is to develop a way to capture these dissipative estimates via a low–medium–high-frequency decomposition.

scholarly journals Global existence of strong solutions for incompressible hydrodynamic flow of liquid crystals with vacuum

Filomat ◽  
2013 ◽  
Vol 27 (7) ◽  
pp. 1247-1257 ◽  
Author(s):  
Shijin Ding ◽  
Jinrui Huang ◽  
Fengguang Xia
Keyword(s):  
Cauchy Problem ◽  
Liquid Crystals ◽  
Global Existence ◽  
Existence And Uniqueness ◽  
Strong Solutions ◽  
Hydrodynamic Flow ◽  
Three Dimensions ◽  
Small Initial Data ◽  
The Cauchy Problem ◽  
Global Existence And Uniqueness

We consider the Cauchy problem for incompressible hydrodynamic flow of nematic liquid crystals in three dimensions. We prove the global existence and uniqueness of the strong solutions with nonnegative p0 and small initial data.

scholarly journals The Cauchy problem of a two-phase flow model for a mixture of non-interacting compressible fluids

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Zhen Cheng ◽  
Wenjun Wang
Keyword(s):  
Cauchy Problem ◽  
Global Existence ◽  
Strong Solutions ◽  
Compressible Fluids ◽  
Existence Theory ◽  
Optimal Time ◽  
Two Phase ◽  
Frequency Decomposition ◽  
The Cauchy Problem ◽  
Global Strong Solutions

<p style='text-indent:20px;'>In this paper, we consider the global existence of the Cauchy problem for a version of one velocity Baer-Nunziato model with dissipation for the mixture of two compressible fluids in <inline-formula><tex-math id="M1">\begin{document}$ \mathbb{R}^3 $\end{document}</tex-math></inline-formula>. We get the existence theory of global strong solutions by using the decaying properties of the solutions. The energy method combined with the low-high-frequency decomposition is used to derive such properties and hence the global existence. As a byproduct, the optimal time decay estimates of all-order spatial derivatives of the pressure and the velocity are obtained.</p>

scholarly journals On the Blow-Up of Solutions of a Weakly Dissipative Modified Two-Component Periodic Camassa-Holm System

2012 ◽  
Vol 2012 ◽  
pp. 1-20 ◽  
Author(s):  
Yongsheng Mi ◽  
Chunlai Mu ◽  
Weian Tao
Keyword(s):  
Cauchy Problem ◽  
Blow Up ◽  
Strong Solutions ◽  
Well Posedness ◽  
Holm Equation ◽  
Two Component ◽  
The Cauchy Problem ◽  
Blow Up Rate

We study the Cauchy problem of a weakly dissipative modified two-component periodic Camassa-Holm equation. We first establish the local well-posedness result. Then we derive the precise blow-up scenario and the blow-up rate for strong solutions to the system. Finally, we present two blow-up results for strong solutions to the system.

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