# American Institute of Mathematical Sciences

• Previous Article
Center manifolds for nonuniform trichotomies and arbitrary growth rates
• CPAA Home
• This Issue
• Next Article
Eigenmodes and eigenfrequencies of vibrating elliptic membranes: a Klein oscillation theorem and numerical calculations
May  2010, 9(3): 625-642. doi: 10.3934/cpaa.2010.9.625

## Flows of weakly compressible viscoelastic fluids through a regular bounded domain with inflow-outflow boundary conditions

 1 Université Paris-Est, Laboratoire d'Analyse et de Mathématiques Appliquées, UMR CNRS 8050, 61 avenue du Général de Gaulle, 94010 Créteil Cedex, France

Received  May 2009 Revised  September 2009 Published  January 2010

We study steady isothermal motions of a nonlinear weakly compressible viscoelastic fluids of Oldroyd type in a bounded domain $\Omega\subset\mathbb{R}^2$, with given non-zero velocities on the boundary of $\Omega$. We suppose that the pressure $p$ and the extra-stress tensor $\tau$ are prescribed on the part of the boundary corresponding to entering velocities. A uniqueness and existence result for the solution $(\mathbf u,p,\tau)$ is established in $W^{2,q}(\Omega)\times W^{1,q}(\Omega)\times W^{1,q}(\Omega)$ with $2 < q < 3$. The proof follows from an analysis of a linearized problem. The fixed point theorem is used to establish the existence of a solution. The solutions of two transport equations for $p$ and $\tau$ are obtained by integration along the streamlines.
Citation: Zaynab Salloum. Flows of weakly compressible viscoelastic fluids through a regular bounded domain with inflow-outflow boundary conditions. Communications on Pure & Applied Analysis, 2010, 9 (3) : 625-642. doi: 10.3934/cpaa.2010.9.625
 [1] Colette Guillopé, Zaynab Salloum, Raafat Talhouk. Regular flows of weakly compressible viscoelastic fluids and the incompressible limit. Discrete & Continuous Dynamical Systems - B, 2010, 14 (3) : 1001-1028. doi: 10.3934/dcdsb.2010.14.1001 [2] Yong Ji, Ercai Chen, Yunping Wang, Cao Zhao. Bowen entropy for fixed-point free flows. Discrete & Continuous Dynamical Systems - A, 2019, 39 (11) : 6231-6239. doi: 10.3934/dcds.2019271 [3] Dou Dou, Meng Fan, Hua Qiu. Topological entropy on subsets for fixed-point free flows. Discrete & Continuous Dynamical Systems - A, 2017, 37 (12) : 6319-6331. doi: 10.3934/dcds.2017273 [4] W. Wei, Yin Li, Zheng-An Yao. Decay of the compressible viscoelastic flows. Communications on Pure & Applied Analysis, 2016, 15 (5) : 1603-1624. doi: 10.3934/cpaa.2016004 [5] Colette Guillopé, Abdelilah Hakim, Raafat Talhouk. Existence of steady flows of slightly compressible viscoelastic fluids of White-Metzner type around an obstacle. Communications on Pure & Applied Analysis, 2005, 4 (1) : 23-43. doi: 10.3934/cpaa.2005.4.23 [6] Van-Sang Ngo, Stefano Scrobogna. Dispersive effects of weakly compressible and fast rotating inviscid fluids. Discrete & Continuous Dynamical Systems - A, 2018, 38 (2) : 749-789. doi: 10.3934/dcds.2018033 [7] Adeolu Taiwo, Lateef Olakunle Jolaoso, Oluwatosin Temitope Mewomo. Viscosity approximation method for solving the multiple-set split equality common fixed-point problems for quasi-pseudocontractive mappings in Hilbert spaces. Journal of Industrial & Management Optimization, 2017, 13 (5) : 0-0. doi: 10.3934/jimo.2020092 [8] Allen Montz, Hamid Bellout, Frederick Bloom. Existence and uniqueness of steady flows of nonlinear bipolar viscous fluids in a cylinder. Discrete & Continuous Dynamical Systems - B, 2015, 20 (7) : 2107-2128. doi: 10.3934/dcdsb.2015.20.2107 [9] Xianpeng Hu, Dehua Wang. The initial-boundary value problem for the compressible viscoelastic flows. Discrete & Continuous Dynamical Systems - A, 2015, 35 (3) : 917-934. doi: 10.3934/dcds.2015.35.917 [10] Gianluca Crippa, Milton C. Lopes Filho, Evelyne Miot, Helena J. Nussenzveig Lopes. Flows of vector fields with point singularities and the vortex-wave system. Discrete & Continuous Dynamical Systems - A, 2016, 36 (5) : 2405-2417. doi: 10.3934/dcds.2016.36.2405 [11] Cristian A. Coclici, Jörg Heiermann, Gh. Moroşanu, W. L. Wendland. Asymptotic analysis of a two--dimensional coupled problem for compressible viscous flows. Discrete & Continuous Dynamical Systems - A, 2004, 10 (1&2) : 137-163. doi: 10.3934/dcds.2004.10.137 [12] Zefu Feng, Changjiang Zhu. Global classical large solution to compressible viscous micropolar and heat-conducting fluids with vacuum. Discrete & Continuous Dynamical Systems - A, 2019, 39 (6) : 3069-3097. doi: 10.3934/dcds.2019127 [13] Yinxia Wang, Hengjun Zhao. Global existence and decay estimate of classical solutions to the compressible viscoelastic flows with self-gravitating. Communications on Pure & Applied Analysis, 2018, 17 (2) : 347-374. doi: 10.3934/cpaa.2018020 [14] Xinghong Pan, Jiang Xu. Global existence and optimal decay estimates of the compressible viscoelastic flows in $L^p$ critical spaces. Discrete & Continuous Dynamical Systems - A, 2019, 39 (4) : 2021-2057. doi: 10.3934/dcds.2019085 [15] Daoyuan Fang, Ting Zhang, Ruizhao Zi. Dispersive effects of the incompressible viscoelastic fluids. Discrete & Continuous Dynamical Systems - A, 2018, 38 (10) : 5261-5295. doi: 10.3934/dcds.2018233 [16] Paolo Secchi. An alpha model for compressible fluids. Discrete & Continuous Dynamical Systems - S, 2010, 3 (2) : 351-359. doi: 10.3934/dcdss.2010.3.351 [17] Xin Liu. Compressible viscous flows in a symmetric domain with complete slip boundary: The nonlinear stability of uniformly rotating states with small angular velocities. Communications on Pure & Applied Analysis, 2019, 18 (2) : 751-794. doi: 10.3934/cpaa.2019037 [18] Nicholas Long. Fixed point shifts of inert involutions. Discrete & Continuous Dynamical Systems - A, 2009, 25 (4) : 1297-1317. doi: 10.3934/dcds.2009.25.1297 [19] Eduard Feireisl. Mathematical theory of viscous fluids: Retrospective and future perspectives. Discrete & Continuous Dynamical Systems - A, 2010, 27 (2) : 533-555. doi: 10.3934/dcds.2010.27.533 [20] Marcelo M. Disconzi. On the existence of solutions and causality for relativistic viscous conformal fluids. Communications on Pure & Applied Analysis, 2019, 18 (4) : 1567-1599. doi: 10.3934/cpaa.2019075

2018 Impact Factor: 0.925