GLOBAL SOLUTIONS FOR SOME OLDROYD MODELS OF NON-NEWTONIAN FLOWS

By Pierre-Louis Lions, Nader Masmoudi. 2000

376 citations recorded by the source corpus.

The authors consider here some Oldroyd models of non-Newtonian flows consisting of a strong coupling between incompressible Navier-Stokes equations and some transport equations. It is proved that there exist global weak solutions for general initial conditions. The existence proof relies upon showing the propagation in time of the compactness of solutions.

Subfields: Applied Mathematics, Computational Mechanics, Fluid Flow and Transfer Processes

View Paper: https://www.semanticscholar.org/paper/341f53e06f83fbd800030bf53742daa7b5415e7a

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Applied Mathematics

1. Nader Masmoudi
2. Peter Constantin
3. Frédéric Rousset
4. Ping Zhang
5. Tarek M. Elgindi
6. Zhifei Zhang
7. Markus Kliegl
8. Pierre-Louis Lions

In Computational Mechanics

1. Nader Masmoudi
2. Peter Constantin
3. Frédéric Rousset
4. Ping Zhang
5. Zhifei Zhang
6. Markus Kliegl
7. Xianpeng Hu
8. Tarek M. Elgindi

In Fluid Flow and Transfer Processes

1. Nader Masmoudi
2. Ping Zhang
3. Pierre-Louis Lions
4. Tarek M. Elgindi
5. Peter Constantin
6. Markus Kliegl

Citations in the corpus

• Global existence of weak solutions to some micro-macro models (2007, positive): “If R is in the whole space, the model reduces to the Hooke model for which φ(R) = k|R| and the model reduces to the Oldroyd-B model (see [3] and [10] for some local and global existence results).”
• About Lifespan of Regular Solutions of Equations Related to Viscoelastic Fluids (2001, background): “[17] P.-L. Lions and N. Masmoudi, Global solutions for some Oldroyd models of non-Newtonian flows, Chinese Ann.”
• Global Well-Posedness for Compressible Viscoelastic Fluids near Equilibrium (2010, background): “When the contribution of the strain rate(symmetric part of ∇v) in the constitutive equation is neglected, Lions and Masmoudi [ 20 ] proved the global existence of a weak solution with general initial data.”
• Note on Global Regularity for Two-Dimensional Oldroyd-B Fluids with Diffusive Stress (2012, background): “Global existence of weak solutions via propagation of compactness was proved under the corotational assumption ([24], [25]) and, very recently, for the full FENE model ([28]).”
• Global well posedness for the Maxwell-Navier-Stokes system in 2D (2010, background): “Moreover, ideas based on the propagation of compactness used in [24] and [25] seem not to work here.”
• Global Regularity for Some Oldroyd‐B Type Models (2013, background): “In the most interesting case where ν > 0 and μ = 0 (which is the classical case), global existence of weak solutions has been shown in the corotational case (with b= 0) only [22].”
• Well‐posedness for the FENE dumbbell model of polymeric flows (2008, methodology): “[6] J.-Y. Chemin and N. Masmoudi.”
• The Global Existence of Small Solutions to the Incompressible Viscoelastic Fluid System in 2 and 3 Space Dimensions (2006, background): “Whereas when the contribution of the strain rate (symmetric part of v in the constitutive equation is neglected, Lions and Masmoudi (2000) proved its global existence of weak solutions with general initial data.”
• Global Existence for the Multi-Dimensional Compressible Viscoelastic flows (2010, background): “On the other hand, the global existence of weak solutions to the incompressible viscoelastic flows with large initial data is still an outstanding open question, although there are some progress in that direction ([16, 18, 19]).”