Optimal time-decay estimates for an Oldroyd-B model with zero viscosity

Barrett, 2011, Existence and approximation of a (regularized) Oldroyd-B model, Math. Models Methods Appl. Sci., 21, 1783, 10.1142/S0218202511005581 Barrett, 2017, Existence of large-data finite-energy global weak solutions to a compressible Oldroyd-B model, Commun. Math. Sci., 15, 1265, 10.4310/CMS.2017.v15.n5.a5 Barrett, 2011, Existence and equilibration of global weak solutions to kinetic models for dilute polymers I: finitely extensible nonlinear bead-spring chains, Math. Models Methods Appl. Sci., 21, 1211, 10.1142/S0218202511005313 Barrett, 2018, Existence of global weak solutions to the kinetic Hookean dumbbell model for incompressible dilute polymeric fluids, Nonlinear Anal., Real World Appl., 39, 362, 10.1016/j.nonrwa.2017.07.012 Bathory, 2021, Large data existence theory for three-dimensional unsteady flows of rate-type viscoelastic fluids with stress diffusion, Adv. Nonlinear Anal., 10, 501, 10.1515/anona-2020-0144 Bhave, 1991, Kinetic theory and rheology of dilute, nonhomogeneous polymer solutions, J. Chem. Phys., 95, 2988, 10.1063/1.460900 Bhave, 1993, A constitutive equation for liquid-crystalline polymer solutions, J. Rheol., 37, 413, 10.1122/1.550452 Boyaval, 2009, Free-energy-dissipative schemes for the Oldroyd-B model, M2AN Math. Model. Numer. Anal., 43, 523, 10.1051/m2an/2009008 Cai, 2019, Vanishing viscosity limit for incompressible viscoelasticity in two dimensions, Commun. Pure Appl. Math., 72, 2063, 10.1002/cpa.21853 Cates, 2006, Rheology of giant micelles, Adv. Phys., 55, 799, 10.1080/00018730601082029 Chen, 2008, Global well-posedness of viscoelastic fluids of Oldroyd type in Besov spaces, Nonlinear Anal., 68, 1928, 10.1016/j.na.2007.01.042 Chupin, 2017, Viscoelastic flows in a rough channel: amultiscale analysis, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, 34, 483, 10.1016/j.anihpc.2016.01.002 Constantin, 2012, Note on global regularity for two dimensional Oldroyd-B fluids stress, Arch. Ration. Mech. Anal., 206, 725, 10.1007/s00205-012-0537-0 Constantin Dhont, 2008, Gradient and vorticity banding, Rheol. Acta, 47, 257, 10.1007/s00397-007-0245-0 Dostalík, 2020, Unconditional finite amplitude stability of a viscoelastic fluid in a mechanically isolated vessel with spatially non-uniform wall temperature, Math. Comput. Simul. E, 2004, Well-posedness for the dumbbell model of polymeric fluids, Commun. Math. Phys., 248, 409, 10.1007/s00220-004-1102-y Elgindi, 2015, Global wellposedness to the generalized Oldroyd type models in R3, J. Differ. Equ., 259, 1958, 10.1016/j.jde.2015.03.026 Elgindi, 2015, Global regularity for some Oldroyd-B type models, Commun. Pure Appl. Math., 68, 2005, 10.1002/cpa.21563 El-Kareh, 1989, Existence of solutions for all Deborah numbers for a non-Newtonian model modified to include diffusion, J. Non-Newton. Fluid Mech., 33, 257, 10.1016/0377-0257(89)80002-3 Fang, 2013, Global existence results for Oldroyd-B fluids in exterior domains: the case of non-small coupling parameters, Math. Ann., 357, 687, 10.1007/s00208-013-0914-5 Fang, 2016, Global solutions to the Oldroyd-B model with a class of large initial data, SIAM J. Math. Anal., 48, 1054, 10.1137/15M1037020 Fernández-Cara, 1998, Some theoretical results concerning non-Newtonian fluids of the Oldroyd kind, Ann. Sc. Norm. Super. Pisa, 26, 1 Guillopé, 1990, Existence results for the flow of viscoelastic fluids with a differential constitutive law, Nonlinear Anal., Theory Methods Appl., 15, 849, 10.1016/0362-546X(90)90097-Z Hall, 2015, Lie Groups, Lie Algebras, and Representations, An Elementary Introduction Hieber, 2012, Global existence results for Oldroyd-B fluids in exterior domains, J. Differ. Equ., 252, 2617, 10.1016/j.jde.2011.09.001 Hieber, 2019, Optimal decay rates for solutions to the incompressible Oldryod-B model in R3, Nonlinearity, 32, 833, 10.1088/1361-6544/aaeec7 Hu, 2007, New entropy estimates for Oldroyd-B and related models, Commun. Math. Sci., 5, 909, 10.4310/CMS.2007.v5.n4.a9 Hu, 2016, Global solutions of two-dimensional incompressible viscoelastic flows with discontinuous initial data, Commun. Pure Appl. Math., LXIX Hu, 2015, Long-time behavior and weak-strong uniqueness for incompressible viscoelastic flows, Discrete Contin. Dyn. Syst., 35, 3437, 10.3934/dcds.2015.35.3437 Hu, 2013, Global existence and optimal decay rates for three-dimensional compressible viscoelastic flows, SIAM J. Math. Anal., 45, 2815, 10.1137/120892350 Kawashima, 1983 La, 2020, On diffusive 2D Fokker-Planck-Navier-Stokes systems, Arch. Ration. Mech. Anal., 235, 1531, 10.1007/s00205-019-01450-0 Lai, 2017, Forward self-similar solutions to the viscoelastic Navier-Stokes equation with damping, SIAM J. Math. Anal., 49, 501, 10.1137/16M1060340 Lei, 2010, On 2D viscoelasticity with small strain, Arch. Ration. Mech. Anal., 198, 13, 10.1007/s00205-010-0346-2 Lei, 2008, Global solutions for incompressible viscoelastic fluids, Arch. Ration. Mech. Anal., 188, 371, 10.1007/s00205-007-0089-x Lin, 2012, Some analytical issues for elastic complex fluids, Commun. Pure Appl. Math., 65, 893, 10.1002/cpa.21402 Lin, 2005, On hydrodynamics of viscoelastic fluids, Commun. Pure Appl. Math., 58, 1437, 10.1002/cpa.20074 Lin, 2008, On the initial-boundary value problem of the incompressible viscoelastic fluid system, Commun. Pure Appl. Math., 61, 539, 10.1002/cpa.20219 Lions, 2000, Global solutions for some Oldroyd models of non-Newtonian flows, Chin. Ann. Math., Ser. B, 21, 131, 10.1142/S0252959900000170 Liu, 1993, Free energy functionals for semiflexible polymer solutions and blends, Macromolecules, 26, 2817, 10.1021/ma00063a028 Lu, 2018, Relative entropy, weak-strong uniqueness and conditional regularity for a compressible Oldroyd-B model, SIAM J. Math. Anal., 50, 557, 10.1137/17M1128654 Málek, 2018, Thermodynamics of viscoelastic rate-type fluids with stress diffusion, Phys. Fluids, 30, 10.1063/1.5018172 Molinet, 2004, On the global and periodic regular flows of viscoelastic fluids with a differential constitutive law, Nonlinear Differ. Equ. Appl., 11, 349, 10.1007/s00030-004-1073-x Oldroyd, 1958, Non-Newtonian effects in steady motion of some idealized elasticoviscous liquids, Proc. R. Soc. Edinb., Sect. A, 245, 278 Rajagopal, 2000, A thermodynamic frame work for rate type fluid models, J. Non-Newton. Fluid Mech., 88, 207, 10.1016/S0377-0257(99)00023-3 Schonbek, 1985, L2 decay for weak solutions of the Navier-Stokes equations, Arch. Ration. Mech. Anal., 88, 209, 10.1007/BF00752111 Schonbek, 1986, Large time behavior of solutions to the Navier-Stokes equations, Commun. Partial Differ. Equ., 11, 753, 10.1080/03605308608820443 Wang, 2020, The Cauchy problem for an Oldroyd-B model in three dimensions, Math. Models Methods Appl. Sci., 30, 139, 10.1142/S0218202520500049 Zhang, 2012, Global existence of strong solution for equations related to the incompressible viscoelastic fluids in the critical Lp framework, SIAM J. Math. Anal., 44, 2266, 10.1137/110851742 Zhu, 2018, Global small solutions of 3D incompressible Oldroyd-B model without damping mechanism, J. Funct. Anal., 274, 2039, 10.1016/j.jfa.2017.09.002 Zi, 2014, Global solution to the incompressible Oldroyd-B model in the critical Lp framework: the case of the non-small coupling parameter, Arch. Ration. Mech. Anal., 213, 651, 10.1007/s00205-014-0732-2 Ziegler, 1987, The derivation of constitutive relations from the free energy and the dissipation function, Adv. Appl. Mech., 25, 183, 10.1016/S0065-2156(08)70278-3