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Stress-diffusive regularizations of non-dissipative rate-type materials

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  • We consider non-dissipative (elastic) rate-type material models that are derived within the Gibbs-potential-based thermodynamic framework. Since the absence of any dissipative mechanism in the model prevents us from establishing even a local-in-time existence result in two spatial dimensions for a spatially periodic problem, we propose two regularisations. For such regularized problems we obtain well-posedness of the planar, spatially periodic problem. In contrast with existing results, we prove ours for a regularizing term present solely in the evolution equation for the stress.

    Mathematics Subject Classification: Primary: 35Q35, 74D10; Secondary: 35A01.


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  •   R. Adams and J. Fournier, Sobolev Spaces, volume 140 of Pure and Applied Mathematics (Amsterdam), Elsevier/Academic Press, Amsterdam, second edition, 2003.
      J. W. Barrett, Y. Lu and E. Süli, Existence of large-data finite-energy global weak solutions to a compressible Oldroyd-B model, Communications in Mathematical Sciences (Accepted, 22 January 2017), Available from: arXiv: 1608.04229, 2016.
      J. W. Barrett  and  S. Boyaval , Existence and approximation of a (regularized) Oldroyd-B model, Mathematical Models and Methods in Applied Sciences, 21 (2011) , 1783-1837.  doi: 10.1142/S0218202511005581.
      J. W. Barrett  and  E. Süli , Existence and equilibration of global weak solutions to kinetic models for dilute polymers Ⅰ: Finitely extensible nonlinear bead-spring chains, Mathematical Models and Methods in Applied Sciences, 21 (2011) , 1211-1289.  doi: 10.1142/S0218202511005313.
      O. Bejaoui  and  M. Majdoub , Global weak solutions for some Oldroyd models, Journal of Differential Equations, 254 (2013) , 660-685.  doi: 10.1016/j.jde.2012.09.010.
      H. Brezis  and  T. Gallouet , Nonlinear Schrödinger evolution equations, Nonlinear Analysis: Theory, Methods & Applications, 4 (1980) , 677-681.  doi: 10.1016/0362-546X(80)90068-1.
      H. Brezis  and  S. Wainger , A note on limiting cases of Sobolev embeddings and convolution inequalities, Communications in Partial Differential Equations, 5 (1980) , 773-789.  doi: 10.1080/03605308008820154.
      J. Chemin  and  N. Masmoudi , About lifespan of regular solutions of equations related to viscoelastic fluids, SIAM Journal on Mathematical Analysis, 33 (2001) , 84-112.  doi: 10.1137/S0036141099359317.
      E. Chiodaroli , E. Feireisl  and  O. Kreml , On the weak solutions to the equations of a compressible heat conducting gas, Ann. Inst. H. Poincaré Anal. Non Linéaire, 32 (2015) , 225-243.  doi: 10.1016/j.anihpc.2013.11.005.
      L. Chupin  and  S. Martin , Stationary Oldroyd model with diffusive stress: Mathematical analysis of the model and vanishing diffusion process, Journal of Non-Newtonian Fluid Mechanics, 218 (2015) , 27-39.  doi: 10.1016/j.jnnfm.2015.01.004.
      P. Constantin  and  M. Kliegl , Note on global regularity for two-dimensional Oldroyd-B fluids with diffusive stress, Archive for Rational Mechanics and Analysis, 206 (2012) , 725-740.  doi: 10.1007/s00205-012-0537-0.
      De Lellis , Székelyhidi  and  Jr. , On admissibility criteria for weak solutions of the Euler equations, Arch. Ration. Mech. Anal., 195 (2010) , 225-260.  doi: 10.1007/s00205-008-0201-x.
      De Lellis , Székelyhidi  and  Jr. , Dissipative continuous Euler flows, Invent. Math., 193 (2013) , 377-407.  doi: 10.1007/s00222-012-0429-9.
      T. M. Elgindi  and  J. Liu , Global wellposedness to the generalized oldroyd type models in R3, Journal of Differential Equations, 259 (2015) , 1958-1966.  doi: 10.1016/j.jde.2015.03.026.
      T. M. Elgindi  and  F. Rousset , Global regularity for some oldroyd-b type models, Communications on Pure and Applied Mathematics, 68 (2015) , 2005-2021.  doi: 10.1002/cpa.21563.
      D. Fang  and  R. Zi , Strong solutions of 3d compressible Oldroyd-B fluids, Mathematical Methods in the Applied Sciences, 36 (2013) , 1423-1439.  doi: 10.1002/mma.2695.
      E. Fernández-Cara, F. Guillén and R. R. Ortega, Mathematical modeling and analysis of viscoelastic fluids of the Oldroyd kind, in Handbook of Numerical Analysis, Handbook of Numerical Analysis, 8, Elsevier, 2002,543-660. 2pt
      C. Foias, O. Manley, R. Rosa and R. Temam, Navier-Stokes Equations and Turbulence, Cambridge University Press, 2001. doi: 10.1017/CBO9780511546754.
      C. Guillopé  and  J. C. Saut , Existence results for the flow of viscoelastic fluids with a differential constitutive law, Nonlinear Anal., 15 (1990) , 849-869.  doi: 10.1016/0362-546X(90)90097-Z.
      N. Gunther , On the motion of fluid in a moving container. izvestia akademia nauk ussr, Seriya Fizicheskaya-Mathematica, 20 (1927) , 1323-1348,1503-1532. 
      T. Kato , On classical solutions of the two-dimensional non-stationary Euler equation, Archive for Rational Mechanics and Analysis, 25 (1967) , 188-200.  doi: 10.1007/BF00251588.
      H. Kozono , T. Ogawa  and  Y. Taniuchi , The critical Sobolev inequalities in Besov spaces and regularity criterion to some semi-linear evolution equations, Mathematische Zeitschrift, 242 (2002) , 251-278.  doi: 10.1007/s002090100332.
      H. Kozono  and  Y. Taniuchi , Limiting case of the Sobolev inequality in BMO, with application to the Euler equations, Communications in Mathematical Physics, 214 (2000) , 191-200.  doi: 10.1007/s002200000267.
      J. Kratochvíl , J. Málek  and  P. Minakowski , A Gibbs-potential-based framework for ideal plasticity of crystalline solids treated as a material flow through an adjustable crystal lattice space and its application to three-dimensional micropillar compression, International Journal of Plasticity, 87 (2016) , 114-129.  doi: 10.1016/j.ijplas.2016.09.006.
      Z. Lei , Global existence of classical solutions for some Oldroyd-B model via the incompressible limit, Chinese Annals of Mathematics, Series B, 27 (2006) , 565-580.  doi: 10.1007/s11401-005-0041-z.
      Z. Lei , Ch. Liu  and  Y. Zhou , Global solutions for incompressible viscoelastic fluids, Archive for Rational Mechanics and Analysis, 188 (2008) , 371-398.  doi: 10.1007/s00205-007-0089-x.
      L. Lichtenstein , Über einige Existenzprobleme der Hydrodynamik homogener, unzusammendrickbarer, reibungsloser Flüssigkeiten und die Helmholtzschen Wirbelsatze, Mathematische Zeitschrift, 23 (1925) , 89-154.  doi: 10.1007/BF01506223.
      F.-H. Lin , Ch. Liu  and  P. Zhang , On hydrodynamics of viscoelastic fluids, Communications on Pure and Applied Mathematics, 58 (2005) , 1437-1471.  doi: 10.1002/cpa.20074.
      P. L. Lions  and  N. Masmoudi , Global solutions for some Oldroyd models of non-Newtonian flows, Chinese Annals of Mathematics. Series B, 21 (2000) , 131-146.  doi: 10.1142/S0252959900000170.
      M. Lukáčová-Medvid'ová , H. Mizerová  and  Š. Nečasová , Global existence and uniqueness result for the diffusive Peterlin viscoelastic model, Nonlinear Analysis: Theory, Methods & Applications, 120 (2015) , 154-170.  doi: 10.1016/j.na.2015.03.001.
      J. Málek, J. Nečas, M. Rokyta and M. Rȯžička, Weak and Measure-Valued Solutions to Evolutionary PDEs, volume 13 of Applied Mathematics and Mathematical Computation, Chapman and Hall, London, 1996. doi: 10.1007/978-1-4899-6824-1.
      J. Málek and K. R. Rajagopal, Chapter 5 -mathematical issues concerning the Navier-Stokes equations and some of its generalizations, in Handbook of Differential Equations Evolutionary Equations (eds. C. M. Dafermos and E. Feireisl), Handbook of Differential Equations: Evolutionary Equations, 2, North-Holland, 2005,371-459. 2pt
      C. Marchioro and M. Pulvirenti, Mathematical Theory of Incompressible Non-Viscous Fluids, Applied Mathematical Sciences, 96, Springer-Verlag, New York, 1994. doi: 10.1007/978-1-4612-4284-0.
      K. R. Rajagopal , The elasticity of elasticity, Z. Angew. Math. Phys., 58 (2007) , 309-317.  doi: 10.1007/s00033-006-6084-5.
      K. R. Rajagopal , On a new class of models in elasticity, Math. Comput. Appl., 15 (2010) , 506-528. 
      K. R. Rajagopal , Conspectus of concepts of elasticity, Math. Mech. Solids, 16 (2011) , 536-562.  doi: 10.1177/1081286510387856.
      K. R. Rajagopal  and  A. R. Srinivasa , On a class of non-dissipative materials that are not hyperelastic, Royal Society of London Proceedings Series A, 465 (2009) , 493-500.  doi: 10.1098/rspa.2008.0319.
      K. R. Rajagopal  and  A. R. Srinivasa , A Gibbs-potential-based formulation for obtaining the response functions for a class of viscoelastic materials, Proc. R. Soc. A, 467 (2011) , 39-58.  doi: 10.1098/rspa.2010.0136.
      J. Simon , Compact sets in the space Lp (0, t; B), Ann. Mat. Pura Appl., 146 (1987) , 65-96.  doi: 10.1007/BF01762360.
      E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Mathematical Series 30, Princeton University Press, 1970.
      R. Sureshkumar  and  A. N. Beris , Effect of artificial stress diffusivity on the stability of numerical calculations and the flow dynamics of time-dependent viscoelastic flows, Journal of Non-Newtonian Fluid Mechanics, 60 (1995) , 53-80.  doi: 10.1016/0377-0257(95)01377-8.
      C. A. Truesdell , Hypo-elasticity, J. Ration. Mech. Anal., 4 (1955) , 83-133. 
      E. Wiedemann , Existence of weak solutions for the incompressible Euler equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 28 (2011) , 727-730.  doi: 10.1016/j.anihpc.2011.05.002.
      W. Wolibner , Un théoréme sur l'existence du mouvement plan d'un fluide parfait, homogene, incompressible, pendant un temps infiniment long, Mathematische Zeitschrift, 37 (1933) , 698-726.  doi: 10.1007/BF01474610.
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