December  2018, 38(12): 6305-6325. doi: 10.3934/dcds.2018269

Attractors for model of polymer solutions motion

1. 

Peoples Friendship University of Russia, 6 Miklukho-Maklaya St., Moscow, 117198, Russian Federation

2. 

Voronezh State University, 1 Universitetskaya sq., Voronezh, 394018, Russian Federation

To Professor Rafael de la Llave

Received  July 2017 Revised  January 2018 Published  September 2018

Existence of trajectory, global and pullback attractors for an incompressible non-Newtonian fluid (namely, for the mathematical model which describes a weak aqueous polymer solutions motion) in 2D and 3D bounded domains is studied in this paper. For this aim the approximating topological method is effectively combined with the theory of attractors of trajectory spaces.

Citation: Andrey Zvyagin. Attractors for model of polymer solutions motion. Discrete & Continuous Dynamical Systems - A, 2018, 38 (12) : 6305-6325. doi: 10.3934/dcds.2018269
References:
[1]

V. B. AmfilokhievYa. I. VoitkunskiyN. P. Mazaeva and Ya. S. Khodorkovskiy, Flows of polymer solutions under convective accelerations, Trudy Leningradskogo Korablestroitel'nogo Instituta, 96 (1975), 3-9.   Google Scholar

[2]

V. B. Amfilokhiev and V. A. Pavlovsky, Experimental data on laminar-turbulent transition for motion of polymer solutions in pipes, Trudy Leningradskogo Korablestroitel'nogo Instituta, 104 (1976), 3-5.   Google Scholar

[3]

M. BoukroucheG. Lukaszewicz and J. Real, On pullback attractors for a class of two-dimensional turbulent shear flows, International Journal of Engineering Science, 44 (2006), 830-844.  doi: 10.1016/j.ijengsci.2006.05.012.  Google Scholar

[4]

C. CaoE. Lunasin and E. S. Titi, Global well-posedness of the three-dimensional viscous and inviscid simplified Bardina turbulence models, Communications in Mathematical Sciences, 4 (2006), 823-848.  doi: 10.4310/CMS.2006.v4.n4.a8.  Google Scholar

[5]

T. CaraballoG. Lukaszewicz and J. Real, Pullback attractors for asymptotically compact non-autonomous dynamical systems, Nonlinear Analysis: Theory, Methods and Applications, 64 (2006), 484-498.  doi: 10.1016/j.na.2005.03.111.  Google Scholar

[6]

V. V. Chepyzhov and M. I. Vishik, Trajectory attractors for evolution equations, Comptes Rendus de l Acad$\acute{e}$mie des Sciences - Series I - Mathematics, 321 (1995), 1309-1314.  doi: 10.1016/S0021-7824(97)89978-3.  Google Scholar

[7]

V. V. Chepyzhov and M. I. Vishik, Evolution equations and their trajectory attractors, Journal de Math$\acute{e}$matiques Pures et Appliqu$\acute{e}$es, 76 (1997), 913-964.  doi: 10.1016/S0021-7824(97)89978-3.  Google Scholar

[8]

V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics, American Mathematical Society. Colloquium Publications, 2002.  Google Scholar

[9]

D. Cioranescu and E. H. Quazar, Nonlinear partial differential equations and their applications: College de france seminar, Vol. VI, H. Brezis and J.L. Lions (eds.), Research Notes in Mathematics, Pitman, Boston, 109 (1984), 178-197. Google Scholar

[10]

H. Crauel and F. Flandoli, Attractors for random dynamical systems, Probability Theory and Related Fields, 100 (1994), 365-393.  doi: 10.1007/BF01193705.  Google Scholar

[11]

A. V. Fursikov, Optimal Control of Distributed Systems. Theorey and Applications, Providence, 2000.  Google Scholar

[12]

G. P. GaldiM. Grobbelaar-Van Dalsen and N. Sauer, Existence and uniqueness of classical solutions of the equations of motion for second-grade fluids, Archive for Rational Mechanics and Analysis, 124 (1993), 221-237.  doi: 10.1007/BF00953067.  Google Scholar

[13]

I. Gyarmati, Non-equilibrium Thermodynamics. Field Theory and Variational Principles, Heidelberg, Springer, 1970. Google Scholar

[14]

P. E. Kloeden and B. Schmalfuv, Nonautonomous systems, cocycle attractors and variable time-step discretization, Numerical Algorithms, 14 (1997), 141-152.  doi: 10.1023/A:1019156812251.  Google Scholar

[15]

O. A. Ladyzhenskaya, On some gaps in two of my papers on the Navier-Stokes equations and the way of closing them, Zapiski Nauchnykh Seminarov POMI, 271 (2000), 151-155.  doi: 10.1023/A:1023321903383.  Google Scholar

[16]

R. de la Llave and R. Obaya, Regularity of the composition operator in spaces of H${\ddot{o}}$lder functions, Discrete and Continuous Dynamical Systems - Series A (DCDS-A), 5 (1999), 157-184.   Google Scholar

[17]

R. de la Llave, Variational methods for quasi-periodic solutions of partial differential equations, Hamiltonian and Celestial Mechanics, 6 (2000), 214-228.  doi: 10.1142/9789812792099_0013.  Google Scholar

[18]

G. Lukaszewicz and J. C. Robinson, Invariant measures for non-autonomous dissipative dynamical systems, Discrete and Continuous Dynamical Systems - Series A (DCDS-A), 34 (2014), 4211-4222.  doi: 10.3934/dcds.2014.34.4211.  Google Scholar

[19]

A. P. Oskolkov, The uniqueness and solvability in the large of boundary value problems for the equations of motion of aueous solutions of polymers, Zapiski Nauchnykh Seminarov Leningradskogo Otdelenia Matematicheskogo Instituta Steklova (LOMI), 38 (1973), 98-136.   Google Scholar

[20]

A. P. Oskolkov, On some quasilinear systems occuring in the study of motion of viscous fluids, Zapiski Nauchnykh Seminarov Leningradskogo Otdelenia Matematicheskogo Instituta Steklova (LOMI), 52 (1975), 128-157.   Google Scholar

[21]

V. A. Pavlovsky, On the theoretical description of weak water solutions of polymers, Doklady Akademii Nauk SSSR, (1971), 809-812.   Google Scholar

[22]

R. S. Rivlin and J. L. Ericksen, Stress-deformation relations for isotropic materials, Journal of Rational Mechanics and Analysis, 4 (1955), 323-425.   Google Scholar

[23]

C. Le Roux, Existence and uniqueness of the flow of second-grade fluids with slip boundary conditions, Archive for Rational Mechanics and Analysis, 148 (1999), 309-356.  doi: 10.1007/s002050050164.  Google Scholar

[24]

G. R. Sell, Global attractors for the three-dimensional Navier-Stokes equations, Journal of Dynamics and Differential Equations, 8 (1996), 1-33.  doi: 10.1007/BF02218613.  Google Scholar

[25]

G. R. Sell and Y. You, Dynamics of Evolutionary Equations, American Mathematical Society, 2002. doi: 10.1007/978-1-4757-5037-9.  Google Scholar

[26]

A. Sequeira and J. Videman, Global existence of classical solutions for the equations of third grade fluids, J. Math. Phys. Sci., 29 (1995), 47-69.   Google Scholar

[27]

J. Simon, Compact sets in the space Lp(0,T;B), Annali di Matematica Pura ed Applicata, 146 (1987), 65-96.  doi: 10.1007/BF01762360.  Google Scholar

[28]

R. Temam, Navier-Stokes Equations. Theory and Nymerical Analysis, Providence, RI, 2001. doi: 10.1090/chel/343.  Google Scholar

[29]

C. Truesdell, A First Course in Rational Continuum Mechanics, The John Hopkins Unirsity, 1972. Google Scholar

[30]

M. I. Vishik and V. V. Chepyzhov, Trajectory attractors of equations of mathematical physics, Russian Mathematical Surveys, 66 (2011), 637-731.  doi: 10.1070/RM2011v066n04ABEH004753.  Google Scholar

[31]

D. A. Vorotnikov and V. G. Zvyagin, Trajectory and global attractors of the boundary value problem for autonomous motion equations of viscoelastic medium, Journal of Mathematical Fluid Mechanics, 10 (2008), 19-44.  doi: 10.1007/s00021-005-0215-1.  Google Scholar

[32]

D. A. Vorotnikov, Asymptotic behaviour of the non-autonomous 3D Navier-Stokes problem with coercive force, Journal of Differential Equations, 251 (2011), 2209-2225.  doi: 10.1016/j.jde.2011.07.008.  Google Scholar

[33]

C. Zhao and S. Zhou, Pullback attractors for a non-autonomous incompressible non-Newtonian fluid, Journal of Differential Equations, 238 (2007), 394-425.  doi: 10.1016/j.jde.2007.04.001.  Google Scholar

[34]

C. ZhaoY. Li and S. Zhou, Regularity of trajectory attractor and upper semicontinuity of global attractor for a 2D non-Newtonian fluid, Journal of Differential Equations, 247 (2009), 2331-2363.  doi: 10.1016/j.jde.2009.07.031.  Google Scholar

[35]

C. Zhao, Pullback asymptotic behavior of solutions for a non-autonomous non-Newtonian fluid on two-dimensional unbounded domains, Journal of Mathematical Physics, 53 (2012), 122702, 22pp. doi: 10.1063/1.4769302.  Google Scholar

[36]

C. ZhaoG. Liu and W. Wang, Smooth pullback attractors for a non-autonomous 2D non-Newtonian fluid and their tempered behaviors, Journal of Mathematical Fluid Mechanics, 16 (2014), 243-262.  doi: 10.1007/s00021-013-0153-2.  Google Scholar

[37]

C. Zhao and L. Yang, Pullback attractors and invariant measures for the non-autonomous globally modified Naviertokes equations, Communications in Mathematical Sciences, 15 (2017), 1565-1580.  doi: 10.4310/CMS.2017.v15.n6.a4.  Google Scholar

[38]

Z. Zhu and C. Zhao, Pullback attractor and invariant measures for the three-dimensional regularized MHD equations, Discrete and Continuous Dynamical Systems - Series A (DCDS-A), 38 (2018), 1461-1477.  doi: 10.3934/dcds.2018060.  Google Scholar

[39]

A. V. Zvyagin, Solvability for the equations of motion of weak aqueous polymer solution with objective derivative, Nonlinear Analysis: Theory, Methods and Applications, 90 (2013), 70-85.  doi: 10.1016/j.na.2013.05.022.  Google Scholar

[40]

A. V. Zvyagin, An optimal control problem with feedback for a mathematical model of the motion of weakly concentrated water polymer solutions with objective derivative, Siberian Mathematical Journal, 54 (2013), 640-655.   Google Scholar

[41]

A. V. Zvyagin, Attractors for a model of polymer motion with objective derivative in the rheological relation, Doklady Mathematics, 88 (2013), 730-733.   Google Scholar

[42]

A. V. Zvyagin and V. G. Zvyagin, Pullback attractors for a model of weakly concentrated aqueous polymer solution motion with a rheological relation satisfying the objectivity principle, Doklady Mathematics, 95 (2017), 247-249.   Google Scholar

[43]

V. G. Zvyagin and S. K. Kondratiev, Attractors of equation of non-Newtinian fluid dynamics, Russian Mathematical Surveys, 69 (2014), 845-913.  doi: 10.4213/rm9615.  Google Scholar

[44]

V. G. Zvyagin and S. K. Kondratiev, Pullback attractors for the model of motion of dilute aqueous polymer solutions, Izvestiya: Mathematics, 79 (2015), 645-667.  doi: 10.4213/im8163.  Google Scholar

[45]

V. G. Zvyagin and M. V. Turbin, The study of initial-boundary value problems for mathematical models of the motion of Kelvin-Voigt fluids, Journal of Mathematical Sciences, 168 (2010), 157-308.  doi: 10.1007/s10958-010-9981-2.  Google Scholar

[46]

V. G. Zvyagin and D. A. Vorotnikov, Approximating-topological methods in some problems of hydrodinamics, Journal of Fixed Point Theory and Applications, 3 (2008), 23-49.  doi: 10.1007/s11784-008-0056-7.  Google Scholar

[47]

V. G. Zvyagin and D. A. Vorotnikov, Topological Approximayion Methods for Evolutionary Problems of Nonlinear Hydrodinamics, De Gruyter Series In Nonlinear Analysis and Applications, 2008. doi: 10.1515/9783110208283.  Google Scholar

show all references

References:
[1]

V. B. AmfilokhievYa. I. VoitkunskiyN. P. Mazaeva and Ya. S. Khodorkovskiy, Flows of polymer solutions under convective accelerations, Trudy Leningradskogo Korablestroitel'nogo Instituta, 96 (1975), 3-9.   Google Scholar

[2]

V. B. Amfilokhiev and V. A. Pavlovsky, Experimental data on laminar-turbulent transition for motion of polymer solutions in pipes, Trudy Leningradskogo Korablestroitel'nogo Instituta, 104 (1976), 3-5.   Google Scholar

[3]

M. BoukroucheG. Lukaszewicz and J. Real, On pullback attractors for a class of two-dimensional turbulent shear flows, International Journal of Engineering Science, 44 (2006), 830-844.  doi: 10.1016/j.ijengsci.2006.05.012.  Google Scholar

[4]

C. CaoE. Lunasin and E. S. Titi, Global well-posedness of the three-dimensional viscous and inviscid simplified Bardina turbulence models, Communications in Mathematical Sciences, 4 (2006), 823-848.  doi: 10.4310/CMS.2006.v4.n4.a8.  Google Scholar

[5]

T. CaraballoG. Lukaszewicz and J. Real, Pullback attractors for asymptotically compact non-autonomous dynamical systems, Nonlinear Analysis: Theory, Methods and Applications, 64 (2006), 484-498.  doi: 10.1016/j.na.2005.03.111.  Google Scholar

[6]

V. V. Chepyzhov and M. I. Vishik, Trajectory attractors for evolution equations, Comptes Rendus de l Acad$\acute{e}$mie des Sciences - Series I - Mathematics, 321 (1995), 1309-1314.  doi: 10.1016/S0021-7824(97)89978-3.  Google Scholar

[7]

V. V. Chepyzhov and M. I. Vishik, Evolution equations and their trajectory attractors, Journal de Math$\acute{e}$matiques Pures et Appliqu$\acute{e}$es, 76 (1997), 913-964.  doi: 10.1016/S0021-7824(97)89978-3.  Google Scholar

[8]

V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics, American Mathematical Society. Colloquium Publications, 2002.  Google Scholar

[9]

D. Cioranescu and E. H. Quazar, Nonlinear partial differential equations and their applications: College de france seminar, Vol. VI, H. Brezis and J.L. Lions (eds.), Research Notes in Mathematics, Pitman, Boston, 109 (1984), 178-197. Google Scholar

[10]

H. Crauel and F. Flandoli, Attractors for random dynamical systems, Probability Theory and Related Fields, 100 (1994), 365-393.  doi: 10.1007/BF01193705.  Google Scholar

[11]

A. V. Fursikov, Optimal Control of Distributed Systems. Theorey and Applications, Providence, 2000.  Google Scholar

[12]

G. P. GaldiM. Grobbelaar-Van Dalsen and N. Sauer, Existence and uniqueness of classical solutions of the equations of motion for second-grade fluids, Archive for Rational Mechanics and Analysis, 124 (1993), 221-237.  doi: 10.1007/BF00953067.  Google Scholar

[13]

I. Gyarmati, Non-equilibrium Thermodynamics. Field Theory and Variational Principles, Heidelberg, Springer, 1970. Google Scholar

[14]

P. E. Kloeden and B. Schmalfuv, Nonautonomous systems, cocycle attractors and variable time-step discretization, Numerical Algorithms, 14 (1997), 141-152.  doi: 10.1023/A:1019156812251.  Google Scholar

[15]

O. A. Ladyzhenskaya, On some gaps in two of my papers on the Navier-Stokes equations and the way of closing them, Zapiski Nauchnykh Seminarov POMI, 271 (2000), 151-155.  doi: 10.1023/A:1023321903383.  Google Scholar

[16]

R. de la Llave and R. Obaya, Regularity of the composition operator in spaces of H${\ddot{o}}$lder functions, Discrete and Continuous Dynamical Systems - Series A (DCDS-A), 5 (1999), 157-184.   Google Scholar

[17]

R. de la Llave, Variational methods for quasi-periodic solutions of partial differential equations, Hamiltonian and Celestial Mechanics, 6 (2000), 214-228.  doi: 10.1142/9789812792099_0013.  Google Scholar

[18]

G. Lukaszewicz and J. C. Robinson, Invariant measures for non-autonomous dissipative dynamical systems, Discrete and Continuous Dynamical Systems - Series A (DCDS-A), 34 (2014), 4211-4222.  doi: 10.3934/dcds.2014.34.4211.  Google Scholar

[19]

A. P. Oskolkov, The uniqueness and solvability in the large of boundary value problems for the equations of motion of aueous solutions of polymers, Zapiski Nauchnykh Seminarov Leningradskogo Otdelenia Matematicheskogo Instituta Steklova (LOMI), 38 (1973), 98-136.   Google Scholar

[20]

A. P. Oskolkov, On some quasilinear systems occuring in the study of motion of viscous fluids, Zapiski Nauchnykh Seminarov Leningradskogo Otdelenia Matematicheskogo Instituta Steklova (LOMI), 52 (1975), 128-157.   Google Scholar

[21]

V. A. Pavlovsky, On the theoretical description of weak water solutions of polymers, Doklady Akademii Nauk SSSR, (1971), 809-812.   Google Scholar

[22]

R. S. Rivlin and J. L. Ericksen, Stress-deformation relations for isotropic materials, Journal of Rational Mechanics and Analysis, 4 (1955), 323-425.   Google Scholar

[23]

C. Le Roux, Existence and uniqueness of the flow of second-grade fluids with slip boundary conditions, Archive for Rational Mechanics and Analysis, 148 (1999), 309-356.  doi: 10.1007/s002050050164.  Google Scholar

[24]

G. R. Sell, Global attractors for the three-dimensional Navier-Stokes equations, Journal of Dynamics and Differential Equations, 8 (1996), 1-33.  doi: 10.1007/BF02218613.  Google Scholar

[25]

G. R. Sell and Y. You, Dynamics of Evolutionary Equations, American Mathematical Society, 2002. doi: 10.1007/978-1-4757-5037-9.  Google Scholar

[26]

A. Sequeira and J. Videman, Global existence of classical solutions for the equations of third grade fluids, J. Math. Phys. Sci., 29 (1995), 47-69.   Google Scholar

[27]

J. Simon, Compact sets in the space Lp(0,T;B), Annali di Matematica Pura ed Applicata, 146 (1987), 65-96.  doi: 10.1007/BF01762360.  Google Scholar

[28]

R. Temam, Navier-Stokes Equations. Theory and Nymerical Analysis, Providence, RI, 2001. doi: 10.1090/chel/343.  Google Scholar

[29]

C. Truesdell, A First Course in Rational Continuum Mechanics, The John Hopkins Unirsity, 1972. Google Scholar

[30]

M. I. Vishik and V. V. Chepyzhov, Trajectory attractors of equations of mathematical physics, Russian Mathematical Surveys, 66 (2011), 637-731.  doi: 10.1070/RM2011v066n04ABEH004753.  Google Scholar

[31]

D. A. Vorotnikov and V. G. Zvyagin, Trajectory and global attractors of the boundary value problem for autonomous motion equations of viscoelastic medium, Journal of Mathematical Fluid Mechanics, 10 (2008), 19-44.  doi: 10.1007/s00021-005-0215-1.  Google Scholar

[32]

D. A. Vorotnikov, Asymptotic behaviour of the non-autonomous 3D Navier-Stokes problem with coercive force, Journal of Differential Equations, 251 (2011), 2209-2225.  doi: 10.1016/j.jde.2011.07.008.  Google Scholar

[33]

C. Zhao and S. Zhou, Pullback attractors for a non-autonomous incompressible non-Newtonian fluid, Journal of Differential Equations, 238 (2007), 394-425.  doi: 10.1016/j.jde.2007.04.001.  Google Scholar

[34]

C. ZhaoY. Li and S. Zhou, Regularity of trajectory attractor and upper semicontinuity of global attractor for a 2D non-Newtonian fluid, Journal of Differential Equations, 247 (2009), 2331-2363.  doi: 10.1016/j.jde.2009.07.031.  Google Scholar

[35]

C. Zhao, Pullback asymptotic behavior of solutions for a non-autonomous non-Newtonian fluid on two-dimensional unbounded domains, Journal of Mathematical Physics, 53 (2012), 122702, 22pp. doi: 10.1063/1.4769302.  Google Scholar

[36]

C. ZhaoG. Liu and W. Wang, Smooth pullback attractors for a non-autonomous 2D non-Newtonian fluid and their tempered behaviors, Journal of Mathematical Fluid Mechanics, 16 (2014), 243-262.  doi: 10.1007/s00021-013-0153-2.  Google Scholar

[37]

C. Zhao and L. Yang, Pullback attractors and invariant measures for the non-autonomous globally modified Naviertokes equations, Communications in Mathematical Sciences, 15 (2017), 1565-1580.  doi: 10.4310/CMS.2017.v15.n6.a4.  Google Scholar

[38]

Z. Zhu and C. Zhao, Pullback attractor and invariant measures for the three-dimensional regularized MHD equations, Discrete and Continuous Dynamical Systems - Series A (DCDS-A), 38 (2018), 1461-1477.  doi: 10.3934/dcds.2018060.  Google Scholar

[39]

A. V. Zvyagin, Solvability for the equations of motion of weak aqueous polymer solution with objective derivative, Nonlinear Analysis: Theory, Methods and Applications, 90 (2013), 70-85.  doi: 10.1016/j.na.2013.05.022.  Google Scholar

[40]

A. V. Zvyagin, An optimal control problem with feedback for a mathematical model of the motion of weakly concentrated water polymer solutions with objective derivative, Siberian Mathematical Journal, 54 (2013), 640-655.   Google Scholar

[41]

A. V. Zvyagin, Attractors for a model of polymer motion with objective derivative in the rheological relation, Doklady Mathematics, 88 (2013), 730-733.   Google Scholar

[42]

A. V. Zvyagin and V. G. Zvyagin, Pullback attractors for a model of weakly concentrated aqueous polymer solution motion with a rheological relation satisfying the objectivity principle, Doklady Mathematics, 95 (2017), 247-249.   Google Scholar

[43]

V. G. Zvyagin and S. K. Kondratiev, Attractors of equation of non-Newtinian fluid dynamics, Russian Mathematical Surveys, 69 (2014), 845-913.  doi: 10.4213/rm9615.  Google Scholar

[44]

V. G. Zvyagin and S. K. Kondratiev, Pullback attractors for the model of motion of dilute aqueous polymer solutions, Izvestiya: Mathematics, 79 (2015), 645-667.  doi: 10.4213/im8163.  Google Scholar

[45]

V. G. Zvyagin and M. V. Turbin, The study of initial-boundary value problems for mathematical models of the motion of Kelvin-Voigt fluids, Journal of Mathematical Sciences, 168 (2010), 157-308.  doi: 10.1007/s10958-010-9981-2.  Google Scholar

[46]

V. G. Zvyagin and D. A. Vorotnikov, Approximating-topological methods in some problems of hydrodinamics, Journal of Fixed Point Theory and Applications, 3 (2008), 23-49.  doi: 10.1007/s11784-008-0056-7.  Google Scholar

[47]

V. G. Zvyagin and D. A. Vorotnikov, Topological Approximayion Methods for Evolutionary Problems of Nonlinear Hydrodinamics, De Gruyter Series In Nonlinear Analysis and Applications, 2008. doi: 10.1515/9783110208283.  Google Scholar

[1]

M. Bulíček, F. Ettwein, P. Kaplický, Dalibor Pražák. The dimension of the attractor for the 3D flow of a non-Newtonian fluid. Communications on Pure & Applied Analysis, 2009, 8 (5) : 1503-1520. doi: 10.3934/cpaa.2009.8.1503

[2]

Hafedh Bousbih. Global weak solutions for a coupled chemotaxis non-Newtonian fluid. Discrete & Continuous Dynamical Systems - B, 2019, 24 (2) : 907-929. doi: 10.3934/dcdsb.2018212

[3]

Jong Yeoul Park, Jae Ug Jeong. Pullback attractors for a $2D$-non-autonomous incompressible non-Newtonian fluid with variable delays. Discrete & Continuous Dynamical Systems - B, 2016, 21 (8) : 2687-2702. doi: 10.3934/dcdsb.2016068

[4]

Guowei Liu, Rui Xue. Pullback dynamic behavior for a non-autonomous incompressible non-Newtonian fluid. Discrete & Continuous Dynamical Systems - B, 2018, 23 (6) : 2193-2216. doi: 10.3934/dcdsb.2018231

[5]

Linfang Liu, Tomás Caraballo, Xianlong Fu. Exponential stability of an incompressible non-Newtonian fluid with delay. Discrete & Continuous Dynamical Systems - B, 2018, 23 (10) : 4285-4303. doi: 10.3934/dcdsb.2018138

[6]

Lars Diening, Michael Růžička. An existence result for non-Newtonian fluids in non-regular domains. Discrete & Continuous Dynamical Systems - S, 2010, 3 (2) : 255-268. doi: 10.3934/dcdss.2010.3.255

[7]

Emil Novruzov. On existence and nonexistence of the positive solutions of non-newtonian filtration equation. Communications on Pure & Applied Analysis, 2011, 10 (2) : 719-730. doi: 10.3934/cpaa.2011.10.719

[8]

I. D. Chueshov, Iryna Ryzhkova. A global attractor for a fluid--plate interaction model. Communications on Pure & Applied Analysis, 2013, 12 (4) : 1635-1656. doi: 10.3934/cpaa.2013.12.1635

[9]

Rodrigo Samprogna, Tomás Caraballo. Pullback attractor for a dynamic boundary non-autonomous problem with Infinite Delay. Discrete & Continuous Dynamical Systems - B, 2018, 23 (2) : 509-523. doi: 10.3934/dcdsb.2017195

[10]

T. Caraballo, J. A. Langa, J. Valero. Structure of the pullback attractor for a non-autonomous scalar differential inclusion. Discrete & Continuous Dynamical Systems - S, 2016, 9 (4) : 979-994. doi: 10.3934/dcdss.2016037

[11]

Yirong Jiang, Nanjing Huang, Zhouchao Wei. Existence of a global attractor for fractional differential hemivariational inequalities. Discrete & Continuous Dynamical Systems - B, 2017, 22 (11) : 0-0. doi: 10.3934/dcdsb.2019216

[12]

Jin Li, Jianhua Huang. Dynamics of a 2D Stochastic non-Newtonian fluid driven by fractional Brownian motion. Discrete & Continuous Dynamical Systems - B, 2012, 17 (7) : 2483-2508. doi: 10.3934/dcdsb.2012.17.2483

[13]

Aneta Wróblewska-Kamińska. Local pressure methods in Orlicz spaces for the motion of rigid bodies in a non-Newtonian fluid with general growth conditions. Discrete & Continuous Dynamical Systems - S, 2013, 6 (5) : 1417-1425. doi: 10.3934/dcdss.2013.6.1417

[14]

Li Fang, Zhenhua Guo. Zero dissipation limit to rarefaction wave with vacuum for a one-dimensional compressible non-Newtonian fluid. Communications on Pure & Applied Analysis, 2017, 16 (1) : 209-242. doi: 10.3934/cpaa.2017010

[15]

Wen Tan. The regularity of pullback attractor for a non-autonomous p-Laplacian equation with dynamical boundary condition. Discrete & Continuous Dynamical Systems - B, 2019, 24 (2) : 529-546. doi: 10.3934/dcdsb.2018194

[16]

Azer Khanmamedov, Sema Simsek. Existence of the global attractor for the plate equation with nonlocal nonlinearity in $ \mathbb{R} ^{n}$. Discrete & Continuous Dynamical Systems - B, 2016, 21 (1) : 151-172. doi: 10.3934/dcdsb.2016.21.151

[17]

Tomás Caraballo, David Cheban. On the structure of the global attractor for non-autonomous dynamical systems with weak convergence. Communications on Pure & Applied Analysis, 2012, 11 (2) : 809-828. doi: 10.3934/cpaa.2012.11.809

[18]

Gabriel Deugoue. Approximation of the trajectory attractor of the 3D MHD System. Communications on Pure & Applied Analysis, 2013, 12 (5) : 2119-2144. doi: 10.3934/cpaa.2013.12.2119

[19]

Vladimir V. Chepyzhov, Mark I. Vishik. Trajectory attractor for reaction-diffusion system with diffusion coefficient vanishing in time. Discrete & Continuous Dynamical Systems - A, 2010, 27 (4) : 1493-1509. doi: 10.3934/dcds.2010.27.1493

[20]

Oleksiy V. Kapustyan, Pavlo O. Kasyanov, José Valero, Michael Z. Zgurovsky. Strong attractors for vanishing viscosity approximations of non-Newtonian suspension flows. Discrete & Continuous Dynamical Systems - B, 2018, 23 (3) : 1155-1176. doi: 10.3934/dcdsb.2018146

2018 Impact Factor: 1.143

Metrics

  • PDF downloads (76)
  • HTML views (115)
  • Cited by (0)

Other articles
by authors

[Back to Top]