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doi: 10.3934/eect.2022011
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Pullback attractors for weak solution to modified Kelvin-Voigt model

Voronezh State University, Universitetskaya sq. 1, Voronezh, 394018, Russia

*Corresponding author: Mikhail Turbin

Received  August 2021 Revised  January 2022 Early access March 2022

The paper is devoted to the investigation of the qualitative dynamics of weak solutions for the modified Kelvin-Voigt model on the base of the theory of pullback attractors for trajectory spaces. At first, for the studied model, an auxiliary problem is considered, its solvability in the weak sense is proved, and some solution estimates are established. Then, on the base of these estimates, a family of trajectory spaces is determined and the existence of trajectory and minimal pullback attractors for the considered trajectory spaces is proved.

Citation: Mikhail Turbin, Anastasiia Ustiuzhaninova. Pullback attractors for weak solution to modified Kelvin-Voigt model. Evolution Equations and Control Theory, doi: 10.3934/eect.2022011
References:
[1]

V. B. Amfilokhiev and V. A. Pavlovsky, Experimental data on laminar-turbulent transition for flows of polymer solutions in pipes, Tr. Leningr. Korablestr. Inst., 104 (1975), 3–5 (in Russian).

[2]

V. B. Amfilokhiev, Y. I. Voitkunskii, N. P. Mazaeva and Y. S. Khodornovskii, Flows of polymer solutions in the case of convective accelerations, Tr. Leningr. Korablestr. Inst., 96 (1975), 3–9 (in Russian).

[3]

C. Amrouche, L. C. Berselli, R. Lewandowski and D. D. Nguyen, Turbulent flows as generalized Kelvin–Voigt materials: Modeling and analysis, Nonlinear Anal., 196 (2020), 111790, 24 pp. doi: 10.1016/j.na.2020.111790.

[4]

A. V. Babin and M. I. Vishik, Attractors of Evolution Equations, Volume 25 of Studies in Mathematics and its Applications, North Holland, New York, NY, 1992.

[5]

M. BoukroucheG. Łukaszewicz and J. Real, On pullback attractors for a class of two-dimensional turbulent shear flows, Internat. J. Engrg. Sci., 44 (2006), 830-844.  doi: 10.1016/j.ijengsci.2006.05.012.

[6]

Y. CaoE. M. Lunasin and E. S. Titi, Global well-posedness of the three-dimensional viscous and inviscid simplified Bardina turbulence models, Commun. Math. Sci., 4 (2006), 823-848.  doi: 10.4310/CMS.2006.v4.n4.a8.

[7]

T. CaraballoG. Łukaszewicz and J. Real, Pullback attractors for asymptotically compact non-autonomous dynamical systems, Nonlinear Anal., 64 (2006), 484-498.  doi: 10.1016/j.na.2005.03.111.

[8]

T. CaraballoG. Łukaszewicz and J. Real, Pullback attractors for non-autonomous 2D-Navier-Stokes equations in some unbounded domains, C. R. Math., 342 (2006), 263-268.  doi: 10.1016/j.crma.2005.12.015.

[9]

A. N. Carvalho, J. A. Langa and J. C. Robinson, Attractors for Infinite-Dimensional Non-Autonomous Dynamical Systems, Springer-Verlag, New York, NY, 2013. doi: 10.1007/978-1-4614-4581-4.

[10]

V. V. Chepyzhov, Approximating the trajectory attractor of the 3D Navier-Stokes system using various $\alpha$-models of fluid dynamics, Sb. Math., 207 (2016), 610-638.  doi: 10.1070/SM8549.

[11]

V. V. ChepyzhovE. S. Titi and M. I. Vishik, On the convergence of solutions of the Leray-$\alpha$ model to the trajectory attractor of the 3D Navier-Stokes system, Discrete Contin. Dyn. Syst., 17 (2007), 481-500.  doi: 10.3934/dcds.2007.17.481.

[12]

V. V. Chepyzhov and M. I. Vishik, Trajectory attractors for evolution equations, C. R. Acad. Sci. Paris Sér. I Math., 321 (1995), 1309-1314. 

[13]

V. V. Chepyzhov and M. I. Vishik, Evolution equations and their trajectory attractors, J. Math. Pures Appl., 76 (1997), 913-964.  doi: 10.1016/S0021-7824(97)89978-3.

[14]

V. Chepyzhov and M. Vishik, A Hausdorff dimension estimate for kernel sections of non-autonomous evolution equations, Indiana Univ. Math. J., 42 (1993), 1057-1076.  doi: 10.1512/iumj.1993.42.42049.

[15]

V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics, American Mathematical Society, Providence, RI, 2002.

[16]

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

[17]

A. V. Fursikov, Optimal Control of Distributed Systems. Theory and Applications, American Mathematical Society, Providence, 2000. doi: 10.1090/mmono/187.

[18]

J. García-LuengoP. Marín-Rubio and J. Real, Pullback attractors for 2D Navier-Stokes equations with delays and their regularity, Adv. Nonlinear Stud., 13 (2013), 331-357.  doi: 10.1515/ans-2013-0205.

[19]

V. K. Kalantarov and E. S. Titi, Global attractors and determining modes for the 3D Navier-Stokes-Voight equations, Chin. Ann. Math. Ser. B, 30 (2009), 697-714.  doi: 10.1007/s11401-009-0205-3.

[20]

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

[21]

O. A. Ladyzhenskaya, The Mathematical Theory of Viscous Incompressible Flow, Gordon and Breach Science Publishers, New-York, NY, 1969.

[22]

O. A. Ladyzhenskaya, A dynamical system generated by the Navier-Stokes equations, Journal of Soviet Mathematics, 3 (1975), 458-479.  doi: 10.1007/BF01084684.

[23]

O. A. Ladyzhenskaya, On the determination of minimal global attractors for the Navier-Stokes and other partial differential equations, Russian Mathematical Surveys, 42 (1987), 27-73.  doi: 10.1070/RM1987v042n06ABEH001503.

[24]

O. A. Ladyzhenskaya, On the global unique solvability of some two-dimensional problems for the water solutions of polymers, J. Math. Sci., 99 (2000), 888-897.  doi: 10.1007/BF02673597.

[25]

O. A. Ladyzhenskaya, On some gaps in two of my papers on the Navier-Stokes equations and the way of closing them, J. Math. Sci., 115 (2003), 2789-2791.  doi: 10.1023/A:1023321903383.

[26]

W. Noll and C. Truesdell, The Non-Linear Field Theories of Mechanics, 3$^{rd}$ edition, Springer-Verlag, Berlin, 2004. doi: 10.1007/978-3-662-10388-3.

[27]

A. P. Oskolkov, Solvability in the large of the first boundary value problem for a certain quasilinear third order system that is encountered in the study of the motion of a viscous fluid, Zap. Naučn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI), 27 (1972), 145–160 (in Russian).

[28]

A. P. Oskolkov, The uniqueness and solvability in the large of boundary value problems for the equations of motion of aqueous solutions of polymers, Zap. Naučn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI), 38 (1973), 98–136 (in Russian).

[29]

A. P. Oskolkov, Some quasilinear systems that arise in the study of the motion of viscous fluids, Zap. Naučn. Sem. Leningrad. Otdel. Mat. Inst. Steklov (LOMI), 52 (1975), 128–157 (in Russian).

[30]

A. P. Oskolkov, Theory of nonstationary flows of Kelvin–Voigt fluids, Journal of Soviet Mathematics, 28 (1985), 751-758.  doi: 10.1007/BF02112340.

[31]

A. P. Oskolkov, Initial-boundary value problems for equations of motion of Kelvin–Voight fluids and Oldroyd fluids, Steklov Inst. Math., 179 (1989), 137-182. 

[32]

V. A. Pavlovsky, On theoretical description of weak aqueous solutions of polymers, Dokl. Akad. Nauk SSSR, 200 (1971), 809–812 (in Russian).

[33]

R. S. Rivlin and J. L. Ericksen, Stress-deformation relations for isotropic materials, J. Rational Mech. Anal., 4 (1955), 323-425.  doi: 10.1512/iumj.1955.4.54011.

[34]

G. R. Sell, Global attractors for the three-dimensional Navier-Stokes equations, J. Dynam. Differential Equations, 8 (1996), 1-33.  doi: 10.1007/BF02218613.

[35]

G. A. Seregin, On a dynamical system generated by the two-dimensional equations of the motion of a Bingham fluid, J. Math. Sci., 70 (1994), 1806-1816.  doi: 10.1007/BF02149150.

[36]

J. Simon, Compact sets in the space $L^p(0, T; B)$, Ann. Mat. Pura Appl., 146 (1986), 65-96.  doi: 10.1007/BF01762360.

[37]

S. L. Sobolev, Some Applications of Functional Analysis in Mathematical Physics, vol. 90, American Mathematical Society, Providence, 1991. doi: 10.1090/mmono/090.

[38]

V. A. Solonnikov, Estimates of Green's tensors for some boundary-value problems, Dokl. Akad. Nauk SSSR, 130 (1960), 988–991 (in Russian).

[39]

R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, 2$^nd$ edition, Springer-Verlag, New York, NY, 1997. doi: 10.1007/978-1-4612-0645-3.

[40]

R. Temam, Navier-Stokes Equations: Theory and Numerical Analysis, AMS Chelsea Publishing, Providence, RI, 2001. doi: 10.1090/chel/343.

[41]

M. V. Turbin and A. S. Ustiuzhaninova, The existence theorem for a weak solution to initial-boundary value problem for system of equations describing the motion of weak aqueous polymer solutions, Russian Math., 63 (2019), 54-69.  doi: 10.3103/S1066369X19080061.

[42]

A. S. Ustiuzhaninova and M. V. Turbin, Feedback control problem for modified Kelvin–Voigt model, Journal of Dynamical and Control Systems, early access, 2021. doi: 10.1007/s10883-021-09539-0.

[43]

A. S. Ustiuzhaninova and M. V. Turbin, Trajectory and global attractors for a modified Kelvin–Voigt model, J. Appl. Ind. Math., 15 (2021), 158-168.  doi: 10.1134/S1990478921010142.

[44]

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

[45]

D. A. Vorotnikov and V. G. Zvyagin, Uniform attractors for non-autonomous motion equations of viscoelastic medium, J. Math. Anal. Appl., 325 (2007), 438-458.  doi: 10.1016/j.jmaa.2006.01.078.

[46]

I. I. Vorovich and V. I. Yudovich, Steady flow of a viscous incompressible fluid, Mat. Sb., 53 (1961), 393–428 (in Russian).

[47]

C. ZhaoW. Sun and C. H. Hsu, Pullback dynamical behaviors of the non-autonomous micropolar fluid flows, Dyn. Partial Differ. Equ., 12 (2015), 265-288.  doi: 10.4310/DPDE.2015.v12.n3.a4.

[48]

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

[49]

A. Zvyagin, Attractors for model of polymer solutions motion, Discrete Contin. Dyn. Syst., 38 (2018), 6305-6325.  doi: 10.3934/dcds.2018269.

[50]

V. Zvyagin and S. Kondratyev, Pullback attractors of the Jeffreys-Oldroyd equations, J. Differential Equations, 260 (2016), 5026-5042.  doi: 10.1016/j.jde.2015.11.038.

[51]

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

[52]

V. G. Zvyagin and M. V. Turbin, Mathematical Problems in Viscoelastic Hydrodynamics, Krasand URSS, Moscow, 2012 (in Russian).

[53]

V. G. Zvyagin and D. A. Vorotnikov, Topological Approximation Methods for Evolutionary Problems of Nonlinear Hydrodynamics, vol. 12, Walter de Gruyter, Berlin, 2008. doi: 10.1515/9783110208283.

show all references

References:
[1]

V. B. Amfilokhiev and V. A. Pavlovsky, Experimental data on laminar-turbulent transition for flows of polymer solutions in pipes, Tr. Leningr. Korablestr. Inst., 104 (1975), 3–5 (in Russian).

[2]

V. B. Amfilokhiev, Y. I. Voitkunskii, N. P. Mazaeva and Y. S. Khodornovskii, Flows of polymer solutions in the case of convective accelerations, Tr. Leningr. Korablestr. Inst., 96 (1975), 3–9 (in Russian).

[3]

C. Amrouche, L. C. Berselli, R. Lewandowski and D. D. Nguyen, Turbulent flows as generalized Kelvin–Voigt materials: Modeling and analysis, Nonlinear Anal., 196 (2020), 111790, 24 pp. doi: 10.1016/j.na.2020.111790.

[4]

A. V. Babin and M. I. Vishik, Attractors of Evolution Equations, Volume 25 of Studies in Mathematics and its Applications, North Holland, New York, NY, 1992.

[5]

M. BoukroucheG. Łukaszewicz and J. Real, On pullback attractors for a class of two-dimensional turbulent shear flows, Internat. J. Engrg. Sci., 44 (2006), 830-844.  doi: 10.1016/j.ijengsci.2006.05.012.

[6]

Y. CaoE. M. Lunasin and E. S. Titi, Global well-posedness of the three-dimensional viscous and inviscid simplified Bardina turbulence models, Commun. Math. Sci., 4 (2006), 823-848.  doi: 10.4310/CMS.2006.v4.n4.a8.

[7]

T. CaraballoG. Łukaszewicz and J. Real, Pullback attractors for asymptotically compact non-autonomous dynamical systems, Nonlinear Anal., 64 (2006), 484-498.  doi: 10.1016/j.na.2005.03.111.

[8]

T. CaraballoG. Łukaszewicz and J. Real, Pullback attractors for non-autonomous 2D-Navier-Stokes equations in some unbounded domains, C. R. Math., 342 (2006), 263-268.  doi: 10.1016/j.crma.2005.12.015.

[9]

A. N. Carvalho, J. A. Langa and J. C. Robinson, Attractors for Infinite-Dimensional Non-Autonomous Dynamical Systems, Springer-Verlag, New York, NY, 2013. doi: 10.1007/978-1-4614-4581-4.

[10]

V. V. Chepyzhov, Approximating the trajectory attractor of the 3D Navier-Stokes system using various $\alpha$-models of fluid dynamics, Sb. Math., 207 (2016), 610-638.  doi: 10.1070/SM8549.

[11]

V. V. ChepyzhovE. S. Titi and M. I. Vishik, On the convergence of solutions of the Leray-$\alpha$ model to the trajectory attractor of the 3D Navier-Stokes system, Discrete Contin. Dyn. Syst., 17 (2007), 481-500.  doi: 10.3934/dcds.2007.17.481.

[12]

V. V. Chepyzhov and M. I. Vishik, Trajectory attractors for evolution equations, C. R. Acad. Sci. Paris Sér. I Math., 321 (1995), 1309-1314. 

[13]

V. V. Chepyzhov and M. I. Vishik, Evolution equations and their trajectory attractors, J. Math. Pures Appl., 76 (1997), 913-964.  doi: 10.1016/S0021-7824(97)89978-3.

[14]

V. Chepyzhov and M. Vishik, A Hausdorff dimension estimate for kernel sections of non-autonomous evolution equations, Indiana Univ. Math. J., 42 (1993), 1057-1076.  doi: 10.1512/iumj.1993.42.42049.

[15]

V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics, American Mathematical Society, Providence, RI, 2002.

[16]

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

[17]

A. V. Fursikov, Optimal Control of Distributed Systems. Theory and Applications, American Mathematical Society, Providence, 2000. doi: 10.1090/mmono/187.

[18]

J. García-LuengoP. Marín-Rubio and J. Real, Pullback attractors for 2D Navier-Stokes equations with delays and their regularity, Adv. Nonlinear Stud., 13 (2013), 331-357.  doi: 10.1515/ans-2013-0205.

[19]

V. K. Kalantarov and E. S. Titi, Global attractors and determining modes for the 3D Navier-Stokes-Voight equations, Chin. Ann. Math. Ser. B, 30 (2009), 697-714.  doi: 10.1007/s11401-009-0205-3.

[20]

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

[21]

O. A. Ladyzhenskaya, The Mathematical Theory of Viscous Incompressible Flow, Gordon and Breach Science Publishers, New-York, NY, 1969.

[22]

O. A. Ladyzhenskaya, A dynamical system generated by the Navier-Stokes equations, Journal of Soviet Mathematics, 3 (1975), 458-479.  doi: 10.1007/BF01084684.

[23]

O. A. Ladyzhenskaya, On the determination of minimal global attractors for the Navier-Stokes and other partial differential equations, Russian Mathematical Surveys, 42 (1987), 27-73.  doi: 10.1070/RM1987v042n06ABEH001503.

[24]

O. A. Ladyzhenskaya, On the global unique solvability of some two-dimensional problems for the water solutions of polymers, J. Math. Sci., 99 (2000), 888-897.  doi: 10.1007/BF02673597.

[25]

O. A. Ladyzhenskaya, On some gaps in two of my papers on the Navier-Stokes equations and the way of closing them, J. Math. Sci., 115 (2003), 2789-2791.  doi: 10.1023/A:1023321903383.

[26]

W. Noll and C. Truesdell, The Non-Linear Field Theories of Mechanics, 3$^{rd}$ edition, Springer-Verlag, Berlin, 2004. doi: 10.1007/978-3-662-10388-3.

[27]

A. P. Oskolkov, Solvability in the large of the first boundary value problem for a certain quasilinear third order system that is encountered in the study of the motion of a viscous fluid, Zap. Naučn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI), 27 (1972), 145–160 (in Russian).

[28]

A. P. Oskolkov, The uniqueness and solvability in the large of boundary value problems for the equations of motion of aqueous solutions of polymers, Zap. Naučn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI), 38 (1973), 98–136 (in Russian).

[29]

A. P. Oskolkov, Some quasilinear systems that arise in the study of the motion of viscous fluids, Zap. Naučn. Sem. Leningrad. Otdel. Mat. Inst. Steklov (LOMI), 52 (1975), 128–157 (in Russian).

[30]

A. P. Oskolkov, Theory of nonstationary flows of Kelvin–Voigt fluids, Journal of Soviet Mathematics, 28 (1985), 751-758.  doi: 10.1007/BF02112340.

[31]

A. P. Oskolkov, Initial-boundary value problems for equations of motion of Kelvin–Voight fluids and Oldroyd fluids, Steklov Inst. Math., 179 (1989), 137-182. 

[32]

V. A. Pavlovsky, On theoretical description of weak aqueous solutions of polymers, Dokl. Akad. Nauk SSSR, 200 (1971), 809–812 (in Russian).

[33]

R. S. Rivlin and J. L. Ericksen, Stress-deformation relations for isotropic materials, J. Rational Mech. Anal., 4 (1955), 323-425.  doi: 10.1512/iumj.1955.4.54011.

[34]

G. R. Sell, Global attractors for the three-dimensional Navier-Stokes equations, J. Dynam. Differential Equations, 8 (1996), 1-33.  doi: 10.1007/BF02218613.

[35]

G. A. Seregin, On a dynamical system generated by the two-dimensional equations of the motion of a Bingham fluid, J. Math. Sci., 70 (1994), 1806-1816.  doi: 10.1007/BF02149150.

[36]

J. Simon, Compact sets in the space $L^p(0, T; B)$, Ann. Mat. Pura Appl., 146 (1986), 65-96.  doi: 10.1007/BF01762360.

[37]

S. L. Sobolev, Some Applications of Functional Analysis in Mathematical Physics, vol. 90, American Mathematical Society, Providence, 1991. doi: 10.1090/mmono/090.

[38]

V. A. Solonnikov, Estimates of Green's tensors for some boundary-value problems, Dokl. Akad. Nauk SSSR, 130 (1960), 988–991 (in Russian).

[39]

R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, 2$^nd$ edition, Springer-Verlag, New York, NY, 1997. doi: 10.1007/978-1-4612-0645-3.

[40]

R. Temam, Navier-Stokes Equations: Theory and Numerical Analysis, AMS Chelsea Publishing, Providence, RI, 2001. doi: 10.1090/chel/343.

[41]

M. V. Turbin and A. S. Ustiuzhaninova, The existence theorem for a weak solution to initial-boundary value problem for system of equations describing the motion of weak aqueous polymer solutions, Russian Math., 63 (2019), 54-69.  doi: 10.3103/S1066369X19080061.

[42]

A. S. Ustiuzhaninova and M. V. Turbin, Feedback control problem for modified Kelvin–Voigt model, Journal of Dynamical and Control Systems, early access, 2021. doi: 10.1007/s10883-021-09539-0.

[43]

A. S. Ustiuzhaninova and M. V. Turbin, Trajectory and global attractors for a modified Kelvin–Voigt model, J. Appl. Ind. Math., 15 (2021), 158-168.  doi: 10.1134/S1990478921010142.

[44]

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

[45]

D. A. Vorotnikov and V. G. Zvyagin, Uniform attractors for non-autonomous motion equations of viscoelastic medium, J. Math. Anal. Appl., 325 (2007), 438-458.  doi: 10.1016/j.jmaa.2006.01.078.

[46]

I. I. Vorovich and V. I. Yudovich, Steady flow of a viscous incompressible fluid, Mat. Sb., 53 (1961), 393–428 (in Russian).

[47]

C. ZhaoW. Sun and C. H. Hsu, Pullback dynamical behaviors of the non-autonomous micropolar fluid flows, Dyn. Partial Differ. Equ., 12 (2015), 265-288.  doi: 10.4310/DPDE.2015.v12.n3.a4.

[48]

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

[49]

A. Zvyagin, Attractors for model of polymer solutions motion, Discrete Contin. Dyn. Syst., 38 (2018), 6305-6325.  doi: 10.3934/dcds.2018269.

[50]

V. Zvyagin and S. Kondratyev, Pullback attractors of the Jeffreys-Oldroyd equations, J. Differential Equations, 260 (2016), 5026-5042.  doi: 10.1016/j.jde.2015.11.038.

[51]

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

[52]

V. G. Zvyagin and M. V. Turbin, Mathematical Problems in Viscoelastic Hydrodynamics, Krasand URSS, Moscow, 2012 (in Russian).

[53]

V. G. Zvyagin and D. A. Vorotnikov, Topological Approximation Methods for Evolutionary Problems of Nonlinear Hydrodynamics, vol. 12, Walter de Gruyter, Berlin, 2008. doi: 10.1515/9783110208283.

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