March  2016, 36(3): 1539-1562. doi: 10.3934/dcds.2016.36.1539

On the existence of global strong solutions to the equations modeling a motion of a rigid body around a viscous fluid

1. 

Mathematical Institute, Academy of Sciences of the Czech Republic, Žitná 25, 115 67 Praha 1

2. 

Department of Mathematics, Humboldt University Berlin, Unter den Linden 6, 10099 Berlin, Germany

Received  August 2014 Revised  June 2015 Published  August 2015

The paper deals with the global existence of strong solution to the equations modeling a motion of a rigid body around viscous fluid. Moreover, the estimates of second gradients of velocity and pressure are given.
Citation: Šárka Nečasová, Joerg Wolf. On the existence of global strong solutions to the equations modeling a motion of a rigid body around a viscous fluid. Discrete and Continuous Dynamical Systems, 2016, 36 (3) : 1539-1562. doi: 10.3934/dcds.2016.36.1539
References:
[1]

W. Borchers, Zur Stabilität und Faktorisienrungsmethode für die Navier-Stokes Gleichungen inkompressibler viskoser Flüssigkeiten, Habilitationsschrift, University of Paderborn, 1992.

[2]

R. Coifman, P. L. Lions, Y. Meyers and S. Semmes, Compensated compacteness and Hardy spaces, J. Math. Pures Appl., 72 (1993), 247-286.

[3]

P. Cumsille and M. Tucsnak, Wellposedness for the Navier-Stokes flow in the exterior of a rotating obstacle, Math. Meth. Appl. Sci., 29 (2006), 595-623. doi: 10.1002/mma.702.

[4]

P. Cumsille and T. Takahashi, Wellposedness for the system modelling the motion of a rigid body of artbitrary form in an incompressible viscous fluid, Czechoslovak Math. J., 58 (2008), 961-992. doi: 10.1007/s10587-008-0063-2.

[5]

E. Dintelmann, M. Geissert and M. Hieber, Strong $L^p$ solutions to the Navier- Stokes flow past moving obstacles: The case of several obstacles and time dependent velocity, Trans. Amer. Math. Soc., 361 (2009), 653-669. doi: 10.1090/S0002-9947-08-04684-9.

[6]

C. Foias, C. Guillopé and R. Temam, New a priori estimates for Navier-Stokes equations in dimension 3, Comm. Partial Differential Equations, 6 (1981), 329-359. doi: 10.1080/03605308108820180.

[7]

H. Fujita and T. Kato, On the Navier-Stokes initial value problem. I, Arch. Rational Mech. Anal., 16 (1964), 269-315. doi: 10.1007/BF00276188.

[8]

G. P. Galdi and A. S. Silvestre, Strong solutions to the Navier-Stokes equations around a rotating obstacle, Arch. Rat. Mech. Anal., 176 (2005), 331-350. doi: 10.1007/s00205-004-0348-z.

[9]

G. P. Galdi and A. S. Silvestre, Strong solution to the problem of motion of a rigid body in a Navier-Stokes liquid under the action of prescribed forces and torques, in Nonlinear Problems in Mathematical Physics and Related Topics (in honor of Prof. O. A. Ladyzhenskaya), Int. Math. Ser. (N.Z.), 1, Kluwer/Plenum, N. Y., 2002, 121-144. doi: 10.1007/978-1-4615-0777-2_8.

[10]

G. P. Galdi, On the motion of a rigid body in a viscous liquid: A mathematical analysis with applications, in Handbook of Mathematical Fluid Dynamics, Vol. 1 (eds. S. Friedlander and D. Serre), Elsevier, 2002, 653-791.

[11]

G. P. Galdi, An Introduction to the Mathematical Theory of the Navier-Stokes Equations, Vol. I: Linearized Steady Problems, Springer-Verlag, New York, 1994. doi: 10.1007/978-1-4612-5364-8.

[12]

M. Geissert, H. Heck and M. Hieber, $L^p$ theory of the Navier-Stokes flow in the exterior of a moving or rotating obstacle, J. Reine Angew. Math., 596 (2006), 45-62. doi: 10.1515/CRELLE.2006.051.

[13]

X. Giga and H. Sohr, Abstract $L^p$ estimates for the Cauchy problem with applications to the Navier-Stokes equations in exterior domains, J. Funct. Anal., 102 (1991), 72-94. doi: 10.1016/0022-1236(91)90136-S.

[14]

C. Grandmont and Y. Maday, Existence for an unsteady fluid-structure interaction problem, M2AN Math. Model. Numer. Anal., 34 (2000), 609-636. doi: 10.1051/m2an:2000159.

[15]

T. Hishida, An existence theorem for the Navier-Stokes flow in the exterior of a rotating obstacle, Arch. Rational Mech. Anal., 150 (1999), 307-348. doi: 10.1007/s002050050190.

[16]

T. Hishida and Y. Shibata, $L_p-L_q$ estimate of Stokes operator and Navier-Stokes flows in the exterior of a rotating obstacle, Arch. Ration. Mech. Anal., 193 (2009), 339-421. doi: 10.1007/s00205-008-0130-8.

[17]

A. Inoue and M. Wakimoto, On existence of solutions of the Navier-Stokes equation in a time dependent domain, J. Fac. Sci. Univ. Tokyo Sect. IA Math., 24 (1977), 303-319.

[18]

O. A. Ladyzhenskaya, An initial-boundary value problem for the Navier-Stokes equations in domains with boundary changing in time, Zap. Naučn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI), 11 (1968), 97-128.

[19]

Š. Nečasová and J. Wolf, On the linear problem arising from motion of fluid around moving rigid body, Matematika Bohemika, 140, (2015), 241-259.

[20]

J. Neustupa, Existence of a weak solution to the Navier-Stokes equation in a general time-varying domain by the Rothe method, Math. Methods Appl. Sci., 32 (2009), 653-683. doi: 10.1002/mma.1059.

[21]

J. Neustupa and P. Penel, A weak solvability of the Navier-Stokes equation with Navier's boundary condition around a ball striking the wall, in Advances in Mathematical Fluid Mechanics, Springer, Berlin, 2010, 385-407. doi: 10.1007/978-3-642-04068-9_24.

[22]

J. Neustupa and P. Penel, A weak solvability of the Navier-Stokes system with Navier's boundary condition around moving and striking bodies, J. Math. Pures Appl., 2010.

[23]

D. Serre, Chute libre d'un solide dans un fluids visqueux incompressible. Existence, Japan J. Appl. Math., 4 (1987), 99-110. doi: 10.1007/BF03167757.

[24]

E. M. Stein, Singular Integrals and Differentaibility Properties of Functions, Princeton University Press, Princeton, 1970.

[25]

T. Takahashi and M. Tucsnak, Global Strong solutions for the two-dimensional motion of an infite cylinder in a viscous fluid, J. Math. Fluid Mech., 6 (2004), 53-77. doi: 10.1007/s00021-003-0083-4.

[26]

T. Takahashi, Existence of strong solution for the problem of a rigid - fluid system, C. R. Acad. Sci. Paris, Ser. I, 336 (2003), 453-458. doi: 10.1016/S1631-073X(03)00081-5.

[27]

T. Takahashi, Analysis of strong solutions for the equations modeling the motion of a rigid-fluid system in a bounded domain, Adv. Differential Equations, 8 (2003), 1499-1532.

[28]

J. L. Vázquez and E. Zuazua, Large time behavior for a simplified 1D model of fluid-solid interaction, Comm. Partial Differential Equations, 28 (2003), 1705-1738. doi: 10.1081/PDE-120024530.

[29]

J. Wolf, On the pressure of strong solutions to the Stokes system in bounded and exterior domains,, , (). 

show all references

References:
[1]

W. Borchers, Zur Stabilität und Faktorisienrungsmethode für die Navier-Stokes Gleichungen inkompressibler viskoser Flüssigkeiten, Habilitationsschrift, University of Paderborn, 1992.

[2]

R. Coifman, P. L. Lions, Y. Meyers and S. Semmes, Compensated compacteness and Hardy spaces, J. Math. Pures Appl., 72 (1993), 247-286.

[3]

P. Cumsille and M. Tucsnak, Wellposedness for the Navier-Stokes flow in the exterior of a rotating obstacle, Math. Meth. Appl. Sci., 29 (2006), 595-623. doi: 10.1002/mma.702.

[4]

P. Cumsille and T. Takahashi, Wellposedness for the system modelling the motion of a rigid body of artbitrary form in an incompressible viscous fluid, Czechoslovak Math. J., 58 (2008), 961-992. doi: 10.1007/s10587-008-0063-2.

[5]

E. Dintelmann, M. Geissert and M. Hieber, Strong $L^p$ solutions to the Navier- Stokes flow past moving obstacles: The case of several obstacles and time dependent velocity, Trans. Amer. Math. Soc., 361 (2009), 653-669. doi: 10.1090/S0002-9947-08-04684-9.

[6]

C. Foias, C. Guillopé and R. Temam, New a priori estimates for Navier-Stokes equations in dimension 3, Comm. Partial Differential Equations, 6 (1981), 329-359. doi: 10.1080/03605308108820180.

[7]

H. Fujita and T. Kato, On the Navier-Stokes initial value problem. I, Arch. Rational Mech. Anal., 16 (1964), 269-315. doi: 10.1007/BF00276188.

[8]

G. P. Galdi and A. S. Silvestre, Strong solutions to the Navier-Stokes equations around a rotating obstacle, Arch. Rat. Mech. Anal., 176 (2005), 331-350. doi: 10.1007/s00205-004-0348-z.

[9]

G. P. Galdi and A. S. Silvestre, Strong solution to the problem of motion of a rigid body in a Navier-Stokes liquid under the action of prescribed forces and torques, in Nonlinear Problems in Mathematical Physics and Related Topics (in honor of Prof. O. A. Ladyzhenskaya), Int. Math. Ser. (N.Z.), 1, Kluwer/Plenum, N. Y., 2002, 121-144. doi: 10.1007/978-1-4615-0777-2_8.

[10]

G. P. Galdi, On the motion of a rigid body in a viscous liquid: A mathematical analysis with applications, in Handbook of Mathematical Fluid Dynamics, Vol. 1 (eds. S. Friedlander and D. Serre), Elsevier, 2002, 653-791.

[11]

G. P. Galdi, An Introduction to the Mathematical Theory of the Navier-Stokes Equations, Vol. I: Linearized Steady Problems, Springer-Verlag, New York, 1994. doi: 10.1007/978-1-4612-5364-8.

[12]

M. Geissert, H. Heck and M. Hieber, $L^p$ theory of the Navier-Stokes flow in the exterior of a moving or rotating obstacle, J. Reine Angew. Math., 596 (2006), 45-62. doi: 10.1515/CRELLE.2006.051.

[13]

X. Giga and H. Sohr, Abstract $L^p$ estimates for the Cauchy problem with applications to the Navier-Stokes equations in exterior domains, J. Funct. Anal., 102 (1991), 72-94. doi: 10.1016/0022-1236(91)90136-S.

[14]

C. Grandmont and Y. Maday, Existence for an unsteady fluid-structure interaction problem, M2AN Math. Model. Numer. Anal., 34 (2000), 609-636. doi: 10.1051/m2an:2000159.

[15]

T. Hishida, An existence theorem for the Navier-Stokes flow in the exterior of a rotating obstacle, Arch. Rational Mech. Anal., 150 (1999), 307-348. doi: 10.1007/s002050050190.

[16]

T. Hishida and Y. Shibata, $L_p-L_q$ estimate of Stokes operator and Navier-Stokes flows in the exterior of a rotating obstacle, Arch. Ration. Mech. Anal., 193 (2009), 339-421. doi: 10.1007/s00205-008-0130-8.

[17]

A. Inoue and M. Wakimoto, On existence of solutions of the Navier-Stokes equation in a time dependent domain, J. Fac. Sci. Univ. Tokyo Sect. IA Math., 24 (1977), 303-319.

[18]

O. A. Ladyzhenskaya, An initial-boundary value problem for the Navier-Stokes equations in domains with boundary changing in time, Zap. Naučn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI), 11 (1968), 97-128.

[19]

Š. Nečasová and J. Wolf, On the linear problem arising from motion of fluid around moving rigid body, Matematika Bohemika, 140, (2015), 241-259.

[20]

J. Neustupa, Existence of a weak solution to the Navier-Stokes equation in a general time-varying domain by the Rothe method, Math. Methods Appl. Sci., 32 (2009), 653-683. doi: 10.1002/mma.1059.

[21]

J. Neustupa and P. Penel, A weak solvability of the Navier-Stokes equation with Navier's boundary condition around a ball striking the wall, in Advances in Mathematical Fluid Mechanics, Springer, Berlin, 2010, 385-407. doi: 10.1007/978-3-642-04068-9_24.

[22]

J. Neustupa and P. Penel, A weak solvability of the Navier-Stokes system with Navier's boundary condition around moving and striking bodies, J. Math. Pures Appl., 2010.

[23]

D. Serre, Chute libre d'un solide dans un fluids visqueux incompressible. Existence, Japan J. Appl. Math., 4 (1987), 99-110. doi: 10.1007/BF03167757.

[24]

E. M. Stein, Singular Integrals and Differentaibility Properties of Functions, Princeton University Press, Princeton, 1970.

[25]

T. Takahashi and M. Tucsnak, Global Strong solutions for the two-dimensional motion of an infite cylinder in a viscous fluid, J. Math. Fluid Mech., 6 (2004), 53-77. doi: 10.1007/s00021-003-0083-4.

[26]

T. Takahashi, Existence of strong solution for the problem of a rigid - fluid system, C. R. Acad. Sci. Paris, Ser. I, 336 (2003), 453-458. doi: 10.1016/S1631-073X(03)00081-5.

[27]

T. Takahashi, Analysis of strong solutions for the equations modeling the motion of a rigid-fluid system in a bounded domain, Adv. Differential Equations, 8 (2003), 1499-1532.

[28]

J. L. Vázquez and E. Zuazua, Large time behavior for a simplified 1D model of fluid-solid interaction, Comm. Partial Differential Equations, 28 (2003), 1705-1738. doi: 10.1081/PDE-120024530.

[29]

J. Wolf, On the pressure of strong solutions to the Stokes system in bounded and exterior domains,, , (). 

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