# American Institute of Mathematical Sciences

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 & Continuous Dynamical Systems - A, 2016, 36 (3) : 1539-1562. doi: 10.3934/dcds.2016.36.1539
##### References:
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Soc., 361 (2009), 653.  doi: 10.1090/S0002-9947-08-04684-9.  Google Scholar [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.  doi: 10.1080/03605308108820180.  Google Scholar [7] H. Fujita and T. Kato, On the Navier-Stokes initial value problem. I,, Arch. Rational Mech. Anal., 16 (1964), 269.  doi: 10.1007/BF00276188.  Google Scholar [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.  doi: 10.1007/s00205-004-0348-z.  Google Scholar [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), (2002), 121.  doi: 10.1007/978-1-4615-0777-2_8.  Google Scholar [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, (2002), 653.   Google Scholar [11] G. P. Galdi, An Introduction to the Mathematical Theory of the Navier-Stokes Equations, Vol. I: Linearized Steady Problems,, Springer-Verlag, (1994).  doi: 10.1007/978-1-4612-5364-8.  Google Scholar [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.  doi: 10.1515/CRELLE.2006.051.  Google Scholar [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.  doi: 10.1016/0022-1236(91)90136-S.  Google Scholar [14] C. Grandmont and Y. Maday, Existence for an unsteady fluid-structure interaction problem,, M2AN Math. Model. Numer. Anal., 34 (2000), 609.  doi: 10.1051/m2an:2000159.  Google Scholar [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.  doi: 10.1007/s002050050190.  Google Scholar [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.  doi: 10.1007/s00205-008-0130-8.  Google Scholar [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.   Google Scholar [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.   Google Scholar [19] Š. Nečasová and J. Wolf, On the linear problem arising from motion of fluid around moving rigid body,, Matematika Bohemika, 140 (2015), 241.   Google Scholar [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.  doi: 10.1002/mma.1059.  Google Scholar [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, (2010), 385.  doi: 10.1007/978-3-642-04068-9_24.  Google Scholar [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).   Google Scholar [23] D. Serre, Chute libre d'un solide dans un fluids visqueux incompressible. Existence,, Japan J. Appl. Math., 4 (1987), 99.  doi: 10.1007/BF03167757.  Google Scholar [24] E. M. Stein, Singular Integrals and Differentaibility Properties of Functions,, Princeton University Press, (1970).   Google Scholar [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.  doi: 10.1007/s00021-003-0083-4.  Google Scholar [26] T. Takahashi, Existence of strong solution for the problem of a rigid - fluid system,, C. R. Acad. Sci. Paris, 336 (2003), 453.  doi: 10.1016/S1631-073X(03)00081-5.  Google Scholar [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.   Google Scholar [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.  doi: 10.1081/PDE-120024530.  Google Scholar [29] J. Wolf, On the pressure of strong solutions to the Stokes system in bounded and exterior domains,, , ().   Google Scholar

show all references

##### References:
 [1] W. Borchers, Zur Stabilität und Faktorisienrungsmethode für die Navier-Stokes Gleichungen inkompressibler viskoser Flüssigkeiten,, Habilitationsschrift, (1992).   Google Scholar [2] R. Coifman, P. L. Lions, Y. Meyers and S. Semmes, Compensated compacteness and Hardy spaces,, J. Math. Pures Appl., 72 (1993), 247.   Google Scholar [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.  doi: 10.1002/mma.702.  Google Scholar [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.  doi: 10.1007/s10587-008-0063-2.  Google Scholar [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.  doi: 10.1090/S0002-9947-08-04684-9.  Google Scholar [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.  doi: 10.1080/03605308108820180.  Google Scholar [7] H. Fujita and T. Kato, On the Navier-Stokes initial value problem. I,, Arch. Rational Mech. Anal., 16 (1964), 269.  doi: 10.1007/BF00276188.  Google Scholar [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.  doi: 10.1007/s00205-004-0348-z.  Google Scholar [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), (2002), 121.  doi: 10.1007/978-1-4615-0777-2_8.  Google Scholar [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, (2002), 653.   Google Scholar [11] G. P. Galdi, An Introduction to the Mathematical Theory of the Navier-Stokes Equations, Vol. I: Linearized Steady Problems,, Springer-Verlag, (1994).  doi: 10.1007/978-1-4612-5364-8.  Google Scholar [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.  doi: 10.1515/CRELLE.2006.051.  Google Scholar [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.  doi: 10.1016/0022-1236(91)90136-S.  Google Scholar [14] C. Grandmont and Y. Maday, Existence for an unsteady fluid-structure interaction problem,, M2AN Math. Model. Numer. Anal., 34 (2000), 609.  doi: 10.1051/m2an:2000159.  Google Scholar [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.  doi: 10.1007/s002050050190.  Google Scholar [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.  doi: 10.1007/s00205-008-0130-8.  Google Scholar [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.   Google Scholar [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.   Google Scholar [19] Š. Nečasová and J. Wolf, On the linear problem arising from motion of fluid around moving rigid body,, Matematika Bohemika, 140 (2015), 241.   Google Scholar [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.  doi: 10.1002/mma.1059.  Google Scholar [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, (2010), 385.  doi: 10.1007/978-3-642-04068-9_24.  Google Scholar [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).   Google Scholar [23] D. Serre, Chute libre d'un solide dans un fluids visqueux incompressible. Existence,, Japan J. Appl. Math., 4 (1987), 99.  doi: 10.1007/BF03167757.  Google Scholar [24] E. M. Stein, Singular Integrals and Differentaibility Properties of Functions,, Princeton University Press, (1970).   Google Scholar [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.  doi: 10.1007/s00021-003-0083-4.  Google Scholar [26] T. Takahashi, Existence of strong solution for the problem of a rigid - fluid system,, C. R. Acad. Sci. Paris, 336 (2003), 453.  doi: 10.1016/S1631-073X(03)00081-5.  Google Scholar [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.   Google Scholar [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.  doi: 10.1081/PDE-120024530.  Google Scholar [29] J. Wolf, On the pressure of strong solutions to the Stokes system in bounded and exterior domains,, , ().   Google Scholar
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