## Topological Methods in Nonlinear Analysis

### Strong solutions in $L^2$ framework for fluid-rigid body interaction problem. Mixed case

#### Abstract

The paper deals with the problem describing the motion of a rigid body inside a viscous incompressible fluid when the mixed boundary conditions are considered. At the fluid-rigid body interface the slip Navier boundary condition is prescribed, having the continuity of velocity just in the normal component, and the Dirichlet condition is given on the boundary of the fluid domain. The existence and uniqueness of the local strong solution is proven by the local transformation and the fixed point argument.

#### Article information

Source
Topol. Methods Nonlinear Anal., Volume 52, Number 1 (2018), 337-350.

Dates
First available in Project Euclid: 9 August 2018

https://projecteuclid.org/euclid.tmna/1533780030

Digital Object Identifier
doi:10.12775/TMNA.2018.028

Mathematical Reviews number (MathSciNet)
MR3867991

Zentralblatt MATH identifier
07029873

#### Citation

Al Baba, Hind; Chemetov, Nikolai V.; Nečasová, Šárka; Muha, Boris. Strong solutions in $L^2$ framework for fluid-rigid body interaction problem. Mixed case. Topol. Methods Nonlinear Anal. 52 (2018), no. 1, 337--350. doi:10.12775/TMNA.2018.028. https://projecteuclid.org/euclid.tmna/1533780030

#### References

• N. Chemetov and Š. Nečasová, The motion of the rigid body in the viscous fluid including collisions. Global solvability result, Nonlinear Anal. Real World Appl. 34 (2017), 416–445.
• C. Conca, J. San Martin and M. Tucsnak, Existence of solutions for the equations modelling the motion of a rigid body in a viscous fluid, Comm. Partial Differential Equations 25 (2000), 1019–1042.
• B. Desjardins and M.J. Esteban, Existence of weak solutions for the motion of rigid bodies in a viscous fluid, Arch. Rational Mech. Anal. 146 (1999), 59–71.
• B. Desjardins and M.J. Esteban, On weak solutions for fluid-rigid structure interaction\rom: Compressible and incompressible models, Comm. Partial Differential Equations 25 (2000), 1399–1413.
• G.P. Galdi, On the motion of a rigid body in a viscous liquid: A mathematical analysis with applications, Handbook of Mathematical Fluid Dynamics, Vol. 1, (Friedlander, D. Serre, ed.), Elsevier, 2002.
• D. Gérard-Varet and M. Hillairet, Existence of weak solutions up to collision for viscous fluid-solid systems with slip, Comm. Pure Appl. Math. 67 (2014), no. 12, 2022–2075.
• D. Gérard-Varet, M. Hillairet and C. Wang, The influence of boundary conditions on the contact problem in a\rom3D Navier–Stokes flow, J. Math. Pures Appl. (9) 103 (2015), no. 1, 1–38.
• M.D. Gunzburger, H. Lee and G. Seregin, Global existence of weak solutions for viscous incompressible flows around a moving rigid body in three dimensions, J. Math. Fluid Mech. 2 (2000), no. 3, 219–266.
• T.I. Hesla, Collision of smooth bodies in a viscous fluid\rom: A mathematical investigation, PhD Thesis, Minnesota, 2005.
• M. Hillairet, Lack of collision between solid bodies in a \rom2D incompressible viscous flow, Comm. Partial Differential Equations 32 (2007), no. 7–9, 1345–1371.
• K.-H. Hoffmann and V.N. Starovoitov, On a motion of a solid body in a viscous fluid. Two dimensional case, Adv. Math. Sci. Appl. 9 (1999), 633–648.
• 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), no. 2, 303–319.
• T. Kato, Fractional powers of dissipative operators, J. Math. Soc. Japan 13 (1961), 246–274.
• T. Kato, Abstract evolution equations of parabolic type in Banach and Hilbert spaces, Nagoya Math. J. 19 (1961), 93–125.
• J. Neustupa and P. Penel, Existence of a weak solution to the Navier–Stokes equation with Navier's boundary condition around striking bodies, Comptes Rendus Mathematique 347 (2009), no. 11–12, 685–690.
• J. Neustupa and P. Penel, A Weak solvability of the Navier–Stokes equation with Navier's boundary condition around a ball striking the wall, Advances in Mathematical Fluid Mechanics: Dedicated to Giovanni Paolo Galdi, Springer–Verlag Berlin, 2010, pp. 385–408.
• Y. Shibata and R. Shimada, On a generalized resolvent estimate for the Stokes system with Robin boundary condition, J. Math. Soc. Japan 59 (2007), no. 2, 469–519.
• 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), no. 12, 1499–1532.
• T. Takahashi and M. Tucsnak, Global strong solutions for the two-dimensional motion of an infinite cylinder in a viscous fluid, J. Math. Fluid Mech. 6 (2004), no. 1, 53–77.
• C. Wang, Strong solutions for the fluid-solid systems in a \rom2D domain, Asymptot. Anal. 89 (2014), no. 3–4, 263–306.