\(L^p\)-\(L^q\) maximal regularity for some operators associated with linearized incompressible fluid-rigid body problems. (English) Zbl 1401.76042
Danchin, Raphaël (ed.) et al., Mathematical analysis in fluid mechanics: selected recent results. International conference on vorticity, rotation and symmetry (IV) – complex fluids and the issue of regularity, CIRM, Luminy, Marseille, France, May 8–12, 2017. Proceedings. Providence, RI: American Mathematical Society (AMS) (ISBN 978-1-4704-3646-9/pbk; 978-1-4704-4807-3/ebook). Contemporary Mathematics 710, 175-201 (2018).
Summary: We study an unbounded operator arising naturally after linearizing the system modelling the motion of a rigid body in a viscous incompressible fluid. We show that this operator is \(\mathcal {R}\) sectorial in \(L^q\) for every \(q\in (1,\infty )\), thus it has the maximal \(L^p\)-\(L^q\) regularity property. Moreover, we show that the generated semigroup is exponentially stable with respect to the \(L^q\) norm. Finally, we use the results to prove the global existence for small initial data, in an \(L^p\)-\(L^q\) setting, for the original nonlinear problem.
For the entire collection see [Zbl 1396.35001].
For the entire collection see [Zbl 1396.35001].
MSC:
| 76D03 | Existence, uniqueness, and regularity theory for incompressible viscous fluids |
| 35Q30 | Navier-Stokes equations |
| 76N10 | Existence, uniqueness, and regularity theory for compressible fluids and gas dynamics |
References:
| [1] | Amann, Herbert, Linear and quasilinear parabolic problems. Vol. I, Monographs in Mathematics 89, xxxvi+335 pp. (1995), Birkh\`“auser Boston, Inc., Boston, MA · Zbl 0819.35001 |
| [2] | Amann, Herbert, On the strong solvability of the Navier-Stokes equations, J. Math. Fluid Mech., 2, 1, 16-98 (2000) · Zbl 0989.35107 |
| [3] | Bensoussan, Alain; Da Prato, Giuseppe; Delfour, Michel C.; Mitter, Sanjoy K., Representation and control of infinite dimensional systems, Systems & Control: Foundations & Applications, xxviii+575 pp. (2007), Birkh\`“auser Boston, Inc., Boston, MA · Zbl 1117.93002 |
| [4] | Cumsille, Patricio; Takahashi, Tak\'eo, Wellposedness for the system modelling the motion of a rigid body of arbitrary form in an incompressible viscous fluid, Czechoslovak Math. J., 58(133), 4, 961-992 (2008) · Zbl 1174.35092 |
| [5] | Denk, Robert; Hieber, Matthias; Pr\`“uss, Jan, \( \mathcal{R} \)-boundedness, Fourier multipliers and problems of elliptic and parabolic type, Mem. Amer. Math. Soc., 166, 788, viii+114 pp. (2003) · Zbl 1274.35002 |
| [6] | Dore, Giovanni, \(L^p\) regularity for abstract differential equations. Functional analysis and related topics, 1991 (Kyoto), Lecture Notes in Math. 1540, 25-38 (1993), Springer, Berlin · Zbl 0818.47044 |
| [7] | Engel, Klaus-Jochen; Nagel, Rainer, One-parameter semigroups for linear evolution equations, Graduate Texts in Mathematics 194, xxii+586 pp. (2000), Springer-Verlag, New York · Zbl 0952.47036 |
| [8] | Fabes, Eugene; Mendez, Osvaldo; Mitrea, Marius, Boundary layers on Sobolev-Besov spaces and Poisson’s equation for the Laplacian in Lipschitz domains, J. Funct. Anal., 159, 2, 323-368 (1998) · Zbl 0930.35045 |
| [9] | Fujiwara, Daisuke; Morimoto, Hiroko, An \(L_r\)-theorem of the Helmholtz decomposition of vector fields, J. Fac. Sci. Univ. Tokyo Sect. IA Math., 24, 3, 685-700 (1977) · Zbl 0386.35038 |
| [10] | Galdi, Giovanni P., On the motion of a rigid body in a viscous liquid: a mathematical analysis with applications. Handbook of mathematical fluid dynamics, Vol. I, 653-791 (2002), North-Holland, Amsterdam · Zbl 1230.76016 |
| [11] | Geissert, Matthias; G\`“otze, Karoline; Hieber, Matthias, \(L^p\)-theory for strong solutions to fluid-rigid body interaction in Newtonian and generalized Newtonian fluids, Trans. Amer. Math. Soc., 365, 3, 1393-1439 (2013) · Zbl 1282.35273 |
| [12] | Geissert, Matthias; Hess, Matthias; Hieber, Matthias; Schwarz, C\'eline; Stavrakidis, Kyriakos, Maximal \(L^p-L^q\)-estimates for the Stokes equation: a short proof of Solonnikov’s theorem, J. Math. Fluid Mech., 12, 1, 47-60 (2010) · Zbl 1261.35120 |
| [13] | G\`“otze, Karoline, Maximal \(L^p\)-regularity for a 2D fluid-solid interaction problem. Spectral theory, mathematical system theory, evolution equations, differential and difference equations, Oper. Theory Adv. Appl. 221, 373-384 (2012), Birkh\'”auser/Springer Basel AG, Basel · Zbl 1254.35190 |
| [14] | Happel, John; Brenner, Howard, Low Reynolds number hydrodynamics with special applications to particulate media, xiii+553 pp. (1965), Prentice-Hall, Inc., Englewood Cliffs, N.J. · Zbl 0612.76032 |
| [15] | Kunstmann, Peer Christian; Weis, Lutz, Perturbation theorems for maximal \(L_p\)-regularity, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 30, 2, 415-435 (2001) · Zbl 1065.47008 |
| [16] | Kunstmann, Peer C.; Weis, Lutz, Maximal \(L_p\)-regularity for parabolic equations, Fourier multiplier theorems and \(H^\infty \)-functional calculus. Functional analytic methods for evolution equations, Lecture Notes in Math. 1855, 65-311 (2004), Springer, Berlin · Zbl 1097.47041 |
| [17] | Maity, Debayan; Raymond, Jean-Pierre, Feedback stabilization of the incompressible Navier-Stokes equations coupled with a damped elastic system in two dimensions, J. Math. Fluid Mech., 19, 4, 773-805 (2017) · Zbl 1386.93153 |
| [18] | D. Maity and M. Tucsnak, A Maximal Regularity Approach to the Analysis of Some Particulate Flows, Springer International Publishing, Cham, 2017, pp. 1-75. · Zbl 1387.35459 |
| [19] | Raymond, J.-P., Stokes and Navier-Stokes equations with nonhomogeneous boundary conditions, Ann. Inst. H. Poincar\'e Anal. Non Lin\'eaire, 24, 6, 921-951 (2007) · Zbl 1136.35070 |
| [20] | Raymond, Jean-Pierre, Feedback stabilization of a fluid-structure model, SIAM J. Control Optim., 48, 8, 5398-5443 (2010) · Zbl 1213.93177 |
| [21] | Takahashi, Tak\'eo, Analysis of strong solutions for the equations modeling the motion of a rigid-fluid system in a bounded domain, Adv. Differential Equations, 8, 12, 1499-1532 (2003) · Zbl 1101.35356 |
| [22] | Takahashi, Tak\'eo; Tucsnak, Marius, Global strong solutions for the two-dimensional motion of an infinite cylinder in a viscous fluid, J. Math. Fluid Mech., 6, 1, 53-77 (2004) · Zbl 1054.35061 |
| [23] | Triebel, Hans, Interpolation theory, function spaces, differential operators, 532 pp. (1995), Johann Ambrosius Barth, Heidelberg · Zbl 0830.46028 |
| [24] | Wang, Yun; Xin, Zhouping, Analyticity of the semigroup associated with the fluid-rigid body problem and local existence of strong solutions, J. Funct. Anal., 261, 9, 2587-2616 (2011) · Zbl 1238.47039 |
| [25] | Weis, Lutz, Operator-valued Fourier multiplier theorems and maximal \(L_p\)-regularity, Math. Ann., 319, 4, 735-758 (2001) · Zbl 0989.47025 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
