×

On some free boundary problem of the Navier-Stokes equations in the maximal \(L_p - L_q\) regularity class. (English) Zbl 1319.35166

The author corrects and reproves his result presented in [Y. Shibata and S. Shimizu, Differ. Integral Equ. 20, No. 3, 241–276 (2007; Zbl 1212.35353)] by a different approach. The evolution of a domain \(\Omega \in \mathbb R^N\) (\(N \geq 2\)) is considered in time when the Navier-Stokes equations are given in \(\Omega\). It is assumed that the boundary of \(\Omega\) consists of two disjoint parts \(\Gamma\) and \(S_t\), where the non-slip condition is fulfilled on the fixed part \(\Gamma\). The zero tension and the zero velocity are given on the free part \(S_t\). The authors prove a local in time unique existence theorem for any initial data, and a global in time unique existence theorem for some small initial data.

MSC:

35Q30 Navier-Stokes equations
76D05 Navier-Stokes equations for incompressible viscous fluids
35R35 Free boundary problems for PDEs

Citations:

Zbl 1212.35353
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Abels, H., The initial-value problem for the Navier-Stokes equations with a free surface in \(L^q\)-Sobolev spaces, Adv. Differential Equations, 10, 45-64 (2005) · Zbl 1105.35072
[2] Allain, G., Small-time existence for Navier-Stokes equations with a free surface, Appl. Math. Optim., 16, 37-50 (1987) · Zbl 0655.76021
[3] Amann, H., Linear and Quasilinear Parabolic Problems (1995), Birkhäuser: Birkhäuser Basel, Vol. I · Zbl 0819.35001
[4] Beale, J. T., The initial value problem for the Navier-Stokes equations with a free surface, Comm. Pure Appl. Math., 34, 359-392 (1981) · Zbl 0464.76028
[5] Beale, T., Large-time regularity of viscous surface waves, Arch. Ration. Mech. Anal., 84, 307-352 (1983/1984) · Zbl 0545.76029
[6] Beale, J. T.; Nishida, T., Large-time behaviour of viscous surface waves, (Recent Topics in Nonlinear PDE, II. Recent Topics in Nonlinear PDE, II, North-Holland Math. Stud., vol. 128 (1985), North-Holland: North-Holland Amsterdam), 1-14
[7] Bourgain, J., Vector-valued singular integrals and the \(H^1\)-BMO duality, (Burkholder, D., Probability Theory and Harmonic Analysis (1986), Marcel Dekker: Marcel Dekker New York), 1-19 · Zbl 0602.42015
[8] Duvaut, G.; Lions, J. L., Inequalities in Mechanics and Physics, Lecture Notes in Math., vol. 393 (1976), Springer-Verlag · Zbl 0331.35002
[9] Enomoto, Y.; Shibata, Y., On the \(R\)-sectoriality and its application to some mathematical viscous compressible fluids, Funkcial. Ekvac., 56, 3, 441-505 (2013) · Zbl 1296.35118
[10] Enomoto, Y.; von Below, L.; Shibata, Y., On some free boundary problem for a compressible barotropic viscous fluid flow, Ann. Univ. Ferrara Sez. VII Sci. Mat., 60, 1, 55-89 (2014) · Zbl 1321.35264
[11] Hataya, Y., Decaying solution of a Navier-Stokes flow without surface tension, J. Math. Kyoto Univ., 49, 691-717 (2009) · Zbl 1382.35190
[12] Mikhlin, S. G., Fourier integrals and multiple singular integrals, Vestnik Leningrad Univ. Ser. Mat., 12, 143-145 (1957), (in Russian) · Zbl 0092.31701
[13] Moglievskiĭ, I. Sh.; Solonnikov, V. A., On the solvability of a free boundary problem for the Navier-Stokes equations in the Hölder space of functions, Nonlinear Analysis. Nonlinear Analysis, Sc. Norm. Super. di Pisa Quaderni, 257-271 (1991) · Zbl 0875.35063
[14] Mucha, P. B.; Zajączkowski, W., On the existence for the Cauchy-Neumann problem for the Stokes system in the \(L_p\)-framework, Studia Math., 143, 1, 75-101 (2000) · Zbl 0970.35107
[15] Mucha, P. B.; Zajączkowski, W., On local existence of solutions of the free boundary problem for an incompressible viscous self-gravitating fluid motion, Appl. Math., 27, 3, 319-333 (2000) · Zbl 0996.35050
[16] Padula, M.; Solonnikov, V. A., On the global existence of nonsteady motions of a fluid drop and their exponential decay to a uniform rigid rotation, Quad. Mat., 10, 185-218 (2002) · Zbl 1284.76095
[17] Schweizer, B., Free boundary fluid systems in a semigroup approach and oscillatory behavior, SIAM J. Math. Anal., 28, 1135-1157 (1997) · Zbl 0889.35075
[18] Shibata, Y., Generalized resolvent estimates of the Stokes equations with first order boundary condition in a general domain, J. Math. Fluid Mech., 15, 1, 1-40 (2013) · Zbl 1278.35202
[19] Shibata, Y., On the \(R\)-boundedness of solution operators for the Stokes equations with free boundary condition, Differential Integral Equations, 27, 313-368 (2014) · Zbl 1324.35151
[20] Shibata, Y.; Shimizu, S., On some free boundary problem for the Navier-Stokes equations, Differential Integral Equations, 20, 241-276 (2007) · Zbl 1212.35353
[21] Shibata, Y.; Shimizu, S., On the \(L_p-L_q\) maximal regularity of the Neumann problem for the Stokes equations in a bounded domain, J. Reine Angew. Math., 615, 157-209 (2008) · Zbl 1145.35053
[22] Solonnikov, V. A., Estimates of solutions of an initial-boundary value problem for the linear nonstationary Navier-Stokes system, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI), 59, 178-254 (1976), (in Russian) · Zbl 0357.76026
[23] Solonnikov, V. A., Solvability of the problem of evolution of an isolated amount of a viscous incompressible capillary fluid, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI), 140, 179-186 (1984), (in Russian) · Zbl 0551.76022
[24] Solonnikov, V. A., Unsteady flow of a finite mass of a fluid bounded by a free surface, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI). Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI), J. Soviet Math., 40, 672-686 (1988), (in Russian); English transl.: · Zbl 0639.76035
[25] Solonnikov, V. A., On the transient motion of an isolated volume of viscous incompressible fluid, Math. USSR-Izv., 31, 2, 381-405 (1988) · Zbl 0850.76180
[26] Solonnikov, V. A., On nonstationary motion of a finite isolated mass of self-gravitating fluid, Algebra i Analiz. Algebra i Analiz, Leningrad Math. J., 1, 227-276 (1990), (in Russian); English transl.: · Zbl 0717.76034
[27] Solonnikov, V. A., Solvability of the problem of evolution of a viscous incompressible fluid bounded by a free surface on a finite time interval, Algebra i Analiz, 3, 191-239 (1991) · Zbl 0731.35079
[28] Solonnikov, V. A., Lectures on evolution free boundary problems: classical solutions, (Ambrosio, L.; etal., Mathematical Aspects of Evolving Interfaces. Mathematical Aspects of Evolving Interfaces, Lecture Notes in Math., vol. 1812 (2003), Springer-Verlag), 123-175 · Zbl 1038.35063
[29] Steiger, O., On Navier-Stokes equations with first order boundary conditions (2004), Universität Zürich, PhD thesis
[30] Strömer, G., About a certain class of parabolic-hyperbolic systems of differential equations, Analysis, 9, 1-39 (1989) · Zbl 0685.35078
[31] Sylvester, D. L.G., Large time existence of small viscous surface waves without surface tension, Comm. Partial Differential Equations, 15, 823-903 (1990) · Zbl 0731.35081
[32] Sylvester, D. L.G., Decay rate for a two-dimensional viscous ocean of finite depth, J. Math. Anal. Appl., 202, 659-666 (1996) · Zbl 0862.76015
[33] Tani, A., Small-time existence for the three-dimensional Navier-Stokes equations for an incompressible fluid with a free surface, Arch. Ration. Mech. Anal., 133, 299-331 (1996) · Zbl 0857.76026
[34] Tani, A.; Tanaka, N., Large-time existence of surface waves in incompressible viscous fluids with or without surface tension, Arch. Ration. Mech. Anal., 130, 303-314 (1995) · Zbl 0844.76025
[35] Weis, L., Operator-valued Fourier multiplier theorems and maximal \(L_p\)-regularity, Math. Ann., 319, 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.