Article Contents
Article Contents

# Local well-posedness of free surface problems for the Navier-Stokes equations in a general domain

• In this paper, we prove the local well-posedness of the free boundary problems of Navier-Stokes equations in a general domain $\Omega\subset\mathbb{R}^N$ ($N \geq 2$). The velocity field is obtained in the maximal regularity class $W^{2,1}_{q,p}(\Omega\times(0, T)) = L_p((0, T), W^2_q(\Omega)^N) \cap W^1_p((0, T), L_q(\Omega)^N)$ ($2 < p < \infty$ and $N < q < \infty$) for any initial data satisfying certain compatibility conditions. The assumption of the domain $\Omega$ is the unique existence of solutions to the weak Dirichlet-Neumann problem as well as some uniformity of covering of the closure of $\Omega$. A bounded domain, a perturbed half space, and a perturbed layer satisfy the conditions for the domain, and therefore drop problems and ocean problems are treated in the uniform manner. Our method is based on the maximal $L_p$-$L_q$ regularity theorem of a linearized problem in a general domain.
Mathematics Subject Classification: Primary: 35R35; Secondary: 35Q30, 76D05, 76D03.

 Citation:

•  [1] H. Abels, The initial-value problem for the Navier-Stokes equations with a free surface in $L_q$ Sobolev spaces, Adv. Differential Equations, 10 (2005), 45-64. [2] G. Allain, Small-time existence for the Navier-Stokes equations with a free surface, Appl. Math. Optim., 16 (1987), 37-50.doi: 10.1007/BF01442184. [3] H. Amann, Linear and Quasilinear Parabolic Problems Vol. I Abstract Linear Theory, Monographs in Math., Vol 89, Birkhäuser Verlag, Basel$\cdot$Boston$\cdot$Berlin, 1995.doi: 10.1007/978-3-0348-9221-6. [4] H. Amann, M. Hieber and G. Simonett, Bounded $H_\infty$-calculus for elliptic operators, Differential Integral Equations, 7 (1994), 613-653. [5] J. T. Beale, The initial value problem for the Navier-Stokes equations with a free boundary, Comm. Pure Appl. Math., 34 (1981), 359-392.doi: 10.1002/cpa.3160340305. [6] J. T. Beale, Large time regularity of viscous surface waves, Arch. Rat. Mech. Anal., 84 (1984), 307-352.doi: 10.1007/BF00250586. [7] J. T. Beale and T. Nishida, Large time behavior of viscous surface waves, Lecture Notes in Numer. Appl. Anal., 128 (1985), 1-14.doi: 10.1016/S0304-0208(08)72355-7. [8] A. P. Calderón, Lebesgue spaces of differentiable functions and distributions, Proc. Symp. in Pure Math., 4 (1961), 33-49. [9] Y. Enomoto and Y. Shibata, On the $\mathcalR$-sectoriality and its application to some mathematical study of the viscous compressible fluids, Funk. Ekvaj., 56 (2013), 441-505.doi: 10.1619/fesi.56.441. [10] G. P. Galdi, An Introduction to the Mathematical Theory of the Navier-Stokes Equations, Steady-State Problems, Second edition, Springer Monographs in Mathematics, Springer, New York, 2011.doi: 10.1007/978-0-387-09620-9. [11] Y. Hataya and S. Kawashima, Decaying solution of the Navier-Stokes flow of infinite volume without surface tension, Nonlinear Anal., 71 (2009), 2535-2539.doi: 10.1016/j.na.2009.05.061. [12] Y. Hataya, A remark on Beal-Nishida's paper, Bull. Inst. Math. Acad. Sin. (N.S.), 6 (2011), 293-303. [13] I. Sh. Mogilevskiĭ and V. A. Solonnikov, On the solvability of a free boundary problem for the Navier-Stokes equations in the Hölder spaces of functions, Nonlinear Analysis. A Tribute in Honour of Giovanni Prodi, Quaderni, Pisa, (1991), 257-272. [14] P. B. Mucha and W. Zajączkowski, On local existence of solutions of the free boundary problem for an incompressible viscous self-gravitating fluid motion, Applicationes Mathematicae, 27 (2000), 319-333. [15] T. Nishida, Equations of fluid dynamics - free surface problems, Comm. Pure Appl. Math., 39 (1986), S221-S238.doi: 10.1002/cpa.3160390712. [16] M. Padula and V. A. Solonnikov, On the local solvability of free boundary problem for the Navier-Stokes equations, J. Math. Sci., 170 (2010), 522-553.doi: 10.1007/s10958-010-0099-3. [17] M. Padula and V. A. Solonnikov, On the global existence of nonsteady motions of a fluid drop and their exponential decay to a uniform rigid rotation, Quad. Mat., 10 (2002), 185-218. [18] H. Saito and Y. Shibata, On the global well posedness of free boundary problem for the Navier-Stokes equations with surface tension, in preparation. [19] B. Schweizer, Free boundary fluid systems in a semigroup approach and oscillatory behavior, SIAM J. Math. Anal., 28 (1997), 1135-1157.doi: 10.1137/S0036141096299892. [20] M. Schonbek and Y. Shibata, On a global well-posedness of Strong Dynamics of Incompressible Nematic Liquid Crystals in $\mathbbR^N$, in preparation. [21] Y. Shibata, On the maximal $L_p$-$L_q$ regularity of the Stokes equations and the one phase free boundary problem for the Navier-Stokes equations, in Mathematical Analysis on the Navier-Stokes Equations and Related Topics, Past and Future - In memory of Prof. T. Miyakawa (ed. T. Adachi et al.), Gakuto International Series, 35, Math. Sci. Appl., 2011, 185-208. [22] Y. Shibata, On some free boundary problem of the Navier-Stokes equations in the maximal $L_p$-$L_q$ regularity class, J. Differential Equations, 258 (2015), 4127-4155.doi: 10.1016/j.jde.2015.01.028. [23] Y. Shibata, On the $\mathcalR$-bounded solution operators in the study of free boundary problem for the Navier-Stokes equations, to appear in the proceedings of the International Conference on Mathematical Fluid Dynamics, Present and Future. [24] Y. Shibata and S. Shimizu, Report on a local in time solvability of free surface problems for the Navier-Stokes equations with surface tension, Applicable Analysis, 90 (2011), 201-214.doi: 10.1080/00036811003735899. [25] V. A. Solonnikov, Unsteady motion of a finite mass of fluid, bounded by a free surface, Zap. Nauchn. Sem. (LOMI), 152 (1986), 137-157 (in Russian); English transl. J. Soviet Math., 40 (1988), 672-686.doi: 10.1007/BF01094193. [26] V. A. Solonnikov, On the transient motion of an isolated volume of viscous incompressible fluid, Izv. Akad. Nauk SSSR Ser. Mat., 51 (1987), 1065-1087 (in Russian); English transl. Math. USSR Izv., 31 (1988), 381-405. [27] V. A. Solonnikov, 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 (1991), 222-257 (in Russian); English transl. St. Petersburg Math. J., 3 (1992), 189-220. [28] V. A. Solonnikov, Lectures on evolution free boundary problems: Classical solutions, Mathematical aspects of evolving interfaces (Funchal, 2000), Lecture Notes in Math., 1812, Springer, Berlin, 2003, 123-175.doi: 10.1007/978-3-540-39189-0_4. [29] D. Sylvester, Large time existence of small viscous surface waves without surface tension, Commun. Partial Differential Equations, 15 (1990), 823-903.doi: 10.1080/03605309908820709. [30] N. Tanaka, Global existence of two phase non-homogeneous viscous incompressible fluid flow, Commun. Partial Differential Equations, 18 (1993), 41-81.doi: 10.1080/03605309308820921. [31] A. Tani, Small-time existence for the three-dimensional incompressible Navier-Stokes equations with a free surface, Arch. Rat. Mech. Anal., 133 (1996), 299-331.doi: 10.1007/BF00375146. [32] A. Tani and N. Tanaka, Large time existence of surface waves in incompressible viscous fluids with or without surface tension, Arch. Rat. Mech. Anal., 130 (1995), 303-314.doi: 10.1007/BF00375142.
Open Access Under a Creative Commons license