# American Institute of Mathematical Sciences

February  2016, 9(1): 315-342. doi: 10.3934/dcdss.2016.9.315

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

 1 Department of Mathematics and Research Institute of Science and Engineering, JST CREST, Waseda University, Ohkubo 3-4-1, Shinjuku-ku, Tokyo 169-8555

Received  September 2014 Revised  February 2015 Published  December 2015

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.
Citation: Yoshihiro Shibata. Local well-posedness of free surface problems for the Navier-Stokes equations in a general domain. Discrete & Continuous Dynamical Systems - S, 2016, 9 (1) : 315-342. doi: 10.3934/dcdss.2016.9.315
##### References:
 [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. Google Scholar [2] G. Allain, Small-time existence for the Navier-Stokes equations with a free surface,, Appl. Math. Optim., 16 (1987), 37. doi: 10.1007/BF01442184. Google Scholar [3] H. Amann, Linear and Quasilinear Parabolic Problems Vol. I Abstract Linear Theory,, Monographs in Math., (1995). doi: 10.1007/978-3-0348-9221-6. Google Scholar [4] H. Amann, M. Hieber and G. Simonett, Bounded $H_\infty$-calculus for elliptic operators,, Differential Integral Equations, 7 (1994), 613. Google Scholar [5] J. T. Beale, The initial value problem for the Navier-Stokes equations with a free boundary,, Comm. Pure Appl. Math., 34 (1981), 359. doi: 10.1002/cpa.3160340305. Google Scholar [6] J. T. Beale, Large time regularity of viscous surface waves,, Arch. Rat. Mech. Anal., 84 (1984), 307. doi: 10.1007/BF00250586. Google Scholar [7] J. T. Beale and T. Nishida, Large time behavior of viscous surface waves,, Lecture Notes in Numer. Appl. Anal., 128 (1985), 1. doi: 10.1016/S0304-0208(08)72355-7. Google Scholar [8] A. P. Calderón, Lebesgue spaces of differentiable functions and distributions,, Proc. Symp. in Pure Math., 4 (1961), 33. Google Scholar [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. doi: 10.1619/fesi.56.441. Google Scholar [10] G. P. Galdi, An Introduction to the Mathematical Theory of the Navier-Stokes Equations, Steady-State Problems,, Second edition, (2011). doi: 10.1007/978-0-387-09620-9. Google Scholar [11] Y. Hataya and S. Kawashima, Decaying solution of the Navier-Stokes flow of infinite volume without surface tension,, Nonlinear Anal., 71 (2009), 2535. doi: 10.1016/j.na.2009.05.061. Google Scholar [12] Y. Hataya, A remark on Beal-Nishida's paper,, Bull. Inst. Math. Acad. Sin. (N.S.), 6 (2011), 293. Google Scholar [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, (1991), 257. Google Scholar [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. Google Scholar [15] T. Nishida, Equations of fluid dynamics - free surface problems,, Comm. Pure Appl. Math., 39 (1986). doi: 10.1002/cpa.3160390712. Google Scholar [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. doi: 10.1007/s10958-010-0099-3. Google Scholar [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. Google Scholar [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., (). Google Scholar [19] B. Schweizer, Free boundary fluid systems in a semigroup approach and oscillatory behavior,, SIAM J. Math. Anal., 28 (1997), 1135. doi: 10.1137/S0036141096299892. Google Scholar [20] M. Schonbek and Y. Shibata, On a global well-posedness of Strong Dynamics of Incompressible Nematic Liquid Crystals in $\mathbbR^N$,, in preparation., (). Google Scholar [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, (2011), 185. Google Scholar [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. doi: 10.1016/j.jde.2015.01.028. Google Scholar [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, (). Google Scholar [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. doi: 10.1080/00036811003735899. Google Scholar [25] V. A. Solonnikov, Unsteady motion of a finite mass of fluid, bounded by a free surface,, Zap. Nauchn. Sem. (LOMI), 152 (1986), 137. doi: 10.1007/BF01094193. Google Scholar [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. Google Scholar [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. Google Scholar [28] V. A. Solonnikov, Lectures on evolution free boundary problems: Classical solutions,, Mathematical aspects of evolving interfaces (Funchal, (2000), 123. doi: 10.1007/978-3-540-39189-0_4. Google Scholar [29] D. Sylvester, Large time existence of small viscous surface waves without surface tension,, Commun. Partial Differential Equations, 15 (1990), 823. doi: 10.1080/03605309908820709. Google Scholar [30] N. Tanaka, Global existence of two phase non-homogeneous viscous incompressible fluid flow,, Commun. Partial Differential Equations, 18 (1993), 41. doi: 10.1080/03605309308820921. Google Scholar [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. doi: 10.1007/BF00375146. Google Scholar [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. doi: 10.1007/BF00375142. Google Scholar

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##### References:
 [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. Google Scholar [2] G. Allain, Small-time existence for the Navier-Stokes equations with a free surface,, Appl. Math. Optim., 16 (1987), 37. doi: 10.1007/BF01442184. Google Scholar [3] H. Amann, Linear and Quasilinear Parabolic Problems Vol. I Abstract Linear Theory,, Monographs in Math., (1995). doi: 10.1007/978-3-0348-9221-6. Google Scholar [4] H. Amann, M. Hieber and G. Simonett, Bounded $H_\infty$-calculus for elliptic operators,, Differential Integral Equations, 7 (1994), 613. Google Scholar [5] J. T. Beale, The initial value problem for the Navier-Stokes equations with a free boundary,, Comm. Pure Appl. Math., 34 (1981), 359. doi: 10.1002/cpa.3160340305. Google Scholar [6] J. T. Beale, Large time regularity of viscous surface waves,, Arch. Rat. Mech. Anal., 84 (1984), 307. doi: 10.1007/BF00250586. Google Scholar [7] J. T. Beale and T. Nishida, Large time behavior of viscous surface waves,, Lecture Notes in Numer. Appl. Anal., 128 (1985), 1. doi: 10.1016/S0304-0208(08)72355-7. Google Scholar [8] A. P. Calderón, Lebesgue spaces of differentiable functions and distributions,, Proc. Symp. in Pure Math., 4 (1961), 33. Google Scholar [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. doi: 10.1619/fesi.56.441. Google Scholar [10] G. P. Galdi, An Introduction to the Mathematical Theory of the Navier-Stokes Equations, Steady-State Problems,, Second edition, (2011). doi: 10.1007/978-0-387-09620-9. Google Scholar [11] Y. Hataya and S. Kawashima, Decaying solution of the Navier-Stokes flow of infinite volume without surface tension,, Nonlinear Anal., 71 (2009), 2535. doi: 10.1016/j.na.2009.05.061. Google Scholar [12] Y. Hataya, A remark on Beal-Nishida's paper,, Bull. Inst. Math. Acad. Sin. (N.S.), 6 (2011), 293. Google Scholar [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, (1991), 257. Google Scholar [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. Google Scholar [15] T. Nishida, Equations of fluid dynamics - free surface problems,, Comm. Pure Appl. Math., 39 (1986). doi: 10.1002/cpa.3160390712. Google Scholar [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. doi: 10.1007/s10958-010-0099-3. Google Scholar [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. Google Scholar [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., (). Google Scholar [19] B. Schweizer, Free boundary fluid systems in a semigroup approach and oscillatory behavior,, SIAM J. Math. Anal., 28 (1997), 1135. doi: 10.1137/S0036141096299892. Google Scholar [20] M. Schonbek and Y. Shibata, On a global well-posedness of Strong Dynamics of Incompressible Nematic Liquid Crystals in $\mathbbR^N$,, in preparation., (). Google Scholar [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, (2011), 185. Google Scholar [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. doi: 10.1016/j.jde.2015.01.028. Google Scholar [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, (). Google Scholar [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. doi: 10.1080/00036811003735899. Google Scholar [25] V. A. Solonnikov, Unsteady motion of a finite mass of fluid, bounded by a free surface,, Zap. Nauchn. Sem. (LOMI), 152 (1986), 137. doi: 10.1007/BF01094193. Google Scholar [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. Google Scholar [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. Google Scholar [28] V. A. Solonnikov, Lectures on evolution free boundary problems: Classical solutions,, Mathematical aspects of evolving interfaces (Funchal, (2000), 123. doi: 10.1007/978-3-540-39189-0_4. Google Scholar [29] D. Sylvester, Large time existence of small viscous surface waves without surface tension,, Commun. Partial Differential Equations, 15 (1990), 823. doi: 10.1080/03605309908820709. Google Scholar [30] N. Tanaka, Global existence of two phase non-homogeneous viscous incompressible fluid flow,, Commun. Partial Differential Equations, 18 (1993), 41. doi: 10.1080/03605309308820921. Google Scholar [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. doi: 10.1007/BF00375146. Google Scholar [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. doi: 10.1007/BF00375142. Google Scholar
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