# American Institute of Mathematical Sciences

March  2018, 7(1): 117-152. doi: 10.3934/eect.2018007

## Global well-posedness of unsteady motion of viscous incompressible capillary liquid bounded by a free surface

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

Received  May 2016 Revised  August 2017 Published  January 2018

Fund Project: Partially supported by JSPS Grant-in-aid for Scientific Research (A) 17H0109 and Top Global University Project. Adjunct faculty member in the Department of Mechanical Engineering and Materials Science, University of Pittsburgh.

In this paper, we prove the global well-posedness of free boundary problems of the Navier-Stokes equations in a bounded domain with surface tension. The velocity field is obtained in the $L_p$ in time $L_q$ in space maximal regularity class, ($2 < p < ∞$, $N < q < ∞$, and $2/p + N/q < 1$), under the assumption that the initial domain is close to a ball and initial data are sufficiently small. The essential point of our approach is to drive the exponential decay theorem in the $L_p$-$L_q$ framework for the linearized equations with the help of maximal $L_p$-$L_q$ regularity theory for the Stokes equations with free boundary conditions and spectral analysis of the Stokes operator and the Laplace-Beltrami operator.

Citation: Yoshihiro Shibata. Global well-posedness of unsteady motion of viscous incompressible capillary liquid bounded by a free surface. Evolution Equations & Control Theory, 2018, 7 (1) : 117-152. doi: 10.3934/eect.2018007
##### 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), 45-64.   Google Scholar [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.  Google Scholar [3] 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.  Google Scholar [4] J. T. Beale, Large time regularity of viscous surface waves, Arch. Rat. Mech. Anal., 84 (1984), 307-352.   Google Scholar [5] J. T. Beale and T. Nishida, Large time behavior of viscous surface waves, Lecture Notes in Numer. Appl. Anal., 8 (1985), 1-14.   Google Scholar [6] 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.  Google Scholar [7] Y. Hataya, A remark on Beal-Nishida's paper, Bull. Inst. Math. Acad. Sin. (N.S.), 6 (2011), 293-303.   Google Scholar [8] M. Köhne, J. Prüss and M. Wilke, Qualitative Behavior of solutions for the two-phase Navier-Stokes equations with surface tension, Math. Ann., 356 (2013), 737-792.  doi: 10.1007/s00208-012-0860-7.  Google Scholar [9] I. Sh. Mogilevskiǐ 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.  Google Scholar [10] 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.   Google Scholar [11] U. Neri, Singular Integrals, Lecutre Notes in Mathematics 200, Springer, New York, 1971.  Google Scholar [12] T. Nishida, Equations of fluid dynamics -free surface problems, Comm. Pure Appl. Math., 39 (1986), 221-238.  doi: 10.1002/cpa.3160390712.  Google Scholar [13] 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.  Google Scholar [14] J. Prüss and G. Simonett, On the two-phase Navier-Stokes equations with surface tension, Interfaces and Free Boundaries, 12 (2010), 311-345.   Google Scholar [15] J. Prüess and G. Simonett, Analytic solutions for the two-phase Navier-Stokes equations with surface tension and gravity, Progress in Nonlinear Differential Equations and Their Applications, 80 (2011), 507-540.   Google Scholar [16] J. Prüess and G. Simonett, Moving Interfaces and Quasilinear Parabolic Evolution Equations, Birkhauser Monographs in Mathematics, 2016.  Google Scholar [17] H. Saito and Y. Shibata, On decay properties of solutions to the Stokes equations with surface tension and gravity in the half space, J. Math. Soc. Japan, 68 (2016), 1559-1614.  doi: 10.2969/jmsj/06841559.  Google Scholar [18] H. Saito and Y. Shibata, On the global wellposedness of free boundary problem for the Navier Stokes systems with surface tension, Preprint. Google Scholar [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.  Google Scholar [20] 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.  Google Scholar [21] Y. Shibata, On the $\mathcal{R}$-bounded solution operators in the study of free boundary problem for the Navier-Stokes equations, in Mathematical Fluid Dynaics, Present and Future, Tokyo, Japna, November 2014 (eds. Y. Shibata and Y. Suzuki), Springer Proceedings in Mathematics & Staistics, 183 (2016), 203–285.  Google Scholar [22] Y. Shibata, Local well-posedness of free surface problems for the Navier-Stokes equations in a general domain, Discrete and Continuous Dynamical Systems, Series S, 9 (2016), 315-342.   Google Scholar [23] Y. Shibata and S. Shimizu, On a free boundary problem for the Navier-Stokes equations, Differential Integral Equations, 20 (2007), 241-276.   Google Scholar [24] 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.  Google Scholar [25] 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.  Google Scholar [26] 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.  Google Scholar [27] V. A. Solonnikov, Lectures on evolution free boundary problems: Classical solutions, L. Ambrosio et al. : Lecture Note in Mathematics (eds. P. Colli and J. F. Rodrigues), SpringerVerlag, Berlin, Heidelberg, 1812 (2003), 123–175.  Google Scholar [28] 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.  Google Scholar [29] N. Tanaka, Global existence of two phase non-homogeneous viscous incompressible weak fluid flow, Commun. Partial Differential Equations, 18 (1993), 41-81.  doi: 10.1080/03605309308820921.  Google Scholar [30] 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.  Google Scholar [31] 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.  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), 45-64.   Google Scholar [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.  Google Scholar [3] 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.  Google Scholar [4] J. T. Beale, Large time regularity of viscous surface waves, Arch. Rat. Mech. Anal., 84 (1984), 307-352.   Google Scholar [5] J. T. Beale and T. Nishida, Large time behavior of viscous surface waves, Lecture Notes in Numer. Appl. Anal., 8 (1985), 1-14.   Google Scholar [6] 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.  Google Scholar [7] Y. Hataya, A remark on Beal-Nishida's paper, Bull. Inst. Math. Acad. Sin. (N.S.), 6 (2011), 293-303.   Google Scholar [8] M. Köhne, J. Prüss and M. Wilke, Qualitative Behavior of solutions for the two-phase Navier-Stokes equations with surface tension, Math. Ann., 356 (2013), 737-792.  doi: 10.1007/s00208-012-0860-7.  Google Scholar [9] I. Sh. Mogilevskiǐ 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.  Google Scholar [10] 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.   Google Scholar [11] U. Neri, Singular Integrals, Lecutre Notes in Mathematics 200, Springer, New York, 1971.  Google Scholar [12] T. Nishida, Equations of fluid dynamics -free surface problems, Comm. Pure Appl. Math., 39 (1986), 221-238.  doi: 10.1002/cpa.3160390712.  Google Scholar [13] 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.  Google Scholar [14] J. Prüss and G. Simonett, On the two-phase Navier-Stokes equations with surface tension, Interfaces and Free Boundaries, 12 (2010), 311-345.   Google Scholar [15] J. Prüess and G. Simonett, Analytic solutions for the two-phase Navier-Stokes equations with surface tension and gravity, Progress in Nonlinear Differential Equations and Their Applications, 80 (2011), 507-540.   Google Scholar [16] J. Prüess and G. Simonett, Moving Interfaces and Quasilinear Parabolic Evolution Equations, Birkhauser Monographs in Mathematics, 2016.  Google Scholar [17] H. Saito and Y. Shibata, On decay properties of solutions to the Stokes equations with surface tension and gravity in the half space, J. Math. Soc. Japan, 68 (2016), 1559-1614.  doi: 10.2969/jmsj/06841559.  Google Scholar [18] H. Saito and Y. Shibata, On the global wellposedness of free boundary problem for the Navier Stokes systems with surface tension, Preprint. Google Scholar [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.  Google Scholar [20] 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.  Google Scholar [21] Y. Shibata, On the $\mathcal{R}$-bounded solution operators in the study of free boundary problem for the Navier-Stokes equations, in Mathematical Fluid Dynaics, Present and Future, Tokyo, Japna, November 2014 (eds. Y. Shibata and Y. Suzuki), Springer Proceedings in Mathematics & Staistics, 183 (2016), 203–285.  Google Scholar [22] Y. Shibata, Local well-posedness of free surface problems for the Navier-Stokes equations in a general domain, Discrete and Continuous Dynamical Systems, Series S, 9 (2016), 315-342.   Google Scholar [23] Y. Shibata and S. Shimizu, On a free boundary problem for the Navier-Stokes equations, Differential Integral Equations, 20 (2007), 241-276.   Google Scholar [24] 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.  Google Scholar [25] 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.  Google Scholar [26] 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.  Google Scholar [27] V. A. Solonnikov, Lectures on evolution free boundary problems: Classical solutions, L. Ambrosio et al. : Lecture Note in Mathematics (eds. P. Colli and J. F. Rodrigues), SpringerVerlag, Berlin, Heidelberg, 1812 (2003), 123–175.  Google Scholar [28] 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.  Google Scholar [29] N. Tanaka, Global existence of two phase non-homogeneous viscous incompressible weak fluid flow, Commun. Partial Differential Equations, 18 (1993), 41-81.  doi: 10.1080/03605309308820921.  Google Scholar [30] 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.  Google Scholar [31] 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.  Google Scholar
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