# American Institute of Mathematical Sciences

July  2018, 17(4): 1681-1721. doi: 10.3934/cpaa.2018081

## On the local wellposedness of free boundary problem for the Navier-Stokes equations in an exterior domain

 1 Department of Mathematics and Reseach Institute of Science and Engineering, Waseda University, Ohkubo 3-4-1, Shinjuku-ku, Tokyo 169-8555, Japan 2 Deparment of Mechanical Engineering and Materials Science University of Pittsburgh, USA

Received  February 2017 Revised  July 2017 Published  April 2018

Fund Project: Partially supported by JSPS@Grant-in-aid for Scientific Research (A) -17H0109, Top Global University Project, and JSPS program of the Japanese-German Graduate Externship.

This paper deals with the local well-posedness of free boundary problems for the Navier-Stokes equations in the case where the fluid initially occupies an exterior domain $Ω$ in $N$-dimensional Euclidian space $\mathbb{R}^N$.

Citation: Yoshihiro Shibata. On the local wellposedness of free boundary problem for the Navier-Stokes equations in an exterior domain. Communications on Pure & Applied Analysis, 2018, 17 (4) : 1681-1721. doi: 10.3934/cpaa.2018081
##### References:
 [1] H. Abels, The initial-value problem for the Navier-Stokes equations with a free surface in $L^q$-Sobolev spaces, Adv. Differential Eqns., 10 (2005), 45-64.   Google Scholar [2] H. Amann, Linear and Quasilinear Parabolic Problems, Vol. Ⅰ. Birkhäuser, Basel, 1995. Google Scholar [3] J. T. Beale, The initial value problem for the Navier-Stokes equations with a free boundary, Comm. Pure Appl. Math., 31 (1980), 359-392.   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] D. Bothe and J. Prüss, $L_p$ theory for a class of non-Newtonian fluids, SIAM J. Math. Anal., 39 (2007), 379-421.   Google Scholar [7] A. P. Calderón, Lebesgue spaces of differentiable functions and distributions, Proc. Symp. in Pure Math., 4 (1961), 33-49.   Google Scholar [8] Y. Enomoto and Y. Shibata, On the $\mathcal{R}$-sectoriality and its application to some mathematical study of the viscous compressible fluids, Funkcial. Ekvac., 56 (2013), 441-505.   Google Scholar [9] G. P. Galdi, An Introduction to the Mathematical Theory of the Navier-Stokes Equations, Steady State Problem, Second Edition, Springer Monographs, Springer, 2011. Google Scholar [10] Y. Hataya and S. Kawashima, Decaying solution of the Navier-Stokes flow of infinite volume without surface tension, Nonlinear Anal., 71 (2009), 2535-2539.   Google Scholar [11] Y. Hataya, A remark on Beal-Nishida's paper, Bull. Inst. Math. Acad. Sin. (N.S.), 6 (2011), 293-303.   Google Scholar [12] I. Sh. Mogilevskii, Estimates of solutions of a general intial-boundary value problem for the linear nonstationary system of Navier-Stokes equations in a half-space, Zap Nauchn. Sem. LOMI., 84 (1979), 147-173.   Google Scholar [13] I. Sh. Mogilevskii, Solvability of a general boundary value problem for a linearized nonstationary system of Navier-Stokes equations, Zap Nauchn. Sem. LOMI., 110 (1981), 105-119.   Google Scholar [14] P. B. Mucha and W. Zajączkowski, On the existence for the Cauchy-Neumann problem for the Stokes system in the Lp-framework, Studia Math., 143 (2000), 75-101.   Google Scholar [15] T. Nishida, Equations of fluid dynamics -free surface problems, Comm. Pure Appl. Math., 39 (1986), 221-238.   Google Scholar [16] J. Prüss and G. Simonett, Moving Interfaces ad Quasilinear Parabolic Evolution Equations, Monographs in Mathematics, vol. 105, Birkhäuser, 2016. Google Scholar [17] T. Runst and W. Sickel, Sobolev Spaces of Fractional Order, Nemytskij Operators, and Nonlinear Partial Differential Equations, Walter de Gruyter, Berlin, New York, 1996. 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] M. Schonbek and Y. Shibata, On a global well-posedness of strong dynamics of incompressible nematic liquid crystals in ${\mathbb{R}^N}$, J. Evol. Equ., (2017), 537-550.  doi: 10.1007/s00028-016-0358-y.  Google Scholar [20] Y. Shibata, On the $\mathcal{R}$-boundedness of solution operators for the Stokes equations with free boundary condition, Diff. Int. Eqns., 27 (2014), 313-368.   Google Scholar [21] Y. Shibata, On some free boundary problem of the Navier-Stokes equations in the maximal Lp-Lq regularity class, J. Differential Equations., 258 (2015), 4127-4155.   Google Scholar [22] Y. Shibata, On the $\mathcal{R}$-bounded solution operators in the study of free boundary problem for the Navier-Stokes equations, in Mathematical Fluid Dynamics, Present and Futureh Tokyo, Japan, November 2014 (ed. Y. Shibata and Y. Suzuki), Springer Proceedings in Mathematics & Statistics, Vol. 183, (2016), 203-285. Google Scholar [23] Y. Shibata, Global well-posedness of unsteady motion of viscous incompressible capillary liquid bounded by a free surface, Evolution Equations and Control Theory, 7 (2018), 117-152.   Google Scholar [24] Y. Shibata, Global wellposedness for the free boundary problem of the Navier-Stokes equations in an exterior domain, Fluid Mech. Res. Int. , 1 (2017), 00008. DOI: 10.15406/fimrij.2017.01.00008. Google Scholar [25] Y. Shibata, On Lp-Lq decay estimate for Stokes equations with free boundary condition in an exterior domain, Accepted for publication in Asymptotic Analysis. Google Scholar [26] Y. Shibata and S. Shimizu, On a resolvent estimate for the Stokes system with Neumann boundary condition, Diff. Int. Eqns., 16 (2003), 385-426.   Google Scholar [27] Y. Shibata and S. Shimizu, Decay properties of the Stokes semigroup in exterior domains with Neumann boundary condition, J. Math. Soc. Japan, 59 (2007), 1-34.   Google Scholar [28] Y. Shibata and S. Shimizu, On the Lp-Lq maximal regularity of the Neumann problem for the Stokes equations in a bounded domain, J. Reine Angew. Math., 615 (2008), 157-209.   Google Scholar [29] C. G. Simader and H. Sohr, The Dirichlet Problem for the Laplacian in Bounded and Unbounded Domain, Pitmann Research Notes in Mathematics Series 360, Addison Wesley Longman Limited, 1996. Google Scholar [30] V. A. Solonnikov, On the transient motion of an isolated volume of viscous incompressible fluid, Math. USSR Izvestiya, 31 (1988), 381-405.   Google Scholar [31] V. Solonnikov, Unsteady motion of a finite mass of fluid, bounded by a free surface, J. Soviet Math., 40 (1988), 672-685.   Google Scholar [32] O. Steiger, On Navier-Stokes equations with first order boundary conditions, J. Math. Fluid Mech., 8 (2006), 456-481.   Google Scholar [33] H. Tanabe, Functional Analytic Methods for Partial Differential Equations, Pure and Applied Mathematics, A Series of Monographs and Textbooks, Marcel Dekker, Inc. New York·Basel, 1997. Google Scholar [34] N. Tanaka, Global existence of two phase non-homogeneous viscous incompressible weak fluid flow, Commun. Partial Differential Equations, 18 (1993), 41-81.   Google Scholar [35] 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.   Google Scholar [36] 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.   Google Scholar [37] L. Weis, Operator-valued Fourier multiplier theorems and maximal Lp-regularity, Math. Ann., 319 (2001), 735-758.   Google Scholar

show all references

##### References:
 [1] H. Abels, The initial-value problem for the Navier-Stokes equations with a free surface in $L^q$-Sobolev spaces, Adv. Differential Eqns., 10 (2005), 45-64.   Google Scholar [2] H. Amann, Linear and Quasilinear Parabolic Problems, Vol. Ⅰ. Birkhäuser, Basel, 1995. Google Scholar [3] J. T. Beale, The initial value problem for the Navier-Stokes equations with a free boundary, Comm. Pure Appl. Math., 31 (1980), 359-392.   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] D. Bothe and J. Prüss, $L_p$ theory for a class of non-Newtonian fluids, SIAM J. Math. Anal., 39 (2007), 379-421.   Google Scholar [7] A. P. Calderón, Lebesgue spaces of differentiable functions and distributions, Proc. Symp. in Pure Math., 4 (1961), 33-49.   Google Scholar [8] Y. Enomoto and Y. Shibata, On the $\mathcal{R}$-sectoriality and its application to some mathematical study of the viscous compressible fluids, Funkcial. Ekvac., 56 (2013), 441-505.   Google Scholar [9] G. P. Galdi, An Introduction to the Mathematical Theory of the Navier-Stokes Equations, Steady State Problem, Second Edition, Springer Monographs, Springer, 2011. Google Scholar [10] Y. Hataya and S. Kawashima, Decaying solution of the Navier-Stokes flow of infinite volume without surface tension, Nonlinear Anal., 71 (2009), 2535-2539.   Google Scholar [11] Y. Hataya, A remark on Beal-Nishida's paper, Bull. Inst. Math. Acad. Sin. (N.S.), 6 (2011), 293-303.   Google Scholar [12] I. Sh. Mogilevskii, Estimates of solutions of a general intial-boundary value problem for the linear nonstationary system of Navier-Stokes equations in a half-space, Zap Nauchn. Sem. LOMI., 84 (1979), 147-173.   Google Scholar [13] I. Sh. Mogilevskii, Solvability of a general boundary value problem for a linearized nonstationary system of Navier-Stokes equations, Zap Nauchn. Sem. LOMI., 110 (1981), 105-119.   Google Scholar [14] P. B. Mucha and W. Zajączkowski, On the existence for the Cauchy-Neumann problem for the Stokes system in the Lp-framework, Studia Math., 143 (2000), 75-101.   Google Scholar [15] T. Nishida, Equations of fluid dynamics -free surface problems, Comm. Pure Appl. Math., 39 (1986), 221-238.   Google Scholar [16] J. Prüss and G. Simonett, Moving Interfaces ad Quasilinear Parabolic Evolution Equations, Monographs in Mathematics, vol. 105, Birkhäuser, 2016. Google Scholar [17] T. Runst and W. Sickel, Sobolev Spaces of Fractional Order, Nemytskij Operators, and Nonlinear Partial Differential Equations, Walter de Gruyter, Berlin, New York, 1996. 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] M. Schonbek and Y. Shibata, On a global well-posedness of strong dynamics of incompressible nematic liquid crystals in ${\mathbb{R}^N}$, J. Evol. Equ., (2017), 537-550.  doi: 10.1007/s00028-016-0358-y.  Google Scholar [20] Y. Shibata, On the $\mathcal{R}$-boundedness of solution operators for the Stokes equations with free boundary condition, Diff. Int. Eqns., 27 (2014), 313-368.   Google Scholar [21] Y. Shibata, On some free boundary problem of the Navier-Stokes equations in the maximal Lp-Lq regularity class, J. Differential Equations., 258 (2015), 4127-4155.   Google Scholar [22] Y. Shibata, On the $\mathcal{R}$-bounded solution operators in the study of free boundary problem for the Navier-Stokes equations, in Mathematical Fluid Dynamics, Present and Futureh Tokyo, Japan, November 2014 (ed. Y. Shibata and Y. Suzuki), Springer Proceedings in Mathematics & Statistics, Vol. 183, (2016), 203-285. Google Scholar [23] Y. Shibata, Global well-posedness of unsteady motion of viscous incompressible capillary liquid bounded by a free surface, Evolution Equations and Control Theory, 7 (2018), 117-152.   Google Scholar [24] Y. Shibata, Global wellposedness for the free boundary problem of the Navier-Stokes equations in an exterior domain, Fluid Mech. Res. Int. , 1 (2017), 00008. DOI: 10.15406/fimrij.2017.01.00008. Google Scholar [25] Y. Shibata, On Lp-Lq decay estimate for Stokes equations with free boundary condition in an exterior domain, Accepted for publication in Asymptotic Analysis. Google Scholar [26] Y. Shibata and S. Shimizu, On a resolvent estimate for the Stokes system with Neumann boundary condition, Diff. Int. Eqns., 16 (2003), 385-426.   Google Scholar [27] Y. Shibata and S. Shimizu, Decay properties of the Stokes semigroup in exterior domains with Neumann boundary condition, J. Math. Soc. Japan, 59 (2007), 1-34.   Google Scholar [28] Y. Shibata and S. Shimizu, On the Lp-Lq maximal regularity of the Neumann problem for the Stokes equations in a bounded domain, J. Reine Angew. Math., 615 (2008), 157-209.   Google Scholar [29] C. G. Simader and H. Sohr, The Dirichlet Problem for the Laplacian in Bounded and Unbounded Domain, Pitmann Research Notes in Mathematics Series 360, Addison Wesley Longman Limited, 1996. Google Scholar [30] V. A. Solonnikov, On the transient motion of an isolated volume of viscous incompressible fluid, Math. USSR Izvestiya, 31 (1988), 381-405.   Google Scholar [31] V. Solonnikov, Unsteady motion of a finite mass of fluid, bounded by a free surface, J. Soviet Math., 40 (1988), 672-685.   Google Scholar [32] O. Steiger, On Navier-Stokes equations with first order boundary conditions, J. Math. Fluid Mech., 8 (2006), 456-481.   Google Scholar [33] H. Tanabe, Functional Analytic Methods for Partial Differential Equations, Pure and Applied Mathematics, A Series of Monographs and Textbooks, Marcel Dekker, Inc. New York·Basel, 1997. Google Scholar [34] N. Tanaka, Global existence of two phase non-homogeneous viscous incompressible weak fluid flow, Commun. Partial Differential Equations, 18 (1993), 41-81.   Google Scholar [35] 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.   Google Scholar [36] 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.   Google Scholar [37] L. Weis, Operator-valued Fourier multiplier theorems and maximal Lp-regularity, Math. Ann., 319 (2001), 735-758.   Google Scholar
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