# American Institute of Mathematical Sciences

October  2021, 41(10): 4609-4643. doi: 10.3934/dcds.2021051

## Unique solvability of elliptic problems associated with two-phase incompressible flows in unbounded domains

 1 Graduate School of Informatics and Engineering, The University of Electro-Communications, 5-1 Chofugaoka 1-chome, Chofu, Tokyo 182-8585, Japan 2 School of Mathematical Sciences, Tongji University, No.1239, Siping Road, Shanghai (200092), China

* Corresponding author: Xin Zhang

Received  January 2020 Revised  January 2021 Published  October 2021 Early access  April 2021

Fund Project: TThe first author was supported by JSPS KAKENHI Grant Number JP17K14224, and the second author was supported by the Top Global University Project

This paper shows the unique solvability of elliptic problems associated with two-phase incompressible flows, which are governed by the two-phase Navier-Stokes equations with a sharp moving interface, in unbounded domains such as the whole space separated by a compact interface and the whole space separated by a non-compact interface. As a by-product, we obtain the Helmholtz-Weyl decomposition for two-phase incompressible flows.

Citation: Hirokazu Saito, Xin Zhang. Unique solvability of elliptic problems associated with two-phase incompressible flows in unbounded domains. Discrete & Continuous Dynamical Systems, 2021, 41 (10) : 4609-4643. doi: 10.3934/dcds.2021051
##### References:
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Galdi, An Introduction to the Mathematical Theory of the Navier-Stokes Equations: Steady-State Problems, $2^{nd}$ edition, Springer Monogr. Math. Springer, New York, 2011. doi: 10.1007/978-0-387-09620-9.  Google Scholar [12] J. Geng and Z. Shen, The Neumann problem and Helmholtz decomposition in convex domains, J. Funct. Anal., 259 (2010), 2147-2164.  doi: 10.1016/j.jfa.2010.07.005.  Google Scholar [13] M. Köhne, J. Prüss, and M. Wilke, Qualitative behaviour 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 [14] S. Maryani and H. Saito, On the $\mathcal{R}$-boundedness of solution operator families for two-phase Stokes resolvent equations, Differential Integral Equations, 30 (2017), 1-52.   Google Scholar [15] V. N. Maslennikova and M. E. Bogovskiǐ, Elliptic boundary value problems in unbounded domains with noncompact and nonsmooth boundaries, Rend. Sem. Mat. Fis. Milano, 56 (1986), 125-138.  doi: 10.1007/BF02925141.  Google Scholar [16] T. Miyakawa, On nonstationary solutions of the Navier-Stokes equations in an exterior domain, Hiroshima Math. J., 12 (1982), 115-140.  doi: 10.32917/hmj/1206133879.  Google Scholar [17] T. Miyakawa, The Helmholtz decomposition of vector fields in some unbounded domains, Math. J. Toyama Univ., 17 (1994), 115-149.   Google Scholar [18] J. Prüss and G. Simonett, Moving Interfaces and Quasilinear Parabolic Evolution Equations, Monographs in Mathematics, 105, Birkhäuser/Springer, Cham, 2016. doi: 10.1007/978-3-319-27698-4.  Google Scholar [19] H. Saito, Global solvability of the Navier-Stokes equations with a free surface in the maximal $L_p$-$L_q$ regularity class, J. Differential Equations, 264 (2018), 1475-1520.  doi: 10.1016/j.jde.2017.09.045.  Google Scholar [20] H. Saito, Y. Shibata and X. Zhang, Some free boundary problem for two phase inhomogeneous incompressible flow, SIAM J. Math. Anal., 52 (2020), 3397-3443.  doi: 10.1137/18M1225239.  Google Scholar [21] K. Schade and Y. Shibata, On strong dynamics of compressible nematic liquid crystals, SIAM J. Math. Anal., 47(5) (2015), 3963-3992.  doi: 10.1137/140970628.  Google Scholar [22] Y. Shibata, Introduction to the Mathematical Theory of Fluid Mechanics (Japanese), in press. Google Scholar [23] Y. Shibata, Generalized resolvent estimates of the Stokes equations with first order boundary condition in a general domain, J. Math. Fluid Mech., 15 (2013), 1-40.  doi: 10.1007/s00021-012-0130-1.  Google Scholar [24] Y. Shibata, On the local wellposedness of free boundary problem for the {N}avier-Stokes equations in an exterior domain, Commun. Pure Appl. Anal., 17 (2018), 1681-1721.  doi: 10.3934/cpaa.2018081.  Google Scholar [25] Y. Shibata and S. Shimizu, On the maximal ${L}_p$-${L}_q$ regularity of the Stokes problem with first order boundary condition; model problems, J. Math. Soc. Japan, 64 (2012), 561-626.  doi: 10.2969/jmsj/06420561.  Google Scholar [26] C. G. Simader and H. Sohr, A new approach to the Helmholtz decomposition and the Neumann problem in $L^q$-spaces for bounded and exterior domains, in Mathematical Problems Relating to the Navier-Stokes Equation, Ser. Adv. Math. Appl. Sci., 11, World Sci. Publ., River Edge, NJ, 1992, 1-35. doi: 10.1142/9789814503594_0001.  Google Scholar [27] C. G. Simader and H. Sohr, The Dirichlet Problem for the Laplacian in Bounded and Unbounded Domains, Pitman Research Notes in Mathematics Series, 360, Longman, Harlow, 1996.  Google Scholar [28] H. Sohr, The Navier-Stokes Equations, An elementary functional analytic approach, Modern Birkhäuser Classics. Birkhäuser/Springer Basel AG, Basel, 2013.  Google Scholar [29] V. A. Solonnikov, Estimates of the solutions of the nonstationary Navier-Stokes system, Zap. Naučn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI), 38 (1973), 153-231.   Google Scholar [30] S. Szufla, On the Hammerstein integral equation with weakly singular kernel, Funkcial. Ekvac., 34 (1991), 279-285.   Google Scholar [31] H. Weyl, The method of orthogonal projection in potential theory, Duke Math. J., 7 (1940), 411-444.  doi: 10.1215/S0012-7094-40-00725-6.  Google Scholar

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##### References:
 [1] T. Abe and Y. Shibata, On a gresolvent estimate of the Stokes equation on an infinite layer. II. $\lambda = 0$ case, J. Math. Fluid Mech., 5 (2003), 245-274.  doi: 10.1007/s00021-003-0075-5.  Google Scholar [2] H. Abels, Generalized Stokes resolvent equations in an infinite layer with mixed boundary conditions, Math. Nachr., 279 (2006), 351-367.  doi: 10.1002/mana.200310365.  Google Scholar [3] R. A. Adams and J. J. F. Fournier, Sobolev Spaces, Pure and Applied Mathematics, 140, $2^{nd}$ edition, Elsevier/Academic Press, Amsterdam, 2003.  Google Scholar [4] E. DiBenedetto, Real Analysis, $2^{nd}$ edition, Birkhäuser Advanced Texts: Basel Textbooks, Birkhäuser, 2016. doi: 10.1007/978-1-4939-4005-9.  Google Scholar [5] E. Fabes, O. Mendez and M. Mitrea., Boundary layers on Sobolev-Besov spaces and Poisson's equation for the Laplacian in Lipschitz domains, J. Funct. Anal., 159 (1998), 323-368.  doi: 10.1006/jfan.1998.3316.  Google Scholar [6] R. Farwig, H. Kozono and H. Sohr, An $L^q$-approach to Stokes and Navier-Stokes equations in general domains, Acta Math., 195 (2005), 21-53.  doi: 10.1007/BF02588049.  Google Scholar [7] R. Farwig, H. Kozono and H. Sohr, On the Helmholtz decomposition in general unbounded domains, Arch. Math. (Basel), 88 (2007), 239-248.  doi: 10.1007/s00013-006-1910-8.  Google Scholar [8] R. Farwig and H. Sohr, Generalized resolvent estimates for the Stokes system in bounded and unbounded domains, J. Math. Soc. Japan, 46 (1994), 607-643.  doi: 10.2969/jmsj/04640607.  Google Scholar [9] R. Farwig and H. Sohr, Helmholtz decomposition and Stokes resolvent system for aperture domains in $L^q$-spaces, Analysis, 16 (1996), 1-26.  doi: 10.1524/anly.1996.16.1.1.  Google Scholar [10] D. Fujiwara and H. Morimoto, An $L_{r}$-theorem of the Helmholtz decomposition of vector fields, J. Fac. Sci. Univ. Tokyo Sect. IA Math., 24 (1977), 685-700.   Google Scholar [11] G. P. Galdi, An Introduction to the Mathematical Theory of the Navier-Stokes Equations: Steady-State Problems, $2^{nd}$ edition, Springer Monogr. Math. Springer, New York, 2011. doi: 10.1007/978-0-387-09620-9.  Google Scholar [12] J. Geng and Z. Shen, The Neumann problem and Helmholtz decomposition in convex domains, J. Funct. Anal., 259 (2010), 2147-2164.  doi: 10.1016/j.jfa.2010.07.005.  Google Scholar [13] M. Köhne, J. Prüss, and M. Wilke, Qualitative behaviour 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 [14] S. Maryani and H. Saito, On the $\mathcal{R}$-boundedness of solution operator families for two-phase Stokes resolvent equations, Differential Integral Equations, 30 (2017), 1-52.   Google Scholar [15] V. N. Maslennikova and M. E. Bogovskiǐ, Elliptic boundary value problems in unbounded domains with noncompact and nonsmooth boundaries, Rend. Sem. Mat. Fis. Milano, 56 (1986), 125-138.  doi: 10.1007/BF02925141.  Google Scholar [16] T. Miyakawa, On nonstationary solutions of the Navier-Stokes equations in an exterior domain, Hiroshima Math. J., 12 (1982), 115-140.  doi: 10.32917/hmj/1206133879.  Google Scholar [17] T. Miyakawa, The Helmholtz decomposition of vector fields in some unbounded domains, Math. J. Toyama Univ., 17 (1994), 115-149.   Google Scholar [18] J. Prüss and G. Simonett, Moving Interfaces and Quasilinear Parabolic Evolution Equations, Monographs in Mathematics, 105, Birkhäuser/Springer, Cham, 2016. doi: 10.1007/978-3-319-27698-4.  Google Scholar [19] H. Saito, Global solvability of the Navier-Stokes equations with a free surface in the maximal $L_p$-$L_q$ regularity class, J. Differential Equations, 264 (2018), 1475-1520.  doi: 10.1016/j.jde.2017.09.045.  Google Scholar [20] H. Saito, Y. Shibata and X. Zhang, Some free boundary problem for two phase inhomogeneous incompressible flow, SIAM J. Math. Anal., 52 (2020), 3397-3443.  doi: 10.1137/18M1225239.  Google Scholar [21] K. Schade and Y. Shibata, On strong dynamics of compressible nematic liquid crystals, SIAM J. Math. Anal., 47(5) (2015), 3963-3992.  doi: 10.1137/140970628.  Google Scholar [22] Y. Shibata, Introduction to the Mathematical Theory of Fluid Mechanics (Japanese), in press. Google Scholar [23] Y. Shibata, Generalized resolvent estimates of the Stokes equations with first order boundary condition in a general domain, J. Math. Fluid Mech., 15 (2013), 1-40.  doi: 10.1007/s00021-012-0130-1.  Google Scholar [24] Y. Shibata, On the local wellposedness of free boundary problem for the {N}avier-Stokes equations in an exterior domain, Commun. Pure Appl. Anal., 17 (2018), 1681-1721.  doi: 10.3934/cpaa.2018081.  Google Scholar [25] Y. Shibata and S. Shimizu, On the maximal ${L}_p$-${L}_q$ regularity of the Stokes problem with first order boundary condition; model problems, J. Math. Soc. Japan, 64 (2012), 561-626.  doi: 10.2969/jmsj/06420561.  Google Scholar [26] C. G. Simader and H. Sohr, A new approach to the Helmholtz decomposition and the Neumann problem in $L^q$-spaces for bounded and exterior domains, in Mathematical Problems Relating to the Navier-Stokes Equation, Ser. Adv. Math. Appl. Sci., 11, World Sci. Publ., River Edge, NJ, 1992, 1-35. doi: 10.1142/9789814503594_0001.  Google Scholar [27] C. G. Simader and H. Sohr, The Dirichlet Problem for the Laplacian in Bounded and Unbounded Domains, Pitman Research Notes in Mathematics Series, 360, Longman, Harlow, 1996.  Google Scholar [28] H. Sohr, The Navier-Stokes Equations, An elementary functional analytic approach, Modern Birkhäuser Classics. Birkhäuser/Springer Basel AG, Basel, 2013.  Google Scholar [29] V. A. Solonnikov, Estimates of the solutions of the nonstationary Navier-Stokes system, Zap. Naučn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI), 38 (1973), 153-231.   Google Scholar [30] S. Szufla, On the Hammerstein integral equation with weakly singular kernel, Funkcial. Ekvac., 34 (1991), 279-285.   Google Scholar [31] H. Weyl, The method of orthogonal projection in potential theory, Duke Math. J., 7 (1940), 411-444.  doi: 10.1215/S0012-7094-40-00725-6.  Google Scholar
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