-
Previous Article
A new approach to MGT-thermoviscoelasticity
- DCDS Home
- This Issue
-
Next Article
Variational relations for metric mean dimension and rate distortion dimension
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 |
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.
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. |
| [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. |
| [3] |
R. A. Adams and J. J. F. Fournier, Sobolev Spaces, Pure and Applied Mathematics, 140, $2^{nd}$
edition, Elsevier/Academic Press, Amsterdam, 2003. |
| [4] |
E. DiBenedetto, Real Analysis, $2^{nd}$ edition, Birkhäuser Advanced Texts: Basel Textbooks, Birkhäuser, 2016.
doi: 10.1007/978-1-4939-4005-9. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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.
|
| [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. |
| [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. |
| [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. |
| [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.
|
| [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. |
| [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. |
| [17] |
T. Miyakawa,
The Helmholtz decomposition of vector fields in some unbounded domains, Math. J. Toyama Univ., 17 (1994), 115-149.
|
| [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. |
| [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. |
| [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. |
| [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. |
| [22] |
Y. Shibata, Introduction to the Mathematical Theory of Fluid Mechanics (Japanese), in press. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [28] |
H. Sohr, The Navier-Stokes Equations, An elementary functional analytic approach, Modern Birkhäuser Classics. Birkhäuser/Springer Basel AG, Basel, 2013. |
| [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.
|
| [30] |
S. Szufla,
On the Hammerstein integral equation with weakly singular kernel, Funkcial. Ekvac., 34 (1991), 279-285.
|
| [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. |
show all references
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. |
| [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. |
| [3] |
R. A. Adams and J. J. F. Fournier, Sobolev Spaces, Pure and Applied Mathematics, 140, $2^{nd}$
edition, Elsevier/Academic Press, Amsterdam, 2003. |
| [4] |
E. DiBenedetto, Real Analysis, $2^{nd}$ edition, Birkhäuser Advanced Texts: Basel Textbooks, Birkhäuser, 2016.
doi: 10.1007/978-1-4939-4005-9. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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.
|
| [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. |
| [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. |
| [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. |
| [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.
|
| [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. |
| [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. |
| [17] |
T. Miyakawa,
The Helmholtz decomposition of vector fields in some unbounded domains, Math. J. Toyama Univ., 17 (1994), 115-149.
|
| [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. |
| [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. |
| [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. |
| [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. |
| [22] |
Y. Shibata, Introduction to the Mathematical Theory of Fluid Mechanics (Japanese), in press. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [28] |
H. Sohr, The Navier-Stokes Equations, An elementary functional analytic approach, Modern Birkhäuser Classics. Birkhäuser/Springer Basel AG, Basel, 2013. |
| [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.
|
| [30] |
S. Szufla,
On the Hammerstein integral equation with weakly singular kernel, Funkcial. Ekvac., 34 (1991), 279-285.
|
| [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. |
| [1] |
Changyan Li, Hui Li. Well-posedness of the two-phase flow problem in incompressible MHD. Discrete and Continuous Dynamical Systems, 2021, 41 (12) : 5609-5632. doi: 10.3934/dcds.2021090 |
| [2] |
Barbara Lee Keyfitz, Richard Sanders, Michael Sever. Lack of hyperbolicity in the two-fluid model for two-phase incompressible flow. Discrete and Continuous Dynamical Systems - B, 2003, 3 (4) : 541-563. doi: 10.3934/dcdsb.2003.3.541 |
| [3] |
Brahim Amaziane, Mladen Jurak, Leonid Pankratov, Anja Vrbaški. Some remarks on the homogenization of immiscible incompressible two-phase flow in double porosity media. Discrete and Continuous Dynamical Systems - B, 2018, 23 (2) : 629-665. doi: 10.3934/dcdsb.2018037 |
| [4] |
Guochun Wu, Yinghui Zhang. Global analysis of strong solutions for the viscous liquid-gas two-phase flow model in a bounded domain. Discrete and Continuous Dynamical Systems - B, 2018, 23 (4) : 1411-1429. doi: 10.3934/dcdsb.2018157 |
| [5] |
Theodore Tachim Medjo. A two-phase flow model with delays. Discrete and Continuous Dynamical Systems - B, 2017, 22 (9) : 3273-3294. doi: 10.3934/dcdsb.2017137 |
| [6] |
Marie Henry, Danielle Hilhorst, Robert Eymard. Singular limit of a two-phase flow problem in porous medium as the air viscosity tends to zero. Discrete and Continuous Dynamical Systems - S, 2012, 5 (1) : 93-113. doi: 10.3934/dcdss.2012.5.93 |
| [7] |
Zhen Cheng, Wenjun Wang. The Cauchy problem of a two-phase flow model for a mixture of non-interacting compressible fluids. Communications on Pure and Applied Analysis, 2021, 20 (12) : 4155-4176. doi: 10.3934/cpaa.2021151 |
| [8] |
Jan Prüss, Jürgen Saal, Gieri Simonett. Singular limits for the two-phase Stefan problem. Discrete and Continuous Dynamical Systems, 2013, 33 (11&12) : 5379-5405. doi: 10.3934/dcds.2013.33.5379 |
| [9] |
Marianne Korten, Charles N. Moore. Regularity for solutions of the two-phase Stefan problem. Communications on Pure and Applied Analysis, 2008, 7 (3) : 591-600. doi: 10.3934/cpaa.2008.7.591 |
| [10] |
Ciprian G. Gal, Maurizio Grasselli. Longtime behavior for a model of homogeneous incompressible two-phase flows. Discrete and Continuous Dynamical Systems, 2010, 28 (1) : 1-39. doi: 10.3934/dcds.2010.28.1 |
| [11] |
Jie Jiang, Yinghua Li, Chun Liu. Two-phase incompressible flows with variable density: An energetic variational approach. Discrete and Continuous Dynamical Systems, 2017, 37 (6) : 3243-3284. doi: 10.3934/dcds.2017138 |
| [12] |
T. Tachim Medjo. Averaging of an homogeneous two-phase flow model with oscillating external forces. Discrete and Continuous Dynamical Systems, 2012, 32 (10) : 3665-3690. doi: 10.3934/dcds.2012.32.3665 |
| [13] |
Esther S. Daus, Josipa-Pina Milišić, Nicola Zamponi. Global existence for a two-phase flow model with cross-diffusion. Discrete and Continuous Dynamical Systems - B, 2020, 25 (3) : 957-979. doi: 10.3934/dcdsb.2019198 |
| [14] |
Theodore Tachim-Medjo. Optimal control of a two-phase flow model with state constraints. Mathematical Control and Related Fields, 2016, 6 (2) : 335-362. doi: 10.3934/mcrf.2016006 |
| [15] |
Daniela De Silva, Fausto Ferrari, Sandro Salsa. Recent progresses on elliptic two-phase free boundary problems. Discrete and Continuous Dynamical Systems, 2019, 39 (12) : 6961-6978. doi: 10.3934/dcds.2019239 |
| [16] |
Jan Prüss, Yoshihiro Shibata, Senjo Shimizu, Gieri Simonett. On well-posedness of incompressible two-phase flows with phase transitions: The case of equal densities. Evolution Equations and Control Theory, 2012, 1 (1) : 171-194. doi: 10.3934/eect.2012.1.171 |
| [17] |
V. S. Manoranjan, Hong-Ming Yin, R. Showalter. On two-phase Stefan problem arising from a microwave heating process. Discrete and Continuous Dynamical Systems, 2006, 15 (4) : 1155-1168. doi: 10.3934/dcds.2006.15.1155 |
| [18] |
Feng Ma, Mingfang Ni. A two-phase method for multidimensional number partitioning problem. Numerical Algebra, Control and Optimization, 2013, 3 (2) : 203-206. doi: 10.3934/naco.2013.3.203 |
| [19] |
Michela Eleuteri, Elisabetta Rocca, Giulio Schimperna. On a non-isothermal diffuse interface model for two-phase flows of incompressible fluids. Discrete and Continuous Dynamical Systems, 2015, 35 (6) : 2497-2522. doi: 10.3934/dcds.2015.35.2497 |
| [20] |
Dieter Bothe, Jan Prüss. Modeling and analysis of reactive multi-component two-phase flows with mass transfer and phase transition the isothermal incompressible case. Discrete and Continuous Dynamical Systems - S, 2017, 10 (4) : 673-696. doi: 10.3934/dcdss.2017034 |
2020 Impact Factor: 1.392
Tools
Metrics
Other articles
by authors
[Back to Top]