|
[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.
|
{{journal.titleEn}} 
DownLoad: