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Strong instability of standing waves for nonlinear Schrödinger equations with a partial confinement
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 |
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 $\mathbb{R}^N$.
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.
|
| [2] |
H. Amann, Linear and Quasilinear Parabolic Problems, Vol. Ⅰ. Birkhäuser, Basel, 1995. |
| [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.
|
| [4] |
J. T. Beale,
Large time regularity of viscous surface waves, Arch. Rat. Mech. Anal., 84 (1984), 307-352.
|
| [5] |
J. T. Beale and T. Nishida,
Large time behavior of viscous surface waves, Lecture Notes in Numer. Appl. Anal., 8 (1985), 1-14.
|
| [6] |
D. Bothe and J. Prüss,
$L_p$ theory for a class of non-Newtonian fluids, SIAM J. Math. Anal., 39 (2007), 379-421.
|
| [7] |
A. P. Calderón,
Lebesgue spaces of differentiable functions and distributions, Proc. Symp. in Pure Math., 4 (1961), 33-49.
|
| [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.
|
| [9] |
G. P. Galdi, An Introduction to the Mathematical Theory of the Navier-Stokes Equations, Steady State Problem, Second Edition, Springer Monographs, Springer, 2011. |
| [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.
|
| [11] |
Y. Hataya,
A remark on Beal-Nishida's paper, Bull. Inst. Math. Acad. Sin. (N.S.), 6 (2011), 293-303.
|
| [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.
|
| [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.
|
| [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.
|
| [15] |
T. Nishida,
Equations of fluid dynamics -free surface problems, Comm. Pure Appl. Math., 39 (1986), 221-238.
|
| [16] |
J. Prüss and G. Simonett, Moving Interfaces ad Quasilinear Parabolic Evolution Equations, Monographs in Mathematics, vol. 105, Birkhäuser, 2016. |
| [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. |
| [18] |
H. Saito and Y. Shibata, On the global wellposedness of free boundary problem for the Navier Stokes systems with surface tension, Preprint. |
| [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. |
| [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.
|
| [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.
|
| [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. |
| [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.
|
| [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. |
| [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. |
| [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.
|
| [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.
|
| [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.
|
| [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. |
| [30] |
V. A. Solonnikov,
On the transient motion of an isolated volume of viscous incompressible fluid, Math. USSR Izvestiya, 31 (1988), 381-405.
|
| [31] |
V. Solonnikov,
Unsteady motion of a finite mass of fluid, bounded by a free surface, J. Soviet Math., 40 (1988), 672-685.
|
| [32] |
O. Steiger,
On Navier-Stokes equations with first order boundary conditions, J. Math. Fluid Mech., 8 (2006), 456-481.
|
| [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. |
| [34] |
N. Tanaka,
Global existence of two phase non-homogeneous viscous incompressible weak fluid flow, Commun. Partial Differential Equations, 18 (1993), 41-81.
|
| [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.
|
| [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.
|
| [37] |
L. Weis,
Operator-valued Fourier multiplier theorems and maximal Lp-regularity, Math. Ann., 319 (2001), 735-758.
|
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.
|
| [2] |
H. Amann, Linear and Quasilinear Parabolic Problems, Vol. Ⅰ. Birkhäuser, Basel, 1995. |
| [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.
|
| [4] |
J. T. Beale,
Large time regularity of viscous surface waves, Arch. Rat. Mech. Anal., 84 (1984), 307-352.
|
| [5] |
J. T. Beale and T. Nishida,
Large time behavior of viscous surface waves, Lecture Notes in Numer. Appl. Anal., 8 (1985), 1-14.
|
| [6] |
D. Bothe and J. Prüss,
$L_p$ theory for a class of non-Newtonian fluids, SIAM J. Math. Anal., 39 (2007), 379-421.
|
| [7] |
A. P. Calderón,
Lebesgue spaces of differentiable functions and distributions, Proc. Symp. in Pure Math., 4 (1961), 33-49.
|
| [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.
|
| [9] |
G. P. Galdi, An Introduction to the Mathematical Theory of the Navier-Stokes Equations, Steady State Problem, Second Edition, Springer Monographs, Springer, 2011. |
| [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.
|
| [11] |
Y. Hataya,
A remark on Beal-Nishida's paper, Bull. Inst. Math. Acad. Sin. (N.S.), 6 (2011), 293-303.
|
| [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.
|
| [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.
|
| [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.
|
| [15] |
T. Nishida,
Equations of fluid dynamics -free surface problems, Comm. Pure Appl. Math., 39 (1986), 221-238.
|
| [16] |
J. Prüss and G. Simonett, Moving Interfaces ad Quasilinear Parabolic Evolution Equations, Monographs in Mathematics, vol. 105, Birkhäuser, 2016. |
| [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. |
| [18] |
H. Saito and Y. Shibata, On the global wellposedness of free boundary problem for the Navier Stokes systems with surface tension, Preprint. |
| [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. |
| [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.
|
| [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.
|
| [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. |
| [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.
|
| [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. |
| [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. |
| [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.
|
| [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.
|
| [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.
|
| [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. |
| [30] |
V. A. Solonnikov,
On the transient motion of an isolated volume of viscous incompressible fluid, Math. USSR Izvestiya, 31 (1988), 381-405.
|
| [31] |
V. Solonnikov,
Unsteady motion of a finite mass of fluid, bounded by a free surface, J. Soviet Math., 40 (1988), 672-685.
|
| [32] |
O. Steiger,
On Navier-Stokes equations with first order boundary conditions, J. Math. Fluid Mech., 8 (2006), 456-481.
|
| [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. |
| [34] |
N. Tanaka,
Global existence of two phase non-homogeneous viscous incompressible weak fluid flow, Commun. Partial Differential Equations, 18 (1993), 41-81.
|
| [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.
|
| [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.
|
| [37] |
L. Weis,
Operator-valued Fourier multiplier theorems and maximal Lp-regularity, Math. Ann., 319 (2001), 735-758.
|
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