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Optimal control for a conserved phase field system with a possibly singular potential
Global well-posedness of unsteady motion of viscous incompressible capillary liquid bounded by a free surface
Department of Mathematics and Research Instituteof Science and Engineering, Waseda University, Ohkubo 3-4-1, Shinjuku-ku, Tokyo 169-8555, Japan |
In this paper, we prove the global well-posedness of free boundary problems of the Navier-Stokes equations in a bounded domain with surface tension. The velocity field is obtained in the $L_p$ in time $L_q$ in space maximal regularity class, ($2 < p < ∞$, $N < q < ∞$, and $2/p + N/q < 1$), under the assumption that the initial domain is close to a ball and initial data are sufficiently small. The essential point of our approach is to drive the exponential decay theorem in the $L_p$-$L_q$ framework for the linearized equations with the help of maximal $L_p$-$L_q$ regularity theory for the Stokes equations with free boundary conditions and spectral analysis of the Stokes operator and the Laplace-Beltrami operator.
References:
[1] |
H. Abels,
The initial-value problem for the Navier-Stokes equations with a free surface in $L_q$ Sobolev spaces, Adv. Differential Equations, 10 (2005), 45-64.
|
[2] |
G. Allain,
Small-time existence for the Navier-Stokes equations with a free surface, Appl. Math. Optim., 16 (1987), 37-50.
doi: 10.1007/BF01442184. |
[3] |
J. T. Beale,
The initial value problem for the Navier-Stokes equations with a free boundary, Comm. Pure Appl. Math., 34 (1981), 359-392.
doi: 10.1002/cpa.3160340305. |
[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] |
Y. Hataya and S. Kawashima,
Decaying solution of the Navier-Stokes flow of infinite volume without surface tension, Nonlinear Anal., 71 (2009), 2535-2539.
doi: 10.1016/j.na.2009.05.061. |
[7] |
Y. Hataya,
A remark on Beal-Nishida's paper, Bull. Inst. Math. Acad. Sin. (N.S.), 6 (2011), 293-303.
|
[8] |
M. Köhne, J. Prüss and M. Wilke,
Qualitative Behavior 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. |
[9] |
I. Sh. Mogilevskiǐ and V. A. Solonnikov, On the solvability of a free boundary problem for
the Navier-Stokes equations in the Hölder spaces of functions, Nonlinear Analysis. A Tribute
in Honour of Giovanni Prodi, Quaderni, Pisa, (1991), 257–272. |
[10] |
P. B. Mucha and W. Zajączkowski,
On local existence of solutions of the free boundary problem for an incompressible viscous self-gravitating fluid motion, Applicationes Mathematicae, 27 (2000), 319-333.
|
[11] |
U. Neri,
Singular Integrals, Lecutre Notes in Mathematics 200, Springer, New York, 1971. |
[12] |
T. Nishida,
Equations of fluid dynamics -free surface problems, Comm. Pure Appl. Math., 39 (1986), 221-238.
doi: 10.1002/cpa.3160390712. |
[13] |
M. Padula and V. A. Solonnikov,
On the local solvability of free boundary problem for the Navier-Stokes equations, J. Math. Sci., 170 (2010), 522-553.
doi: 10.1007/s10958-010-0099-3. |
[14] |
J. Prüss and G. Simonett,
On the two-phase Navier-Stokes equations with surface tension, Interfaces and Free Boundaries, 12 (2010), 311-345.
|
[15] |
J. Prüess and G. Simonett,
Analytic solutions for the two-phase Navier-Stokes equations with surface tension and gravity, Progress in Nonlinear Differential Equations and Their Applications, 80 (2011), 507-540.
|
[16] |
J. Prüess and G. Simonett,
Moving Interfaces and Quasilinear Parabolic Evolution Equations, Birkhauser Monographs in Mathematics, 2016. |
[17] |
H. Saito and Y. Shibata,
On decay properties of solutions to the Stokes equations with surface tension and gravity in the half space, J. Math. Soc. Japan, 68 (2016), 1559-1614.
doi: 10.2969/jmsj/06841559. |
[18] |
H. Saito and Y. Shibata, On the global wellposedness of free boundary problem for the Navier Stokes systems with surface tension, Preprint. |
[19] |
B. Schweizer,
Free boundary fluid systems in a semigroup approach and oscillatory behavior, SIAM J. Math. Anal., 28 (1997), 1135-1157.
doi: 10.1137/S0036141096299892. |
[20] |
Y. Shibata,
On some free boundary problem of the Navier-Stokes equations in the maximal $L_p$
-$L_q$ regularity class, J. Differential Equations, 258 (2015), 4127-4155.
doi: 10.1016/j.jde.2015.01.028. |
[21] |
Y. Shibata, On the $\mathcal{R}$-bounded solution operators in the study of free boundary problem
for the Navier-Stokes equations, in Mathematical Fluid Dynaics, Present and Future, Tokyo,
Japna, November 2014 (eds. Y. Shibata and Y. Suzuki), Springer Proceedings in Mathematics & Staistics, 183 (2016), 203–285. |
[22] |
Y. Shibata,
Local well-posedness of free surface problems for the Navier-Stokes equations in a general domain, Discrete and Continuous Dynamical Systems, Series S, 9 (2016), 315-342.
|
[23] |
Y. Shibata and S. Shimizu,
On a free boundary problem for the Navier-Stokes equations, Differential Integral Equations, 20 (2007), 241-276.
|
[24] |
V. A. Solonnikov, Unsteady motion of a finite mass of fluid, bounded by a free surface, Zap.
Nauchn. Sem. (LOMI), 152 (1986), 137–157 (in Russian); English transl. : J. Soviet Math.,
40 (1988), 672–686. |
[25] |
V. A. Solonnikov, On the transient motion of an isolated volume of viscous incompressible
fluid, Izv. Akad. Nauk SSSR Ser. Mat., 51 (1987), 1065–1087 (in Russian); English transl. :
Math. USSR Izv. , 31 (1988), 381–405. |
[26] |
V. A. Solonnikov, Solvability of the problem of evolution of a viscous incompressible fluid
bounded by a free surface on a finite time interval, Algebra i Analiz, 3 (1991), 222–257 (in
Russian); English transl. : St. Petersburg Math. J. , 3 (1992), 189–220. |
[27] |
V. A. Solonnikov, Lectures on evolution free boundary problems: Classical solutions, L.
Ambrosio et al. : Lecture Note in Mathematics (eds. P. Colli and J. F. Rodrigues), SpringerVerlag, Berlin, Heidelberg, 1812 (2003), 123–175. |
[28] |
D. Sylvester,
Large time existence of small viscous surface waves without surface tension, Commun. Partial Differential Equations, 15 (1990), 823-903.
doi: 10.1080/03605309908820709. |
[29] |
N. Tanaka,
Global existence of two phase non-homogeneous viscous incompressible weak fluid flow, Commun. Partial Differential Equations, 18 (1993), 41-81.
doi: 10.1080/03605309308820921. |
[30] |
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.
doi: 10.1007/BF00375146. |
[31] |
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.
doi: 10.1007/BF00375142. |
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 Equations, 10 (2005), 45-64.
|
[2] |
G. Allain,
Small-time existence for the Navier-Stokes equations with a free surface, Appl. Math. Optim., 16 (1987), 37-50.
doi: 10.1007/BF01442184. |
[3] |
J. T. Beale,
The initial value problem for the Navier-Stokes equations with a free boundary, Comm. Pure Appl. Math., 34 (1981), 359-392.
doi: 10.1002/cpa.3160340305. |
[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] |
Y. Hataya and S. Kawashima,
Decaying solution of the Navier-Stokes flow of infinite volume without surface tension, Nonlinear Anal., 71 (2009), 2535-2539.
doi: 10.1016/j.na.2009.05.061. |
[7] |
Y. Hataya,
A remark on Beal-Nishida's paper, Bull. Inst. Math. Acad. Sin. (N.S.), 6 (2011), 293-303.
|
[8] |
M. Köhne, J. Prüss and M. Wilke,
Qualitative Behavior 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. |
[9] |
I. Sh. Mogilevskiǐ and V. A. Solonnikov, On the solvability of a free boundary problem for
the Navier-Stokes equations in the Hölder spaces of functions, Nonlinear Analysis. A Tribute
in Honour of Giovanni Prodi, Quaderni, Pisa, (1991), 257–272. |
[10] |
P. B. Mucha and W. Zajączkowski,
On local existence of solutions of the free boundary problem for an incompressible viscous self-gravitating fluid motion, Applicationes Mathematicae, 27 (2000), 319-333.
|
[11] |
U. Neri,
Singular Integrals, Lecutre Notes in Mathematics 200, Springer, New York, 1971. |
[12] |
T. Nishida,
Equations of fluid dynamics -free surface problems, Comm. Pure Appl. Math., 39 (1986), 221-238.
doi: 10.1002/cpa.3160390712. |
[13] |
M. Padula and V. A. Solonnikov,
On the local solvability of free boundary problem for the Navier-Stokes equations, J. Math. Sci., 170 (2010), 522-553.
doi: 10.1007/s10958-010-0099-3. |
[14] |
J. Prüss and G. Simonett,
On the two-phase Navier-Stokes equations with surface tension, Interfaces and Free Boundaries, 12 (2010), 311-345.
|
[15] |
J. Prüess and G. Simonett,
Analytic solutions for the two-phase Navier-Stokes equations with surface tension and gravity, Progress in Nonlinear Differential Equations and Their Applications, 80 (2011), 507-540.
|
[16] |
J. Prüess and G. Simonett,
Moving Interfaces and Quasilinear Parabolic Evolution Equations, Birkhauser Monographs in Mathematics, 2016. |
[17] |
H. Saito and Y. Shibata,
On decay properties of solutions to the Stokes equations with surface tension and gravity in the half space, J. Math. Soc. Japan, 68 (2016), 1559-1614.
doi: 10.2969/jmsj/06841559. |
[18] |
H. Saito and Y. Shibata, On the global wellposedness of free boundary problem for the Navier Stokes systems with surface tension, Preprint. |
[19] |
B. Schweizer,
Free boundary fluid systems in a semigroup approach and oscillatory behavior, SIAM J. Math. Anal., 28 (1997), 1135-1157.
doi: 10.1137/S0036141096299892. |
[20] |
Y. Shibata,
On some free boundary problem of the Navier-Stokes equations in the maximal $L_p$
-$L_q$ regularity class, J. Differential Equations, 258 (2015), 4127-4155.
doi: 10.1016/j.jde.2015.01.028. |
[21] |
Y. Shibata, On the $\mathcal{R}$-bounded solution operators in the study of free boundary problem
for the Navier-Stokes equations, in Mathematical Fluid Dynaics, Present and Future, Tokyo,
Japna, November 2014 (eds. Y. Shibata and Y. Suzuki), Springer Proceedings in Mathematics & Staistics, 183 (2016), 203–285. |
[22] |
Y. Shibata,
Local well-posedness of free surface problems for the Navier-Stokes equations in a general domain, Discrete and Continuous Dynamical Systems, Series S, 9 (2016), 315-342.
|
[23] |
Y. Shibata and S. Shimizu,
On a free boundary problem for the Navier-Stokes equations, Differential Integral Equations, 20 (2007), 241-276.
|
[24] |
V. A. Solonnikov, Unsteady motion of a finite mass of fluid, bounded by a free surface, Zap.
Nauchn. Sem. (LOMI), 152 (1986), 137–157 (in Russian); English transl. : J. Soviet Math.,
40 (1988), 672–686. |
[25] |
V. A. Solonnikov, On the transient motion of an isolated volume of viscous incompressible
fluid, Izv. Akad. Nauk SSSR Ser. Mat., 51 (1987), 1065–1087 (in Russian); English transl. :
Math. USSR Izv. , 31 (1988), 381–405. |
[26] |
V. A. Solonnikov, Solvability of the problem of evolution of a viscous incompressible fluid
bounded by a free surface on a finite time interval, Algebra i Analiz, 3 (1991), 222–257 (in
Russian); English transl. : St. Petersburg Math. J. , 3 (1992), 189–220. |
[27] |
V. A. Solonnikov, Lectures on evolution free boundary problems: Classical solutions, L.
Ambrosio et al. : Lecture Note in Mathematics (eds. P. Colli and J. F. Rodrigues), SpringerVerlag, Berlin, Heidelberg, 1812 (2003), 123–175. |
[28] |
D. Sylvester,
Large time existence of small viscous surface waves without surface tension, Commun. Partial Differential Equations, 15 (1990), 823-903.
doi: 10.1080/03605309908820709. |
[29] |
N. Tanaka,
Global existence of two phase non-homogeneous viscous incompressible weak fluid flow, Commun. Partial Differential Equations, 18 (1993), 41-81.
doi: 10.1080/03605309308820921. |
[30] |
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.
doi: 10.1007/BF00375146. |
[31] |
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.
doi: 10.1007/BF00375142. |
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