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December  2014, 3(4): 627-644. doi: 10.3934/eect.2014.3.627

On a linear problem arising in dynamic boundaries

1. 

Department of Mathematics, Vanderbilt University, 326 Stevenson Center, Nashville, TN, 37240, United States

Received  March 2014 Revised  May 2014 Published  October 2014

We study a linear problem that arises in the study of dynamic boundaries, in particular in free boundary problems in connection with fluid dynamics. The equations are also very natural and of interest on their own.
Citation: Marcelo Disconzi. On a linear problem arising in dynamic boundaries. Evolution Equations and Control Theory, 2014, 3 (4) : 627-644. doi: 10.3934/eect.2014.3.627
References:
[1]

R. A. Adams and J. J. F. Fournier, Sobolev Spaces, Second Edition (Pure and Applied Mathematics), 140. Elsevier/Academic Press, Amsterdam, 2003.

[2]

D. M. Ambrose, Well-posedness of vortex sheets with surface tension, SIAM J. Math. Anal., 35 (2003), 211-244. doi: 10.1137/S0036141002403869.

[3]

D. M. Ambrose and N. Masmoudi, The zero surface tension limit of two-dimensional water waves, Comm. Pure Appl. Math., 58 (2005), 1287-1315. doi: 10.1002/cpa.20085.

[4]

W. Arendt, C. Batty, M. Hieber and F. Neubrander, Vector-valued Laplace Transforms and Cauchy Problems, Monographs in Mathematics, 96. Birkhäuser Verlag, Basel, 2001. doi: 10.1007/978-3-0348-5075-9.

[5]

J. P. Bourguignon and H. Brezis, Remarks on the Euler equation, Journal of Functional Analysis, 15 (1974), 341-363. doi: 10.1016/0022-1236(74)90027-5.

[6]

W. Craig, An existence theory for water waves and the Boussinesq and Korteweg-de Vries scaling limits, Comm. Partial Differential Equations, 10 (1985), 787-1003. doi: 10.1080/03605308508820396.

[7]

D. Christodoulou and H. Lindblad, On the motion of the free surface of a liquid, Comm. Pure Appl. Math., 53 (2000), 1536-1602. doi: 10.1002/1097-0312(200012)53:12<1536::AID-CPA2>3.0.CO;2-Q.

[8]

D. Coutand and S. Shkoller, Well-posedness of the free-surface incompressible Euler equations with or without surface tension, J. Amer. Math. Soc., 20 (2007), 829-930. doi: 10.1090/S0894-0347-07-00556-5.

[9]

D. Coutand and S. Shkoller, A simple proof of well-posedness for the free-surface incompressible Euler equations, Discrete Contin. Dyn. Syst. Ser. S, 3 (2010), 429-449. doi: 10.3934/dcdss.2010.3.429.

[10]

D. Coutand and S. Shkoller, Well-Posedness in Smooth Function Spaces for the Moving-Boundary Three-Dimensional Compressible Euler Equations in Physical Vacuum, Arch. Ration. Mech. Anal., 206 (2012), 515-616. doi: 10.1007/s00205-012-0536-1.

[11]

D. Coutand and S. Shkoller, Well-posedness in smooth function spaces for moving-boundary 1-D compressible Euler equations in physical vacuum, Comm. Pure Appl. Math., 64 (2011), 328-366. doi: 10.1002/cpa.20344.

[12]

D. Coutand and S. Shkoller, On the finite-time splash and splat singularities for the 3-D free-surface Euler equations, Commun. Math. Phys., 325 (2014), 143-183. doi: 10.1007/s00220-013-1855-2.

[13]

D. Coutand, J. Hole and S. Shkoller, Well-posedness of the free-boundary compressible 3-D Euler equations with surface tension and the zero surface tension limit, SIAM J. Math. Anal., 45 (2013), 3690-3767. doi: 10.1137/120888697.

[14]

D. Coutand, H. Lindblad and S. Shkoller, A priori estimates for the free-boundary 3D compressible Euler equations in physical vacuum, Comm. Math. Phys., 296 (2010), 559-587. doi: 10.1007/s00220-010-1028-5.

[15]

E. Di Nezza, G. Palatucci and E. Valdinoci, Hitchhiker's Guide to the fractional Sobolev Spaces,, arXiv:1104.4345 [math.FA], (). 

[16]

M. M. Disconzi and D. G. Ebin, On the limit of large surface tension for a fluid motion with free boundary, Communications in Partial Differential Equations, 39 (2014), 740-779. doi: 10.1080/03605302.2013.865058.

[17]

M. M. Disconzi and D. G. Ebin, The Free Boundary Euler Equations with Large Surface Tension,, In preparation., (). 

[18]

D. G. Ebin, The manifold of Riemannian metrics, 1970 Global Analysis, (Proc. Sympos. Pure Math., Vol. XV, Berkeley, Calif., 1968) pp. 11-40 Amer. Math. Soc., Providence, R.I.

[19]

D. G. Ebin, The equations of motion of a perfect fluid with free boundary are not well posed, Comm. in Partial Diff. Eq., 12 (1987), 1175-1201. doi: 10.1080/03605308708820523.

[20]

D. G. Ebin, Espace des Metrique Riemanniennes et Mouvement des Fluids via les Varietes D'applications, Ecole Polytechnique, Paris, 1972.

[21]

D. G. Ebin, The motion of slightly compressible fluids viewed as a motion with strong constraining force, Annals of Math., 105 (1977), 141-200. doi: 10.2307/1971029.

[22]

D. G. Ebin, The initial boundary value problem for sub-sonic fluid motion, Comm. on Pure and Applied Math., 32 (1979), 1-19. doi: 10.1002/cpa.3160320102.

[23]

D. G. Ebin, Geodesics on the symplectomorphism group, GAFA, 22 (2012), 202-212. doi: 10.1007/s00039-012-0150-2.

[24]

D. G. Ebin, Motion of slightly compressible fluids in a bounded domain I, Comm. Pure Appl. Math., 35 (1982), 451-485. doi: 10.1002/cpa.3160350402.

[25]

D. G. Ebin and M. M. Disconzi, Motion of Slightly Compressible Fluids II, arXiv: 1309.0477 [math.AP] (2013). 49 pages.

[26]

D. G. Ebin and J. Marsden, Groups of diffeomorphisms and the motion of an incompressible fluid, Annals of Math., 92 (1970), 102-163. doi: 10.2307/1970699.

[27]

J. Escher, The Dirichlet-Neumann operator on continuous functions, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 21 (1994), 235-266.

[28]

A. Favini, G. R. Goldstein, J. A. Goldstein and S. Romanelli, $C^0$-semigroups generated by second order differential operators with general Wentzell boundary conditions, Proc. Amer. Math. Soc., 128 (2000), 1981-1989. doi: 10.1090/S0002-9939-00-05486-1.

[29]

A. Favini, G. R. Goldstein, J. A. Goldstein and S. Romanelli, On some classes of differential operators generating analytic semigroups. Evolution equations and their applications in physical and life sciences (Bad Herrenalb, 1998), Lecture Notes in Pure and Appl. Math., Dekker, New York, 215 (2001), 105-120.

[30]

A. Favini, G. R. Goldstein, J. A. Goldstein and S. Romanelli, The heat equation with generalized Wentzell boundary condition, J. Evol. Equ., 2 (2002), 1-19. doi: 10.1007/s00028-002-8077-y.

[31]

A. Favini, G. R. Goldstein, J. A. Goldstein and S. Romanelli, The heat equation with nonlinear general Wentzell boundary condition, Adv. Differential Equations, 11 (2006), 481-510.

[32]

A. Favini, G. R. Goldstein, J. A. Goldstein and S. Romanelli, Classification of general Wentzell boundary conditions for fourth order operators in one space dimension, J. Math. Anal. Appl., 333 (2007), 219-235. doi: 10.1016/j.jmaa.2006.11.058.

[33]

T. Hintermann, Evolution equations with dynamic boundary conditions, Proc. Roy. Soc. Edinburgh Sect. A, 113 (1989), 43-60. doi: 10.1017/S0308210500023945.

[34]

T. Kato, The Cauchy problem for quasi-linear symmetric hyperbolic systems, Arch. Rational Mech. Anal., 58 (1975), 181-205. doi: 10.1007/BF00280740.

[35]

M. Köhne, J. Prüss and W. 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.

[36]

D. Lannes, Well-posedness of the water-waves equations, J. Amer. Math. Soc., 18 (2005), 605-654. doi: 10.1090/S0894-0347-05-00484-4.

[37]

H. Lindblad, Well-posedness for the motion of an incompressible liquid with free surface boundary, Annals of Mathematics, 162 (2005), 109-194. doi: 10.4007/annals.2005.162.109.

[38]

H. Lindblad, Well-posedness for the linearized motion of an incompressible liquid with free surface boundary, Comm. Pure Appl. Math., 56 (2003), 153-197. doi: 10.1002/cpa.10055.

[39]

H. Lindblad and K. Nordgren, A priori estimates for the motion of a self-gravitating incompressible liquid with free surface boundary, J. Hyperbolic Differ. Eq., 6 (2009), 407-432. doi: 10.1142/S021989160900185X.

[40]

J. Marsden, D. G. Ebin and A. E. Fischer, Diffeomorphism groups, hydrodynamics and relativity, Proceedings of the Thirteenth Biennial Seminar of the Canadian Mathematical Congress Differential Geometry and Applications,(Dalhousie Univ., Halifax, N. S., 1971), Canad. Math. Congr., Montreal, Que., 1 (1972), 135-279.

[41]

T. Makino, On a local existence theorem for the evolution equation of gaseous stars, in Patterns and Waves, Stud. Math. Appl., North-Holland, Amsterdam, 18 (1986), 459-479. doi: 10.1016/S0168-2024(08)70142-5.

[42]

I. S. Mogilevskii and V. A. Solonnikov, On the solvability of an evolution free boundary problem for the Navier-Stokes equations in Hölder spaces of functions, Mathematical problems relating to the Navier-Stokes equation, Ser. Adv. Math. Appl. Sci., World Sci. Publ., River Edge, NJ, 11 (1992), 105-181. doi: 10.1142/9789814503594_0004.

[43]

V. I. Nalimov, The Cauchy-Poisson Problem (in Russian), Dynamika Splosh. Sredy, 18 (1974), 104-210.

[44]

T. Nishida, Equations of fluid dynamics - free surface problems, Frontiers of the mathematical sciences: 1985 (New York, 1985). Comm. Pure Appl. Math., 39 (1986), S221-S238. doi: 10.1002/cpa.3160390712.

[45]

R. S. Palais, Seminar on the Atiyah-Singer Index Theorem, Ann. of Math. Studies No. 57, Princeton, 1965.

[46]

J. Prüss and G. Simonett, On the two-phase Navier-Stokes equations with surface tension, Interfaces Free Bound., 12 (2010), 311-345. doi: 10.4171/IFB/237.

[47]

B. Schweizer, On the three-dimensional Euler equations with a free boundary subject to surface tension, Ann. I. H. Poincaré - AN, 22 (2005), 753-781. doi: 10.1016/j.anihpc.2004.11.001.

[48]

P. Secchi, On the uniqueness of motion of viscous gaseous stars, Math. Methods Appl. Sci., 13 (1990), 391-404. doi: 10.1002/mma.1670130504.

[49]

P. Secchi, On the motion of gaseous stars in the presence of radiation, Commun. Part. Diff. Eqs., 15 (1990), 185-204. doi: 10.1080/03605309908820683.

[50]

P. Secchi, On the evolution equations of viscous gaseous stars, Ann. Scuola Norm. Sup. Pisa, 18 (1991), 295-318.

[51]

J. Shatah and C. Zeng, Geometry and a priori estimates for free boundary problems of the Euler's equation, Communications on Pure and Applied Mathematics, 61 (2008), 698-744. doi: 10.1002/cpa.20213.

[52]

Y. Shibata and S. Shimizu, Report on a local in time solvability of free surface problems for the Navier-Stokes equations with surface tension, Appl. Anal. 90, no. 1, (2011) 201-214.

[53]

V. A. Solonnikov, Solvability of the problem of evolution of an isolated amount of a viscous incompressible capillary fluid. (Russian) Mathematical questions in the theory of wave propagation, 14. Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 140 (1984), 179-186.

[54]

V. A. Solonnikov, Unsteady flow of a finite mass of a fluid bounded by a free surface, (Russian. English summary) Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov., (LOMI) 152 (1986), 137-157. Translation in J. Soviet Math., 40 (1988), 672-686. doi: 10.1007/BF01094193.

[55]

V. A. Solonnikov, Unsteady motions of a finite isolated mass of a self-gravitating fluid, (Russian) Algebra i Analiz, 1 (1989), 207-249. Translation in Leningrad Math. J., 1 (1990), 227-276.

[56]

V. A. Solonnikov, Solvability of a problem on the evolution of a viscous incompressible fluid, bounded by a free surface, on a finite time interval, (Russian) Algebra i Analiz, 3 (1991), 222-257. Translation in St. Petersburg Math. J., 3 (1992), 189-220.

[57]

V. A. Solonnikov, On the quasistationary approximation in the problem of motion of a capillary drop, Topics in Nonlinear Analysis, The Herbert Amann Anniversary Volume, (J. Escher, G. Simonett, eds.) Birkhäuser, Basel, 35 (1999), 643-671.

[58]

V. A. Solonnikov, $L^q$-estimates for a solution to the problem about the evolution of an isolated amount of a fluid, J. Math. Sci. (N. Y.), 117 (2003), 4237-4259. doi: 10.1023/A:1024872705127.

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V. A. Solonnikov, Lectures on evolution free boundary problems: Classical solutions, Mathematical aspects of evolving interfaces, (Funchal, 2000), Lecture Notes in Math., Springer, Berlin, 1812 (2003), 123-175. doi: 10.1007/978-3-540-39189-0_4.

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C. C. Travis and G. F. Webb, Cosine families and abstract nonlinear second order differential equations, Acta Mathematica Ademiae Scientiarum Hungaricae Tomus, 32 (1978), 75-96. doi: 10.1007/BF01902205.

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G. F. Webb, Existence and asymptotic behavior for a strongly damped nonlinear wave equation, Can. J. Math., 32 (1980), 631-643. doi: 10.4153/CJM-1980-049-5.

[62]

F. White, Fluid Mechanics, Mcgraw Hill Higher Education. 7th edition, 2011.

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S. Wu, Well-posedness in Sobolev spaces of the full water wave problem in 3-D, J. Amer. Math. Soc., 12 (1999), 445-495. doi: 10.1090/S0894-0347-99-00290-8.

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K. Yosida, Functional Analysis, Springer, 1980.

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H. Yosihara, Gravity Waves on the Free Surface of an Incompressible Perfect Fluid, Publ. RIMS Kyoto Univ., 18 (1982), 49-96. doi: 10.2977/prims/1195184016.

show all references

References:
[1]

R. A. Adams and J. J. F. Fournier, Sobolev Spaces, Second Edition (Pure and Applied Mathematics), 140. Elsevier/Academic Press, Amsterdam, 2003.

[2]

D. M. Ambrose, Well-posedness of vortex sheets with surface tension, SIAM J. Math. Anal., 35 (2003), 211-244. doi: 10.1137/S0036141002403869.

[3]

D. M. Ambrose and N. Masmoudi, The zero surface tension limit of two-dimensional water waves, Comm. Pure Appl. Math., 58 (2005), 1287-1315. doi: 10.1002/cpa.20085.

[4]

W. Arendt, C. Batty, M. Hieber and F. Neubrander, Vector-valued Laplace Transforms and Cauchy Problems, Monographs in Mathematics, 96. Birkhäuser Verlag, Basel, 2001. doi: 10.1007/978-3-0348-5075-9.

[5]

J. P. Bourguignon and H. Brezis, Remarks on the Euler equation, Journal of Functional Analysis, 15 (1974), 341-363. doi: 10.1016/0022-1236(74)90027-5.

[6]

W. Craig, An existence theory for water waves and the Boussinesq and Korteweg-de Vries scaling limits, Comm. Partial Differential Equations, 10 (1985), 787-1003. doi: 10.1080/03605308508820396.

[7]

D. Christodoulou and H. Lindblad, On the motion of the free surface of a liquid, Comm. Pure Appl. Math., 53 (2000), 1536-1602. doi: 10.1002/1097-0312(200012)53:12<1536::AID-CPA2>3.0.CO;2-Q.

[8]

D. Coutand and S. Shkoller, Well-posedness of the free-surface incompressible Euler equations with or without surface tension, J. Amer. Math. Soc., 20 (2007), 829-930. doi: 10.1090/S0894-0347-07-00556-5.

[9]

D. Coutand and S. Shkoller, A simple proof of well-posedness for the free-surface incompressible Euler equations, Discrete Contin. Dyn. Syst. Ser. S, 3 (2010), 429-449. doi: 10.3934/dcdss.2010.3.429.

[10]

D. Coutand and S. Shkoller, Well-Posedness in Smooth Function Spaces for the Moving-Boundary Three-Dimensional Compressible Euler Equations in Physical Vacuum, Arch. Ration. Mech. Anal., 206 (2012), 515-616. doi: 10.1007/s00205-012-0536-1.

[11]

D. Coutand and S. Shkoller, Well-posedness in smooth function spaces for moving-boundary 1-D compressible Euler equations in physical vacuum, Comm. Pure Appl. Math., 64 (2011), 328-366. doi: 10.1002/cpa.20344.

[12]

D. Coutand and S. Shkoller, On the finite-time splash and splat singularities for the 3-D free-surface Euler equations, Commun. Math. Phys., 325 (2014), 143-183. doi: 10.1007/s00220-013-1855-2.

[13]

D. Coutand, J. Hole and S. Shkoller, Well-posedness of the free-boundary compressible 3-D Euler equations with surface tension and the zero surface tension limit, SIAM J. Math. Anal., 45 (2013), 3690-3767. doi: 10.1137/120888697.

[14]

D. Coutand, H. Lindblad and S. Shkoller, A priori estimates for the free-boundary 3D compressible Euler equations in physical vacuum, Comm. Math. Phys., 296 (2010), 559-587. doi: 10.1007/s00220-010-1028-5.

[15]

E. Di Nezza, G. Palatucci and E. Valdinoci, Hitchhiker's Guide to the fractional Sobolev Spaces,, arXiv:1104.4345 [math.FA], (). 

[16]

M. M. Disconzi and D. G. Ebin, On the limit of large surface tension for a fluid motion with free boundary, Communications in Partial Differential Equations, 39 (2014), 740-779. doi: 10.1080/03605302.2013.865058.

[17]

M. M. Disconzi and D. G. Ebin, The Free Boundary Euler Equations with Large Surface Tension,, In preparation., (). 

[18]

D. G. Ebin, The manifold of Riemannian metrics, 1970 Global Analysis, (Proc. Sympos. Pure Math., Vol. XV, Berkeley, Calif., 1968) pp. 11-40 Amer. Math. Soc., Providence, R.I.

[19]

D. G. Ebin, The equations of motion of a perfect fluid with free boundary are not well posed, Comm. in Partial Diff. Eq., 12 (1987), 1175-1201. doi: 10.1080/03605308708820523.

[20]

D. G. Ebin, Espace des Metrique Riemanniennes et Mouvement des Fluids via les Varietes D'applications, Ecole Polytechnique, Paris, 1972.

[21]

D. G. Ebin, The motion of slightly compressible fluids viewed as a motion with strong constraining force, Annals of Math., 105 (1977), 141-200. doi: 10.2307/1971029.

[22]

D. G. Ebin, The initial boundary value problem for sub-sonic fluid motion, Comm. on Pure and Applied Math., 32 (1979), 1-19. doi: 10.1002/cpa.3160320102.

[23]

D. G. Ebin, Geodesics on the symplectomorphism group, GAFA, 22 (2012), 202-212. doi: 10.1007/s00039-012-0150-2.

[24]

D. G. Ebin, Motion of slightly compressible fluids in a bounded domain I, Comm. Pure Appl. Math., 35 (1982), 451-485. doi: 10.1002/cpa.3160350402.

[25]

D. G. Ebin and M. M. Disconzi, Motion of Slightly Compressible Fluids II, arXiv: 1309.0477 [math.AP] (2013). 49 pages.

[26]

D. G. Ebin and J. Marsden, Groups of diffeomorphisms and the motion of an incompressible fluid, Annals of Math., 92 (1970), 102-163. doi: 10.2307/1970699.

[27]

J. Escher, The Dirichlet-Neumann operator on continuous functions, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 21 (1994), 235-266.

[28]

A. Favini, G. R. Goldstein, J. A. Goldstein and S. Romanelli, $C^0$-semigroups generated by second order differential operators with general Wentzell boundary conditions, Proc. Amer. Math. Soc., 128 (2000), 1981-1989. doi: 10.1090/S0002-9939-00-05486-1.

[29]

A. Favini, G. R. Goldstein, J. A. Goldstein and S. Romanelli, On some classes of differential operators generating analytic semigroups. Evolution equations and their applications in physical and life sciences (Bad Herrenalb, 1998), Lecture Notes in Pure and Appl. Math., Dekker, New York, 215 (2001), 105-120.

[30]

A. Favini, G. R. Goldstein, J. A. Goldstein and S. Romanelli, The heat equation with generalized Wentzell boundary condition, J. Evol. Equ., 2 (2002), 1-19. doi: 10.1007/s00028-002-8077-y.

[31]

A. Favini, G. R. Goldstein, J. A. Goldstein and S. Romanelli, The heat equation with nonlinear general Wentzell boundary condition, Adv. Differential Equations, 11 (2006), 481-510.

[32]

A. Favini, G. R. Goldstein, J. A. Goldstein and S. Romanelli, Classification of general Wentzell boundary conditions for fourth order operators in one space dimension, J. Math. Anal. Appl., 333 (2007), 219-235. doi: 10.1016/j.jmaa.2006.11.058.

[33]

T. Hintermann, Evolution equations with dynamic boundary conditions, Proc. Roy. Soc. Edinburgh Sect. A, 113 (1989), 43-60. doi: 10.1017/S0308210500023945.

[34]

T. Kato, The Cauchy problem for quasi-linear symmetric hyperbolic systems, Arch. Rational Mech. Anal., 58 (1975), 181-205. doi: 10.1007/BF00280740.

[35]

M. Köhne, J. Prüss and W. 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.

[36]

D. Lannes, Well-posedness of the water-waves equations, J. Amer. Math. Soc., 18 (2005), 605-654. doi: 10.1090/S0894-0347-05-00484-4.

[37]

H. Lindblad, Well-posedness for the motion of an incompressible liquid with free surface boundary, Annals of Mathematics, 162 (2005), 109-194. doi: 10.4007/annals.2005.162.109.

[38]

H. Lindblad, Well-posedness for the linearized motion of an incompressible liquid with free surface boundary, Comm. Pure Appl. Math., 56 (2003), 153-197. doi: 10.1002/cpa.10055.

[39]

H. Lindblad and K. Nordgren, A priori estimates for the motion of a self-gravitating incompressible liquid with free surface boundary, J. Hyperbolic Differ. Eq., 6 (2009), 407-432. doi: 10.1142/S021989160900185X.

[40]

J. Marsden, D. G. Ebin and A. E. Fischer, Diffeomorphism groups, hydrodynamics and relativity, Proceedings of the Thirteenth Biennial Seminar of the Canadian Mathematical Congress Differential Geometry and Applications,(Dalhousie Univ., Halifax, N. S., 1971), Canad. Math. Congr., Montreal, Que., 1 (1972), 135-279.

[41]

T. Makino, On a local existence theorem for the evolution equation of gaseous stars, in Patterns and Waves, Stud. Math. Appl., North-Holland, Amsterdam, 18 (1986), 459-479. doi: 10.1016/S0168-2024(08)70142-5.

[42]

I. S. Mogilevskii and V. A. Solonnikov, On the solvability of an evolution free boundary problem for the Navier-Stokes equations in Hölder spaces of functions, Mathematical problems relating to the Navier-Stokes equation, Ser. Adv. Math. Appl. Sci., World Sci. Publ., River Edge, NJ, 11 (1992), 105-181. doi: 10.1142/9789814503594_0004.

[43]

V. I. Nalimov, The Cauchy-Poisson Problem (in Russian), Dynamika Splosh. Sredy, 18 (1974), 104-210.

[44]

T. Nishida, Equations of fluid dynamics - free surface problems, Frontiers of the mathematical sciences: 1985 (New York, 1985). Comm. Pure Appl. Math., 39 (1986), S221-S238. doi: 10.1002/cpa.3160390712.

[45]

R. S. Palais, Seminar on the Atiyah-Singer Index Theorem, Ann. of Math. Studies No. 57, Princeton, 1965.

[46]

J. Prüss and G. Simonett, On the two-phase Navier-Stokes equations with surface tension, Interfaces Free Bound., 12 (2010), 311-345. doi: 10.4171/IFB/237.

[47]

B. Schweizer, On the three-dimensional Euler equations with a free boundary subject to surface tension, Ann. I. H. Poincaré - AN, 22 (2005), 753-781. doi: 10.1016/j.anihpc.2004.11.001.

[48]

P. Secchi, On the uniqueness of motion of viscous gaseous stars, Math. Methods Appl. Sci., 13 (1990), 391-404. doi: 10.1002/mma.1670130504.

[49]

P. Secchi, On the motion of gaseous stars in the presence of radiation, Commun. Part. Diff. Eqs., 15 (1990), 185-204. doi: 10.1080/03605309908820683.

[50]

P. Secchi, On the evolution equations of viscous gaseous stars, Ann. Scuola Norm. Sup. Pisa, 18 (1991), 295-318.

[51]

J. Shatah and C. Zeng, Geometry and a priori estimates for free boundary problems of the Euler's equation, Communications on Pure and Applied Mathematics, 61 (2008), 698-744. doi: 10.1002/cpa.20213.

[52]

Y. Shibata and S. Shimizu, Report on a local in time solvability of free surface problems for the Navier-Stokes equations with surface tension, Appl. Anal. 90, no. 1, (2011) 201-214.

[53]

V. A. Solonnikov, Solvability of the problem of evolution of an isolated amount of a viscous incompressible capillary fluid. (Russian) Mathematical questions in the theory of wave propagation, 14. Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 140 (1984), 179-186.

[54]

V. A. Solonnikov, Unsteady flow of a finite mass of a fluid bounded by a free surface, (Russian. English summary) Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov., (LOMI) 152 (1986), 137-157. Translation in J. Soviet Math., 40 (1988), 672-686. doi: 10.1007/BF01094193.

[55]

V. A. Solonnikov, Unsteady motions of a finite isolated mass of a self-gravitating fluid, (Russian) Algebra i Analiz, 1 (1989), 207-249. Translation in Leningrad Math. J., 1 (1990), 227-276.

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