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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 & 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, (2003).   Google Scholar

[2]

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

[3]

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

[4]

W. Arendt, C. Batty, M. Hieber and F. Neubrander, Vector-valued Laplace Transforms and Cauchy Problems,, Monographs in Mathematics, (2001).  doi: 10.1007/978-3-0348-5075-9.  Google Scholar

[5]

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

[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.  doi: 10.1080/03605308508820396.  Google Scholar

[7]

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

[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.  doi: 10.1090/S0894-0347-07-00556-5.  Google Scholar

[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.  doi: 10.3934/dcdss.2010.3.429.  Google Scholar

[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.  doi: 10.1007/s00205-012-0536-1.  Google Scholar

[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.  doi: 10.1002/cpa.20344.  Google Scholar

[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.  doi: 10.1007/s00220-013-1855-2.  Google Scholar

[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.  doi: 10.1137/120888697.  Google Scholar

[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.  doi: 10.1007/s00220-010-1028-5.  Google Scholar

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

[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.  doi: 10.1080/03605302.2013.865058.  Google Scholar

[17]

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

[18]

D. G. Ebin, The manifold of Riemannian metrics,, 1970 Global Analysis, (1968), 11.   Google Scholar

[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.  doi: 10.1080/03605308708820523.  Google Scholar

[20]

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

[21]

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

[22]

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

[23]

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

[24]

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

[25]

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

[26]

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

[27]

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

[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.  doi: 10.1090/S0002-9939-00-05486-1.  Google Scholar

[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, 215 (2001), 105.   Google Scholar

[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.  doi: 10.1007/s00028-002-8077-y.  Google Scholar

[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.   Google Scholar

[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.  doi: 10.1016/j.jmaa.2006.11.058.  Google Scholar

[33]

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

[34]

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

[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.  doi: 10.1007/s00208-012-0860-7.  Google Scholar

[36]

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

[37]

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

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

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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.  doi: 10.1142/S021989160900185X.  Google Scholar

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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, 1 (1972), 135.   Google Scholar

[41]

T. Makino, On a local existence theorem for the evolution equation of gaseous stars,, in Patterns and Waves, 18 (1986), 459.  doi: 10.1016/S0168-2024(08)70142-5.  Google Scholar

[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, 11 (1992), 105.  doi: 10.1142/9789814503594_0004.  Google Scholar

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

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T. Nishida, Equations of fluid dynamics - free surface problems,, Frontiers of the mathematical sciences: 1985 (New York, 39 (1986).  doi: 10.1002/cpa.3160390712.  Google Scholar

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

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

[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.  doi: 10.1016/j.anihpc.2004.11.001.  Google Scholar

[48]

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

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

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

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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.  doi: 10.1002/cpa.20213.  Google Scholar

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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, 90 (2011), 201.   Google Scholar

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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, 140 (1984), 179.   Google Scholar

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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., 152 (1986), 137.  doi: 10.1007/BF01094193.  Google Scholar

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V. A. Solonnikov, Unsteady motions of a finite isolated mass of a self-gravitating fluid, (Russian), Algebra i Analiz, 1 (1989), 207.   Google Scholar

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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.   Google Scholar

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V. A. Solonnikov, On the quasistationary approximation in the problem of motion of a capillary drop,, Topics in Nonlinear Analysis, 35 (1999), 643.   Google Scholar

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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, (2003).   Google Scholar

[2]

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

[3]

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

[4]

W. Arendt, C. Batty, M. Hieber and F. Neubrander, Vector-valued Laplace Transforms and Cauchy Problems,, Monographs in Mathematics, (2001).  doi: 10.1007/978-3-0348-5075-9.  Google Scholar

[5]

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

[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.  doi: 10.1080/03605308508820396.  Google Scholar

[7]

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

[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.  doi: 10.1090/S0894-0347-07-00556-5.  Google Scholar

[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.  doi: 10.3934/dcdss.2010.3.429.  Google Scholar

[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.  doi: 10.1007/s00205-012-0536-1.  Google Scholar

[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.  doi: 10.1002/cpa.20344.  Google Scholar

[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.  doi: 10.1007/s00220-013-1855-2.  Google Scholar

[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.  doi: 10.1137/120888697.  Google Scholar

[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.  doi: 10.1007/s00220-010-1028-5.  Google Scholar

[15]

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

[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.  doi: 10.1080/03605302.2013.865058.  Google Scholar

[17]

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

[18]

D. G. Ebin, The manifold of Riemannian metrics,, 1970 Global Analysis, (1968), 11.   Google Scholar

[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.  doi: 10.1080/03605308708820523.  Google Scholar

[20]

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

[21]

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

[22]

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

[23]

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

[24]

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

[25]

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

[26]

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

[27]

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

[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.  doi: 10.1090/S0002-9939-00-05486-1.  Google Scholar

[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, 215 (2001), 105.   Google Scholar

[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.  doi: 10.1007/s00028-002-8077-y.  Google Scholar

[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.   Google Scholar

[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.  doi: 10.1016/j.jmaa.2006.11.058.  Google Scholar

[33]

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

[34]

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

[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.  doi: 10.1007/s00208-012-0860-7.  Google Scholar

[36]

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

[37]

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

[38]

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

[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.  doi: 10.1142/S021989160900185X.  Google Scholar

[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, 1 (1972), 135.   Google Scholar

[41]

T. Makino, On a local existence theorem for the evolution equation of gaseous stars,, in Patterns and Waves, 18 (1986), 459.  doi: 10.1016/S0168-2024(08)70142-5.  Google Scholar

[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, 11 (1992), 105.  doi: 10.1142/9789814503594_0004.  Google Scholar

[43]

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

[44]

T. Nishida, Equations of fluid dynamics - free surface problems,, Frontiers of the mathematical sciences: 1985 (New York, 39 (1986).  doi: 10.1002/cpa.3160390712.  Google Scholar

[45]

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

[46]

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

[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.  doi: 10.1016/j.anihpc.2004.11.001.  Google Scholar

[48]

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

[49]

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

[50]

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

[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.  doi: 10.1002/cpa.20213.  Google Scholar

[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, 90 (2011), 201.   Google Scholar

[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, 140 (1984), 179.   Google Scholar

[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., 152 (1986), 137.  doi: 10.1007/BF01094193.  Google Scholar

[55]

V. A. Solonnikov, Unsteady motions of a finite isolated mass of a self-gravitating fluid, (Russian), Algebra i Analiz, 1 (1989), 207.   Google Scholar

[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.   Google Scholar

[57]

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