# American Institute of Mathematical Sciences

September  2012, 11(5): 2055-2078. doi: 10.3934/cpaa.2012.11.2055

## On singular Navier-Stokes equations and irreversible phase transitions

 1 Departamento de Matemática, Instituto de Matem, Brazil 2 Departamento de Matem, Brazil 3 Departamento de Matemática, IMECC - UNICAMP, Rua Sergio Buarque de Holanda, 651, 13083-859 Campinas, SP

Received  April 2010 Revised  December 2011 Published  March 2012

We analyze a singular system of partial differential equations corresponding to a model for the evolution of an irreversible solidification of certain pure materials by taking into account the effects of fluid flow in the molten regions. The model consists of a system of highly non-linear free-boundary parabolic equations and includes: a heat equation, a doubly nonlinear inclusion for the phase change and Navier-Stokes equations singularly perturbed by a Carman-Kozeny type term to take care of the flow in the mushy region and a Boussinesq term for the buoyancy forces due to thermal differences. Our approach to show existence of generalized solutions of this system involves time-discretization, a suitable regularization procedure and fixed point arguments.
Citation: José Luiz Boldrini, Luís H. de Miranda, Gabriela Planas. On singular Navier-Stokes equations and irreversible phase transitions. Communications on Pure and Applied Analysis, 2012, 11 (5) : 2055-2078. doi: 10.3934/cpaa.2012.11.2055
##### References:
 [1] R. A. Adams and J. J. F. Fournier, "Sobolev Spaces," 2nd edition, Academic Press, New York, 2003. [2] M. Aso, M. Frémond and N. Kenmochi, Parabolic systems with the unknown dependent constraints arising in phase transitions, "Free Boundary Problems: Theory and Applications" (eds. I. N. Figueiredo, J. N. Rodrigues and L. Santos), doi: 10.1007/978-3-7643-7719-9_5. [3] V. Barbu, "Nonlinear Semigroups and Differential Equations in Banach Spaces," Noordhoff, Leyden, 1976. [4] J. L. Boldrini and G. Planas, A tridimensional phase-field model with convection for phase change of an alloy, Discrete Continuous Dynam. Systems - A, 13 (2005), 429-450. doi: 10.3934/dcds.2005.13.429. [5] J. L. Boldrini and G. Planas, Some thoughts on mathematical modeling of solidification and melting, Boletín de la Sociedad Española de Matemática Aplicada, 41 (2007), 77-90. [6] E. Bonetti, Global solution to a nonlinear phase transition model with dissipation, Adv. Math. Sci. Appl., 12 (2002), 355-376. [7] G. Bonfanti, M. Frémond and F. Luterotti, Global solution to a nonlinear system for irreversible phase changes, Adv. Math. Sci. Appl., 10 (2000), 1-24. [8] H. Brezis, "Opératours Maximaux Monotones et Semi-Groupes de Contractions dans les Espaces de Hilbert," North-Holland Math. Studies, 5, North Holland, Amsterdan, 1973. [9] P. Colli, F. Luterotti, G. Schimperna and U. Stefanelli, Global existence for a class of generalized systems for irreversible phase changes, NoDEA, Nonlinear Differ. Equ. Appl., 9 (2002), 255-276. doi: 10.1007/s00030-002-8127-8. [10] K-H. Hoffmann and L. Jiang, Optimal control of a phase field model for solidification, Numer. Funct. Anal. and Optim., 13 (1992), 11-27. doi: 10.1080/01630569208816458. [11] P. Laurençot, G. Schimperma and U. Stefanelli, Global existence of a strong solution to the one-dimensional full model for irreversible phase transitions, J. Math. Anal. Appl., 271 (2002), 426-442. doi: 10.1016/S0022-247X(02)00127-0. [12] F. Luterotti, G. Schimperma and U. Stefanelli, Global solution to a phase field model with irreversible and constrained phase evolution, Quart. Appl. Math., 60 (2002), 301-316. [13] G. Planas and J. L. Boldrini, A bidimensional phase-field model with convection for change phase of an alloy, J. Math. Anal. Appl., 303 (2005), 669-687. doi: 10.1016/j.jmaa.2004.08.068. [14] J. Simon, Compact sets in the space $L^p(0,T;B)$, Ann. Mat. Pura Appl., 146 (1987), 65-96. doi: 10.1007/BF01762360.

show all references

##### References:
 [1] R. A. Adams and J. J. F. Fournier, "Sobolev Spaces," 2nd edition, Academic Press, New York, 2003. [2] M. Aso, M. Frémond and N. Kenmochi, Parabolic systems with the unknown dependent constraints arising in phase transitions, "Free Boundary Problems: Theory and Applications" (eds. I. N. Figueiredo, J. N. Rodrigues and L. Santos), doi: 10.1007/978-3-7643-7719-9_5. [3] V. Barbu, "Nonlinear Semigroups and Differential Equations in Banach Spaces," Noordhoff, Leyden, 1976. [4] J. L. Boldrini and G. Planas, A tridimensional phase-field model with convection for phase change of an alloy, Discrete Continuous Dynam. Systems - A, 13 (2005), 429-450. doi: 10.3934/dcds.2005.13.429. [5] J. L. Boldrini and G. Planas, Some thoughts on mathematical modeling of solidification and melting, Boletín de la Sociedad Española de Matemática Aplicada, 41 (2007), 77-90. [6] E. Bonetti, Global solution to a nonlinear phase transition model with dissipation, Adv. Math. Sci. Appl., 12 (2002), 355-376. [7] G. Bonfanti, M. Frémond and F. Luterotti, Global solution to a nonlinear system for irreversible phase changes, Adv. Math. Sci. Appl., 10 (2000), 1-24. [8] H. Brezis, "Opératours Maximaux Monotones et Semi-Groupes de Contractions dans les Espaces de Hilbert," North-Holland Math. Studies, 5, North Holland, Amsterdan, 1973. [9] P. Colli, F. Luterotti, G. Schimperna and U. Stefanelli, Global existence for a class of generalized systems for irreversible phase changes, NoDEA, Nonlinear Differ. Equ. Appl., 9 (2002), 255-276. doi: 10.1007/s00030-002-8127-8. [10] K-H. Hoffmann and L. Jiang, Optimal control of a phase field model for solidification, Numer. Funct. Anal. and Optim., 13 (1992), 11-27. doi: 10.1080/01630569208816458. [11] P. Laurençot, G. Schimperma and U. Stefanelli, Global existence of a strong solution to the one-dimensional full model for irreversible phase transitions, J. Math. Anal. Appl., 271 (2002), 426-442. doi: 10.1016/S0022-247X(02)00127-0. [12] F. Luterotti, G. Schimperma and U. Stefanelli, Global solution to a phase field model with irreversible and constrained phase evolution, Quart. Appl. Math., 60 (2002), 301-316. [13] G. Planas and J. L. Boldrini, A bidimensional phase-field model with convection for change phase of an alloy, J. Math. Anal. Appl., 303 (2005), 669-687. doi: 10.1016/j.jmaa.2004.08.068. [14] J. Simon, Compact sets in the space $L^p(0,T;B)$, Ann. Mat. Pura Appl., 146 (1987), 65-96. doi: 10.1007/BF01762360.
 [1] Hantaek Bae. Solvability of the free boundary value problem of the Navier-Stokes equations. Discrete and Continuous Dynamical Systems, 2011, 29 (3) : 769-801. doi: 10.3934/dcds.2011.29.769 [2] J. I. Díaz, J. F. Padial. On a free-boundary problem modeling the action of a limiter on a plasma. Conference Publications, 2007, 2007 (Special) : 313-322. doi: 10.3934/proc.2007.2007.313 [3] Zilai Li, Zhenhua Guo. On free boundary problem for compressible navier-stokes equations with temperature-dependent heat conductivity. Discrete and Continuous Dynamical Systems - B, 2017, 22 (10) : 3903-3919. doi: 10.3934/dcdsb.2017201 [4] Xulong Qin, Zheng-An Yao. Global solutions of the free boundary problem for the compressible Navier-Stokes equations with density-dependent viscosity. Communications on Pure and Applied Analysis, 2010, 9 (4) : 1041-1052. doi: 10.3934/cpaa.2010.9.1041 [5] Yoshihiro Shibata. On the local wellposedness of free boundary problem for the Navier-Stokes equations in an exterior domain. Communications on Pure and Applied Analysis, 2018, 17 (4) : 1681-1721. doi: 10.3934/cpaa.2018081 [6] Panagiota Daskalopoulos, Eunjai Rhee. Free-boundary regularity for generalized porous medium equations. Communications on Pure and Applied Analysis, 2003, 2 (4) : 481-494. doi: 10.3934/cpaa.2003.2.481 [7] Yoshikazu Giga. A remark on a Liouville problem with boundary for the Stokes and the Navier-Stokes equations. Discrete and Continuous Dynamical Systems - S, 2013, 6 (5) : 1277-1289. doi: 10.3934/dcdss.2013.6.1277 [8] Boris Muha, Zvonimir Tutek. Note on evolutionary free piston problem for Stokes equations with slip boundary conditions. Communications on Pure and Applied Analysis, 2014, 13 (4) : 1629-1639. doi: 10.3934/cpaa.2014.13.1629 [9] J. F. Padial. Existence and estimate of the location of the free-boundary for a non local inverse elliptic-parabolic problem arising in nuclear fusion. Conference Publications, 2011, 2011 (Special) : 1176-1185. doi: 10.3934/proc.2011.2011.1176 [10] Marcelo M. Disconzi, Igor Kukavica. A priori estimates for the 3D compressible free-boundary Euler equations with surface tension in the case of a liquid. Evolution Equations and Control Theory, 2019, 8 (3) : 503-542. doi: 10.3934/eect.2019025 [11] Zhenhua Guo, Zilai Li. Global existence of weak solution to the free boundary problem for compressible Navier-Stokes. Kinetic and Related Models, 2016, 9 (1) : 75-103. doi: 10.3934/krm.2016.9.75 [12] Feimin Huang, Xiaoding Shi, Yi Wang. Stability of viscous shock wave for compressible Navier-Stokes equations with free boundary. Kinetic and Related Models, 2010, 3 (3) : 409-425. doi: 10.3934/krm.2010.3.409 [13] Donatella Donatelli, Tessa Thorsen, Konstantina Trivisa. Weak dissipative solutions to a free-boundary problem for finitely extensible bead-spring chain molecules: Variable viscosity coefficients. Kinetic and Related Models, 2020, 13 (5) : 1047-1070. doi: 10.3934/krm.2020037 [14] Anna Lisa Amadori. Contour enhancement via a singular free boundary problem. Conference Publications, 2007, 2007 (Special) : 44-53. doi: 10.3934/proc.2007.2007.44 [15] Giovanna Bonfanti, Fabio Luterotti. A well-posedness result for irreversible phase transitions with a nonlinear heat flux law. Discrete and Continuous Dynamical Systems - S, 2013, 6 (2) : 331-351. doi: 10.3934/dcdss.2013.6.331 [16] Xavier Fernández-Real, Xavier Ros-Oton. On global solutions to semilinear elliptic equations related to the one-phase free boundary problem. Discrete and Continuous Dynamical Systems, 2019, 39 (12) : 6945-6959. doi: 10.3934/dcds.2019238 [17] Avner Friedman. Free boundary problems for systems of Stokes equations. Discrete and Continuous Dynamical Systems - B, 2016, 21 (5) : 1455-1468. doi: 10.3934/dcdsb.2016006 [18] Junde Wu, Shangbin Cui. Asymptotic behavior of solutions of a free boundary problem modelling the growth of tumors with Stokes equations. Discrete and Continuous Dynamical Systems, 2009, 24 (2) : 625-651. doi: 10.3934/dcds.2009.24.625 [19] Renjun Duan, Xiongfeng Yang. Stability of rarefaction wave and boundary layer for outflow problem on the two-fluid Navier-Stokes-Poisson equations. Communications on Pure and Applied Analysis, 2013, 12 (2) : 985-1014. doi: 10.3934/cpaa.2013.12.985 [20] Michal Beneš. Mixed initial-boundary value problem for the three-dimensional Navier-Stokes equations in polyhedral domains. Conference Publications, 2011, 2011 (Special) : 135-144. doi: 10.3934/proc.2011.2011.135

2021 Impact Factor: 1.273