American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Lyapunov function computation for autonomous linear stochastic differential equations using sum-of-squares programming
  • DCDS-B Home
  • This Issue
  • Next Article
    Linear type centers of polynomial Hamiltonian systems with nonlinearities of degree 4 symmetric with respect to the y-axis
March  2018, 23(2): 913-938. doi: 10.3934/dcdsb.2018048

Large deviation principle for the micropolar, magneto-micropolar fluid systems

Kazuo Yamazaki

Department of Mathematics, University of Rochester, Rochester, NY, 14627, USA

Received  January 2017 Revised  August 2017 Published  December 2017

Micropolar fluid and magneto-micropolar fluid systems are systems of equations with distinctive feature in its applicability and also mathematical difficulty. The purpose of this work is to follow the approach of [8] and show that another general class of systems of equations, that includes the two-dimensional micropolar and magneto-micropolar fluid systems, is well-posed and satisfies the Laplace principle, and consequently the large deviation principle, with the same rate function.

Keywords: Laplace principle, large deviation principle, micropolar fluid system, magneto-micropolar fluid system, well-posedness.
Mathematics Subject Classification: Primary: 35Q35, 37L55; Secondary: 60F10.
Citation: Kazuo Yamazaki. Large deviation principle for the micropolar, magneto-micropolar fluid systems. Discrete & Continuous Dynamical Systems - B, 2018, 23 (2) : 913-938. doi: 10.3934/dcdsb.2018048
References:
[1]

G. Ahmadi and M. Shahinpoor, Universal stability of magneto-micropolar fluid motions, Int. J. Engng. Sci., 12 (1974), 657-663.  doi: 10.1016/0020-7225(74)90042-1.  Google Scholar

[2]

A. Bensoussan, Stochastic Navier-Stokes equations, Acta Appl. Math., 38 (1995), 267-304.  doi: 10.1007/BF00996149.  Google Scholar

[3]

J. L. Boldrini, M. A. Rojas-Medar and E. Fernández-Cara, Semi-Galerkin approximation and strong solutions to the equations of the nonhomogeneous asymmetric fluids, J. Math. Pures Appl., 82 (2003), 1499-1525.  doi: 10.1016/j.matpur.2003.09.005.  Google Scholar

[4]

A. Budhiraja and P. Dupuis, A variational representation for positive functionals of infinite dimensional Brownian motion, Probab. Math. Statist., 20 (2000), 39-61.   Google Scholar

[5]

A. Budhiraja, P. Dupuis and V. Maroulas, Large deviations for infinite dimensional stochastic dynamical systems, Ann. Probab., 36 (2008), 1390-1420.  doi: 10.1214/07-AOP362.  Google Scholar

[6]

S. Cerrai and M. Röckner, Large deviations for stochastic reaction-diffusion systems with multiplicative noise and non-Lipschitz reaction term, Ann. Probab., 32 (2004), 1100-1139.  doi: 10.1214/aop/1079021473.  Google Scholar

[7]

M.-H. Chang, Large deviation for Navier-Stokes equations with small stochastic perturbation, App. Math. Comput., 76 (1996), 65-93.  doi: 10.1016/0096-3003(95)00150-6.  Google Scholar

[8]

I. Chueshov and A. Millet, Stochastic 2D hydrodynamical type systems: Well posedness and large deviations, Appl. Math. Optim., 61 (2010), 379-420.  doi: 10.1007/s00245-009-9091-z.  Google Scholar

[9]

I. Chueshov and A. Millet, Stochastic two-dimensional hydrodynamical systems: Wong-Zakai approximation and support theorem, Stoch. Anal. Appl., 29 (2011), 570-611.  doi: 10.1080/07362994.2011.581081.  Google Scholar

[10]

P. Constantin and C. Foias, Navier-Stokes Equations, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, 1988.  Google Scholar

[11]

N. J. Cutland and B. Enright, Stochastic nonhomogeneous incompressible Navier-Stokes equations, J. Differential Equations, 228 (2006), 140-170.  doi: 10.1016/j.jde.2006.04.009.  Google Scholar

[12]

G. Da Prato and J. Zabczyk Stochastic Equations in Infinite Dimensions, Cambridge University Press, U. K., 2014.  Google Scholar

[13]

A. Dembo and O. Zeitouni Large Deviations Techniques and Applications, Springer-Verlag, Berlin, Heidelberg, 1998.  Google Scholar

[14]

B.-Q. Dong and Z. Zhang, Global regularity of the 2D micropolar fluid flows with zero angular viscosity, J. Differential Equations, 249 (2010), 200-213.  doi: 10.1016/j.jde.2010.03.016.  Google Scholar

[15]

J. Duan and A. Millet, Large deviations for the Boussinesq equations under random influences, Stochastic Process. Appl., 119 (2009), 2052-2081.  doi: 10.1016/j.spa.2008.10.004.  Google Scholar

[16]

P. Dupuis and R. S. Ellis A Weak Convergence Approach to the Theory of Large Deviations, John Wiley & Sons, New York, 1997.  Google Scholar

[17]

A. C. Eringen, Simple microfluids, Int. J. Engng. Sci., 2 (1964), 205-217.  doi: 10.1016/0020-7225(64)90005-9.  Google Scholar

[18]

A. C. Eringen, Theory of micropolar fluids, J. Math. Mech., 16 (1966), 1-18.   Google Scholar

[19]

F. Flandoli and D. Gatarek, Martingale and stationary solutions for stochastic Navier-Stokes equations, Probab. Theory Relat. Fields, 102 (1995), 367-391.  doi: 10.1007/BF01192467.  Google Scholar

[20]

M. I. Freidlin and A. D. Wentzell, Random Perturbations of Dynamical Systems, Third Edition, Springer, Heidelberg, New York, Dordrecht, London, 2012.  Google Scholar

[21]

G. P. Galdi and S. Rionero, A note on the existence and uniqueness of solutions of the micropolar fluid equations, Int. J. Engng. Sci., 15 (1977), 105-108.  doi: 10.1016/0020-7225(77)90025-8.  Google Scholar

[22]

T. Hmidi, S. Keraani and R. Rousset, Global well-posedness for a Boussinesq-Navier-Stokes system with critical dissipation, J. Differential Equations, 249 (2010), 2147-2174.  doi: 10.1016/j.jde.2010.07.008.  Google Scholar

[23]

I. Karatzas and S. E. Shreve, Brownian Motion and Stochastic Calculus, Third Edition, Springer, New York, 1991.  Google Scholar

[24]

H. Kunita, Stochastic Flows and Stochastic Differential Equations, Cambridge University Press, United Kingdom, 1990.  Google Scholar

[25]

J. L. Lions, Quelquels Méthodes de Réesolution des Problémes aux Limites Non Linéaires, Dunod, Gauthiers-Villars, Paris, 1969.  Google Scholar

[26]

G. Lukaszewicz, Micropolar Fluids, Theory and Applications, Birkh mat$\ddot h$rmauser, Boston, 1999.  Google Scholar

[27]

S. J. A. Malham, Regularity Assumptions and Length Scales for the Navier-Stokes Equations, Ph. D thesis, University of London, 1993. Google Scholar

[28]

U. Manna, S. S. Sritharan and P. Sundar, Large deviations for the stochastic shell model of turbulence, NoDEA Nonlinear Differential Equations Appl., 16 (2009), 493-521.  doi: 10.1007/s00030-009-0023-z.  Google Scholar

[29]

J.-L. Menaldi and S. Sritharan, Stochastic 2-D Navier-Stokes equation, Appl. Math. Optim., 46 (2002), 31-53.  doi: 10.1007/s00245-002-0734-6.  Google Scholar

[30]

E. E. Ortega-Torres and M. A. Rojas-Medar, Magneto-micropolar fluid motion: Global existence of strong solutions, Abstr. Appl. Anal., 4 (1999), 109-125.  doi: 10.1155/S1085337599000287.  Google Scholar

[31]

M. Röckner, B. Schmuland and X. Zhang, Yamada-Watanabe theorem for stochastic evolution equations in infinite dimensions, Cond. Matt. Phys., 11 (2008), 247-259.   Google Scholar

[32]

M. A. Rojas-Medar, Magneto-micropolar fluid motion: Existence and uniqueness of strong solutions, Math. Nachr., 188 (1997), 301-319.  doi: 10.1002/mana.19971880116.  Google Scholar

[33]

M. Sango, Density dependent stochastic Navier-Stokes equations with non-Lipschitz random forcing, Rev. Math. Phys., 22 (2010), 669-697.  doi: 10.1142/S0129055X10004041.  Google Scholar

[34]

A. V. Skorokhod, Limit theorems for stochastic processes, Theory Probab. Appl., 1 (1956), 261-290.  doi: 10.1137/1101022.  Google Scholar

[35]

S. S. Sritharan and P. Sundar, Large deviations for the two-dimensional Navier-Stokes equations with multiplicative noise, Stochastic Process. Appl., 116 (2006), 1636-1659.  doi: 10.1016/j.spa.2006.04.001.  Google Scholar

[36]

R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Second Edition, Springer-Verlag New York, Inc., 1997.  Google Scholar

[37]

K. Yamazaki, 3-D stochastic micropolar and magneto-micropolar fluid systems with non-Lipschitz multiplicative noise, Commun. Stoch. Anal., 8 (2014), 413-437.   Google Scholar

[38]

K. Yamazaki, Global regularity of the two-dimensional magneto-micropolar fluid system with zero angular viscosity, Discrete Contin. Dyn. Syst., 35 (2015), 2193-2207.  doi: 10.3934/dcds.2015.35.2193.  Google Scholar

[39]

K. Yamazaki, Recent developments on the micropolar and magneto-micropolar fluid systems: Deterministic and stochastic perspectives, in Stochastic Equations for Complex Systems: Theoretical and Computational Topics (eds. S. Heinz and H. Bessaih), Springer International Publishing, (2015), 85-103.  doi: 10.1007/978-3-319-18206-3_4.  Google Scholar

[40]

K. Yamazaki, Global martingale solution to the stochastic nonhomogeneous magnetohydrodynamics system, Adv. Differential Equations, 21 (2016), 1085-1116.   Google Scholar

[41]

K. Yamazaki, Exponential convergence of the stochastic micropolar and magneto-micropolar fluid systems, Commun. Stoch. Anal., 10 (2016), 271-295.   Google Scholar

show all references

References:
[1]

G. Ahmadi and M. Shahinpoor, Universal stability of magneto-micropolar fluid motions, Int. J. Engng. Sci., 12 (1974), 657-663.  doi: 10.1016/0020-7225(74)90042-1.  Google Scholar

[2]

A. Bensoussan, Stochastic Navier-Stokes equations, Acta Appl. Math., 38 (1995), 267-304.  doi: 10.1007/BF00996149.  Google Scholar

[3]

J. L. Boldrini, M. A. Rojas-Medar and E. Fernández-Cara, Semi-Galerkin approximation and strong solutions to the equations of the nonhomogeneous asymmetric fluids, J. Math. Pures Appl., 82 (2003), 1499-1525.  doi: 10.1016/j.matpur.2003.09.005.  Google Scholar

[4]

A. Budhiraja and P. Dupuis, A variational representation for positive functionals of infinite dimensional Brownian motion, Probab. Math. Statist., 20 (2000), 39-61.   Google Scholar

[5]

A. Budhiraja, P. Dupuis and V. Maroulas, Large deviations for infinite dimensional stochastic dynamical systems, Ann. Probab., 36 (2008), 1390-1420.  doi: 10.1214/07-AOP362.  Google Scholar

[6]

S. Cerrai and M. Röckner, Large deviations for stochastic reaction-diffusion systems with multiplicative noise and non-Lipschitz reaction term, Ann. Probab., 32 (2004), 1100-1139.  doi: 10.1214/aop/1079021473.  Google Scholar

[7]

M.-H. Chang, Large deviation for Navier-Stokes equations with small stochastic perturbation, App. Math. Comput., 76 (1996), 65-93.  doi: 10.1016/0096-3003(95)00150-6.  Google Scholar

[8]

I. Chueshov and A. Millet, Stochastic 2D hydrodynamical type systems: Well posedness and large deviations, Appl. Math. Optim., 61 (2010), 379-420.  doi: 10.1007/s00245-009-9091-z.  Google Scholar

[9]

I. Chueshov and A. Millet, Stochastic two-dimensional hydrodynamical systems: Wong-Zakai approximation and support theorem, Stoch. Anal. Appl., 29 (2011), 570-611.  doi: 10.1080/07362994.2011.581081.  Google Scholar

[10]

P. Constantin and C. Foias, Navier-Stokes Equations, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, 1988.  Google Scholar

[11]

N. J. Cutland and B. Enright, Stochastic nonhomogeneous incompressible Navier-Stokes equations, J. Differential Equations, 228 (2006), 140-170.  doi: 10.1016/j.jde.2006.04.009.  Google Scholar

[12]

G. Da Prato and J. Zabczyk Stochastic Equations in Infinite Dimensions, Cambridge University Press, U. K., 2014.  Google Scholar

[13]

A. Dembo and O. Zeitouni Large Deviations Techniques and Applications, Springer-Verlag, Berlin, Heidelberg, 1998.  Google Scholar

[14]

B.-Q. Dong and Z. Zhang, Global regularity of the 2D micropolar fluid flows with zero angular viscosity, J. Differential Equations, 249 (2010), 200-213.  doi: 10.1016/j.jde.2010.03.016.  Google Scholar

[15]

J. Duan and A. Millet, Large deviations for the Boussinesq equations under random influences, Stochastic Process. Appl., 119 (2009), 2052-2081.  doi: 10.1016/j.spa.2008.10.004.  Google Scholar

[16]

P. Dupuis and R. S. Ellis A Weak Convergence Approach to the Theory of Large Deviations, John Wiley & Sons, New York, 1997.  Google Scholar

[17]

A. C. Eringen, Simple microfluids, Int. J. Engng. Sci., 2 (1964), 205-217.  doi: 10.1016/0020-7225(64)90005-9.  Google Scholar

[18]

A. C. Eringen, Theory of micropolar fluids, J. Math. Mech., 16 (1966), 1-18.   Google Scholar

[19]

F. Flandoli and D. Gatarek, Martingale and stationary solutions for stochastic Navier-Stokes equations, Probab. Theory Relat. Fields, 102 (1995), 367-391.  doi: 10.1007/BF01192467.  Google Scholar

[20]

M. I. Freidlin and A. D. Wentzell, Random Perturbations of Dynamical Systems, Third Edition, Springer, Heidelberg, New York, Dordrecht, London, 2012.  Google Scholar

[21]

G. P. Galdi and S. Rionero, A note on the existence and uniqueness of solutions of the micropolar fluid equations, Int. J. Engng. Sci., 15 (1977), 105-108.  doi: 10.1016/0020-7225(77)90025-8.  Google Scholar

[22]

T. Hmidi, S. Keraani and R. Rousset, Global well-posedness for a Boussinesq-Navier-Stokes system with critical dissipation, J. Differential Equations, 249 (2010), 2147-2174.  doi: 10.1016/j.jde.2010.07.008.  Google Scholar

[23]

I. Karatzas and S. E. Shreve, Brownian Motion and Stochastic Calculus, Third Edition, Springer, New York, 1991.  Google Scholar

[24]

H. Kunita, Stochastic Flows and Stochastic Differential Equations, Cambridge University Press, United Kingdom, 1990.  Google Scholar

[25]

J. L. Lions, Quelquels Méthodes de Réesolution des Problémes aux Limites Non Linéaires, Dunod, Gauthiers-Villars, Paris, 1969.  Google Scholar

[26]

G. Lukaszewicz, Micropolar Fluids, Theory and Applications, Birkh mat$\ddot h$rmauser, Boston, 1999.  Google Scholar

[27]

S. J. A. Malham, Regularity Assumptions and Length Scales for the Navier-Stokes Equations, Ph. D thesis, University of London, 1993. Google Scholar

[28]

U. Manna, S. S. Sritharan and P. Sundar, Large deviations for the stochastic shell model of turbulence, NoDEA Nonlinear Differential Equations Appl., 16 (2009), 493-521.  doi: 10.1007/s00030-009-0023-z.  Google Scholar

[29]

J.-L. Menaldi and S. Sritharan, Stochastic 2-D Navier-Stokes equation, Appl. Math. Optim., 46 (2002), 31-53.  doi: 10.1007/s00245-002-0734-6.  Google Scholar

[30]

E. E. Ortega-Torres and M. A. Rojas-Medar, Magneto-micropolar fluid motion: Global existence of strong solutions, Abstr. Appl. Anal., 4 (1999), 109-125.  doi: 10.1155/S1085337599000287.  Google Scholar

[31]

M. Röckner, B. Schmuland and X. Zhang, Yamada-Watanabe theorem for stochastic evolution equations in infinite dimensions, Cond. Matt. Phys., 11 (2008), 247-259.   Google Scholar

[32]

M. A. Rojas-Medar, Magneto-micropolar fluid motion: Existence and uniqueness of strong solutions, Math. Nachr., 188 (1997), 301-319.  doi: 10.1002/mana.19971880116.  Google Scholar

[33]

M. Sango, Density dependent stochastic Navier-Stokes equations with non-Lipschitz random forcing, Rev. Math. Phys., 22 (2010), 669-697.  doi: 10.1142/S0129055X10004041.  Google Scholar

[34]

A. V. Skorokhod, Limit theorems for stochastic processes, Theory Probab. Appl., 1 (1956), 261-290.  doi: 10.1137/1101022.  Google Scholar

[35]

S. S. Sritharan and P. Sundar, Large deviations for the two-dimensional Navier-Stokes equations with multiplicative noise, Stochastic Process. Appl., 116 (2006), 1636-1659.  doi: 10.1016/j.spa.2006.04.001.  Google Scholar

[36]

R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Second Edition, Springer-Verlag New York, Inc., 1997.  Google Scholar

[37]

K. Yamazaki, 3-D stochastic micropolar and magneto-micropolar fluid systems with non-Lipschitz multiplicative noise, Commun. Stoch. Anal., 8 (2014), 413-437.   Google Scholar

[38]

K. Yamazaki, Global regularity of the two-dimensional magneto-micropolar fluid system with zero angular viscosity, Discrete Contin. Dyn. Syst., 35 (2015), 2193-2207.  doi: 10.3934/dcds.2015.35.2193.  Google Scholar

[39]

K. Yamazaki, Recent developments on the micropolar and magneto-micropolar fluid systems: Deterministic and stochastic perspectives, in Stochastic Equations for Complex Systems: Theoretical and Computational Topics (eds. S. Heinz and H. Bessaih), Springer International Publishing, (2015), 85-103.  doi: 10.1007/978-3-319-18206-3_4.  Google Scholar

[40]

K. Yamazaki, Global martingale solution to the stochastic nonhomogeneous magnetohydrodynamics system, Adv. Differential Equations, 21 (2016), 1085-1116.   Google Scholar

[41]

K. Yamazaki, Exponential convergence of the stochastic micropolar and magneto-micropolar fluid systems, Commun. Stoch. Anal., 10 (2016), 271-295.   Google Scholar

[1]

Hiroshi Inoue, Kei Matsuura, Mitsuharu Ôtani. Strong solutions of magneto-micropolar fluid equation. Conference Publications, 2003, 2003 (Special) : 439-448. doi: 10.3934/proc.2003.2003.439
[2]

Kazuo Yamazaki. Global regularity of the two-dimensional magneto-micropolar fluid system with zero angular viscosity. Discrete & Continuous Dynamical Systems - A, 2015, 35 (5) : 2193-2207. doi: 10.3934/dcds.2015.35.2193
[3]

Cung The Anh, Vu Manh Toi. Local exact controllability to trajectories of the magneto-micropolar fluid equations. Evolution Equations & Control Theory, 2017, 6 (3) : 357-379. doi: 10.3934/eect.2017019
[4]

Jinbo Geng, Xiaochun Chen, Sadek Gala. On regularity criteria for the 3D magneto-micropolar fluid equations in the critical Morrey-Campanato space. Communications on Pure & Applied Analysis, 2011, 10 (2) : 583-592. doi: 10.3934/cpaa.2011.10.583
[5]

George Avalos, Roberto Triggiani. Semigroup well-posedness in the energy space of a parabolic-hyperbolic coupled Stokes-Lamé PDE system of fluid-structure interaction. Discrete & Continuous Dynamical Systems - S, 2009, 2 (3) : 417-447. doi: 10.3934/dcdss.2009.2.417
[6]

Yueling Li, Yingchao Xie, Xicheng Zhang. Large deviation principle for stochastic heat equation with memory. Discrete & Continuous Dynamical Systems - A, 2015, 35 (11) : 5221-5237. doi: 10.3934/dcds.2015.35.5221
[7]

Haibo Cui, Haiyan Yin. Stability of the composite wave for the inflow problem on the micropolar fluid model. Communications on Pure & Applied Analysis, 2017, 16 (4) : 1265-1292. doi: 10.3934/cpaa.2017062
[8]

George Avalos, Pelin G. Geredeli, Justin T. Webster. Semigroup well-posedness of a linearized, compressible fluid with an elastic boundary. Discrete & Continuous Dynamical Systems - B, 2018, 23 (3) : 1267-1295. doi: 10.3934/dcdsb.2018151
[9]

Vanessa Barros, Felipe Linares. A remark on the well-posedness of a degenerated Zakharov system. Communications on Pure & Applied Analysis, 2015, 14 (4) : 1259-1274. doi: 10.3934/cpaa.2015.14.1259
[10]

Baoquan Yuan, Xiao Li. Blow-up criteria of smooth solutions to the three-dimensional micropolar fluid equations in Besov space. Discrete & Continuous Dynamical Systems - S, 2016, 9 (6) : 2167-2179. doi: 10.3934/dcdss.2016090
[11]

Bo-Qing Dong, Zhi-Min Chen. Asymptotic profiles of solutions to the 2D viscous incompressible micropolar fluid flows. Discrete & Continuous Dynamical Systems - A, 2009, 23 (3) : 765-784. doi: 10.3934/dcds.2009.23.765
[12]

Jihong Zhao, Ting Zhang, Qiao Liu. Global well-posedness for the dissipative system modeling electro-hydrodynamics with large vertical velocity component in critical Besov space. Discrete & Continuous Dynamical Systems - A, 2015, 35 (1) : 555-582. doi: 10.3934/dcds.2015.35.555
[13]

Changjie Fang, Weimin Han. Well-posedness and optimal control of a hemivariational inequality for nonstationary Stokes fluid flow. Discrete & Continuous Dynamical Systems - A, 2016, 36 (10) : 5369-5386. doi: 10.3934/dcds.2016036
[14]

Jishan Fan, Kun Zhao. Improved extensibility criteria and global well-posedness of a coupled chemotaxis-fluid model on bounded domains. Discrete & Continuous Dynamical Systems - B, 2018, 23 (9) : 3949-3967. doi: 10.3934/dcdsb.2018119
[15]

Gustavo Ponce, Jean-Claude Saut. Well-posedness for the Benney-Roskes/Zakharov- Rubenchik system. Discrete & Continuous Dynamical Systems - A, 2005, 13 (3) : 811-825. doi: 10.3934/dcds.2005.13.811
[16]

Hung Luong. Local well-posedness for the Zakharov system on the background of a line soliton. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2657-2682. doi: 10.3934/cpaa.2018126
[17]

Anna Amirdjanova, Jie Xiong. Large deviation principle for a stochastic navier-Stokes equation in its vorticity form for a two-dimensional incompressible flow. Discrete & Continuous Dynamical Systems - B, 2006, 6 (4) : 651-666. doi: 10.3934/dcdsb.2006.6.651
[18]

A. Guillin, R. Liptser. Examples of moderate deviation principle for diffusion processes. Discrete & Continuous Dynamical Systems - B, 2006, 6 (4) : 803-828. doi: 10.3934/dcdsb.2006.6.803
[19]

Wenjing Zhao. Local well-posedness and blow-up criteria of magneto-viscoelastic flows. Discrete & Continuous Dynamical Systems - A, 2018, 38 (9) : 4637-4655. doi: 10.3934/dcds.2018203
[20]

Qiao Liu, Ting Zhang, Jihong Zhao. Well-posedness for the 3D incompressible nematic liquid crystal system in the critical $L^p$ framework. Discrete & Continuous Dynamical Systems - A, 2016, 36 (1) : 371-402. doi: 10.3934/dcds.2016.36.371

2018 Impact Factor: 1.008

Tools

Metrics

  • PDF downloads (43)
  • HTML views (186)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences