October  2021, 20(10): 3379-3401. doi: 10.3934/cpaa.2021110

On a free boundary model for three-dimensional MEMS with a hinged top plate II: Parabolic case

Faculty of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, A–1090 Vienna, Austria

Received  January 2021 Revised  May 2021 Published  October 2021 Early access  June 2021

A parabolic free boundary problem modeling a three-dimensional electrostatic MEMS device is investigated. The device is made of a rigid ground plate and an elastic top plate which is hinged at its boundary, the plates being held at different voltages. The model couples a fourth-order semilinear parabolic equation for the deformation of the top plate to a Laplace equation for the electrostatic potential in the device. The strength of the coupling is tuned by a parameter $ \lambda $ which is proportional to the square of the applied voltage difference. It is proven that the model is locally well-posed in time and that, for $ \lambda $ sufficiently small, solutions exist globally in time. In addition, touchdown of the top plate on the ground plate is shown to be the only possible finite time singularity.

Citation: Katerina Nik. On a free boundary model for three-dimensional MEMS with a hinged top plate II: Parabolic case. Communications on Pure & Applied Analysis, 2021, 20 (10) : 3379-3401. doi: 10.3934/cpaa.2021110
References:
[1]

H. Amann, Nonhomogeneous linear and quasilinear elliptic and parabolic boundary value problems, in Function Spaces, Differential Operators and Nonlinear Analysis (Friedrichroda, 1992), vol. 133 of Teubner-Texte Math. Teubner, Stuttgart, (1993), 9–126. doi: 10.1007/978-3-663-11336-2\_1.  Google Scholar

[2]

H. Amann, Linear and Quasilinear Parabolic Problems, Volume I: Abstract Linear Theory, Birkhäuser, 1995. doi: 10.1007/978-3-0348-9221-6.  Google Scholar

[3]

J. EscherPh. Laurençot and Ch. Walker, A parabolic free boundary problem modeling electrostatic MEMS, Arch. Ration. Mech. Anal., 211 (2014), 389-417.  doi: 10.1007/s00205-013-0656-2.  Google Scholar

[4]

P. Grisvard, Elliptic Problems in Nonsmooth Domains, Pitman, Boston, 1985.  Google Scholar

[5]

D. Guidetti, On elliptic problems in Besov spaces, Math. Nachr., 152 (1991), 247-275.  doi: 10.1002/mana.19911520120.  Google Scholar

[6]

D. Guidetti, On interpolation with boundary conditions, Math. Z., 207 (1991), 439-460.  doi: 10.1007/BF02571401.  Google Scholar

[7]

D. Guidetti, On elliptic systems in $L^1$, Osaka J. Math., 30 (1993), 397-429.   Google Scholar

[8]

T. Kato, Perturbation Theory for Linear Operators, Springer, Berlin, 1995.  Google Scholar

[9]

Ph. Laurençot and Ch. Walker, A free boundary problem modeling electrostatic MEMS: I. Linear bending effects, Math. Ann., 360 (2014), 307-349.  doi: 10.1007/s00208-014-1032-8.  Google Scholar

[10]

Ph. Laurençot and Ch. Walker, A free boundary problem modeling electrostatic MEMS: II. Nonlinear bending, Math. Models Methods Appl. Sci., 24 (2014), 2549-2568.  doi: 10.1142/S0218202514500298.  Google Scholar

[11]

Ph. Laurençot and Ch. Walker, The time singular limit for a fourth-order damped wave equation for MEMS, Springer Proc. Math. Stat., 119 (2015), 233-246.  doi: 10.1007/978-3-319-12547-3\_10.  Google Scholar

[12]

Ph. Laurençot and Ch. Walker, On a three-dimensional free boundary problem modeling electrostatic MEMS, Interfaces Free Bound., 18 (2016), 393-411.  doi: 10.4171/IFB/368.  Google Scholar

[13]

Ph. Laurençot and Ch. Walker, A variational approach to a stationary free boundary problem modeling MEMS, ESAIM Control Optim. Calc. Var., 22 (2016), 417-438.  doi: 10.1051/cocv/2015012.  Google Scholar

[14]

Ph. Laurençot and Ch. Walker, Shape derivative of the Dirichlet energy for a transmission problem, Arch. Ration. Mech. Anal., 237 (2020), 447-496.  doi: 10.1007/s00205-020-01512-8.  Google Scholar

[15]

Ph. Laurençot and Ch. Walker, Touchdown is the only finite time singularity in a three-dimensional MEMS model, Ann. Math. Blaise Pascal, 27 (2020), 65-81.   Google Scholar

[16]

A. F. Marques, R. C. Castelló and A. M. Shkel, Modelling the electrostatic actuation of MEMS: state of the art 2005, Technical Report, Universitat Politècnica de Catalunya, (2005). Google Scholar

[17]

K. Nik, On a free boundary model for three-dimensional MEMS with a hinged top plate I: Stationary sase, preprint, arXiv: 2103.06772. Google Scholar

[18]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, vol. 44 of Applied Mathematical Sciences, Springer-Verlag, New York, 1983.  Google Scholar

[19] J. A. Pelesko and D. H. Bernstein, Semigroups of Linear Operators and Applications to Partial Differential Equations, Chapman & Hall/CRC Press, Boca Raton, FL, 2003.   Google Scholar
[20]

G. Sweers and K. Vassi, Positivity for a hinged convex plate with stress, SIAM J. Math. Anal., 50 (2018), 1163-1174.  doi: 10.1137/17M1138790.  Google Scholar

[21]

H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, 2$^nd$ edition, Johann Ambrosius Barth, Heidelberg, 1995. Google Scholar

[22]

M. I. Younis, MEMS Linear and Nonlinear Statics and Dynamics, Springer, New York, 2011. Google Scholar

show all references

References:
[1]

H. Amann, Nonhomogeneous linear and quasilinear elliptic and parabolic boundary value problems, in Function Spaces, Differential Operators and Nonlinear Analysis (Friedrichroda, 1992), vol. 133 of Teubner-Texte Math. Teubner, Stuttgart, (1993), 9–126. doi: 10.1007/978-3-663-11336-2\_1.  Google Scholar

[2]

H. Amann, Linear and Quasilinear Parabolic Problems, Volume I: Abstract Linear Theory, Birkhäuser, 1995. doi: 10.1007/978-3-0348-9221-6.  Google Scholar

[3]

J. EscherPh. Laurençot and Ch. Walker, A parabolic free boundary problem modeling electrostatic MEMS, Arch. Ration. Mech. Anal., 211 (2014), 389-417.  doi: 10.1007/s00205-013-0656-2.  Google Scholar

[4]

P. Grisvard, Elliptic Problems in Nonsmooth Domains, Pitman, Boston, 1985.  Google Scholar

[5]

D. Guidetti, On elliptic problems in Besov spaces, Math. Nachr., 152 (1991), 247-275.  doi: 10.1002/mana.19911520120.  Google Scholar

[6]

D. Guidetti, On interpolation with boundary conditions, Math. Z., 207 (1991), 439-460.  doi: 10.1007/BF02571401.  Google Scholar

[7]

D. Guidetti, On elliptic systems in $L^1$, Osaka J. Math., 30 (1993), 397-429.   Google Scholar

[8]

T. Kato, Perturbation Theory for Linear Operators, Springer, Berlin, 1995.  Google Scholar

[9]

Ph. Laurençot and Ch. Walker, A free boundary problem modeling electrostatic MEMS: I. Linear bending effects, Math. Ann., 360 (2014), 307-349.  doi: 10.1007/s00208-014-1032-8.  Google Scholar

[10]

Ph. Laurençot and Ch. Walker, A free boundary problem modeling electrostatic MEMS: II. Nonlinear bending, Math. Models Methods Appl. Sci., 24 (2014), 2549-2568.  doi: 10.1142/S0218202514500298.  Google Scholar

[11]

Ph. Laurençot and Ch. Walker, The time singular limit for a fourth-order damped wave equation for MEMS, Springer Proc. Math. Stat., 119 (2015), 233-246.  doi: 10.1007/978-3-319-12547-3\_10.  Google Scholar

[12]

Ph. Laurençot and Ch. Walker, On a three-dimensional free boundary problem modeling electrostatic MEMS, Interfaces Free Bound., 18 (2016), 393-411.  doi: 10.4171/IFB/368.  Google Scholar

[13]

Ph. Laurençot and Ch. Walker, A variational approach to a stationary free boundary problem modeling MEMS, ESAIM Control Optim. Calc. Var., 22 (2016), 417-438.  doi: 10.1051/cocv/2015012.  Google Scholar

[14]

Ph. Laurençot and Ch. Walker, Shape derivative of the Dirichlet energy for a transmission problem, Arch. Ration. Mech. Anal., 237 (2020), 447-496.  doi: 10.1007/s00205-020-01512-8.  Google Scholar

[15]

Ph. Laurençot and Ch. Walker, Touchdown is the only finite time singularity in a three-dimensional MEMS model, Ann. Math. Blaise Pascal, 27 (2020), 65-81.   Google Scholar

[16]

A. F. Marques, R. C. Castelló and A. M. Shkel, Modelling the electrostatic actuation of MEMS: state of the art 2005, Technical Report, Universitat Politècnica de Catalunya, (2005). Google Scholar

[17]

K. Nik, On a free boundary model for three-dimensional MEMS with a hinged top plate I: Stationary sase, preprint, arXiv: 2103.06772. Google Scholar

[18]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, vol. 44 of Applied Mathematical Sciences, Springer-Verlag, New York, 1983.  Google Scholar

[19] J. A. Pelesko and D. H. Bernstein, Semigroups of Linear Operators and Applications to Partial Differential Equations, Chapman & Hall/CRC Press, Boca Raton, FL, 2003.   Google Scholar
[20]

G. Sweers and K. Vassi, Positivity for a hinged convex plate with stress, SIAM J. Math. Anal., 50 (2018), 1163-1174.  doi: 10.1137/17M1138790.  Google Scholar

[21]

H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, 2$^nd$ edition, Johann Ambrosius Barth, Heidelberg, 1995. Google Scholar

[22]

M. I. Younis, MEMS Linear and Nonlinear Statics and Dynamics, Springer, New York, 2011. Google Scholar

Figure 1.  Cross section of the idealized MEMS device
[1]

Mircea Sofonea, Yi-bin Xiao. Tykhonov well-posedness of a viscoplastic contact problem. Evolution Equations & Control Theory, 2020, 9 (4) : 1167-1185. doi: 10.3934/eect.2020048

[2]

Fujun Zhou, Shangbin Cui. Well-posedness and stability of a multidimensional moving boundary problem modeling the growth of tumor cord. Discrete & Continuous Dynamical Systems, 2008, 21 (3) : 929-943. doi: 10.3934/dcds.2008.21.929

[3]

Joachim Escher, Anca-Voichita Matioc. Well-posedness and stability analysis for a moving boundary problem modelling the growth of nonnecrotic tumors. Discrete & Continuous Dynamical Systems - B, 2011, 15 (3) : 573-596. doi: 10.3934/dcdsb.2011.15.573

[4]

Boling Guo, Jun Wu. Well-posedness of the initial-boundary value problem for the fourth-order nonlinear Schrödinger equation. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021205

[5]

Ugur G. Abdulla. On the optimal control of the free boundary problems for the second order parabolic equations. I. Well-posedness and convergence of the method of lines. Inverse Problems & Imaging, 2013, 7 (2) : 307-340. doi: 10.3934/ipi.2013.7.307

[6]

Qi Wang. On some touchdown behaviors of the generalized MEMS device equation. Communications on Pure & Applied Analysis, 2016, 15 (6) : 2447-2456. doi: 10.3934/cpaa.2016043

[7]

Igor Chueshov, Irena Lasiecka, Justin Webster. Flow-plate interactions: Well-posedness and long-time behavior. Discrete & Continuous Dynamical Systems - S, 2014, 7 (5) : 925-965. doi: 10.3934/dcdss.2014.7.925

[8]

Barbara Kaltenbacher, Irena Lasiecka. Well-posedness of the Westervelt and the Kuznetsov equation with nonhomogeneous Neumann boundary conditions. Conference Publications, 2011, 2011 (Special) : 763-773. doi: 10.3934/proc.2011.2011.763

[9]

Iñigo U. Erneta. Well-posedness for boundary value problems for coagulation-fragmentation equations. Kinetic & Related Models, 2020, 13 (4) : 815-835. doi: 10.3934/krm.2020028

[10]

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

[11]

Ivonne Rivas, Muhammad Usman, Bing-Yu Zhang. Global well-posedness and asymptotic behavior of a class of initial-boundary-value problem of the Korteweg-De Vries equation on a finite domain. Mathematical Control & Related Fields, 2011, 1 (1) : 61-81. doi: 10.3934/mcrf.2011.1.61

[12]

Zhaohui Huo, Boling Guo. The well-posedness of Cauchy problem for the generalized nonlinear dispersive equation. Discrete & Continuous Dynamical Systems, 2005, 12 (3) : 387-402. doi: 10.3934/dcds.2005.12.387

[13]

Hongmei Cao, Hao-Guang Li, Chao-Jiang Xu, Jiang Xu. Well-posedness of Cauchy problem for Landau equation in critical Besov space. Kinetic & Related Models, 2019, 12 (4) : 829-884. doi: 10.3934/krm.2019032

[14]

Changyan Li, Hui Li. Well-posedness of the two-phase flow problem in incompressible MHD. Discrete & Continuous Dynamical Systems, 2021, 41 (12) : 5609-5632. doi: 10.3934/dcds.2021090

[15]

Orazio Arena. A problem of boundary controllability for a plate. Evolution Equations & Control Theory, 2013, 2 (4) : 557-562. doi: 10.3934/eect.2013.2.557

[16]

Yoshihiro Shibata. Global well-posedness of unsteady motion of viscous incompressible capillary liquid bounded by a free surface. Evolution Equations & Control Theory, 2018, 7 (1) : 117-152. doi: 10.3934/eect.2018007

[17]

Jiali Lian. Global well-posedness of the free-interface incompressible Euler equations with damping. Discrete & Continuous Dynamical Systems, 2020, 40 (4) : 2061-2087. doi: 10.3934/dcds.2020106

[18]

Daniel Coutand, Steve Shkoller. A simple proof of well-posedness for the free-surface incompressible Euler equations. Discrete & Continuous Dynamical Systems - S, 2010, 3 (3) : 429-449. doi: 10.3934/dcdss.2010.3.429

[19]

Yoshihiro Shibata. Local well-posedness of free surface problems for the Navier-Stokes equations in a general domain. Discrete & Continuous Dynamical Systems - S, 2016, 9 (1) : 315-342. doi: 10.3934/dcdss.2016.9.315

[20]

Irena Lasiecka, Roberto Triggiani. A sharp trace result on a thermo-elastic plate equation with coupled hinged/Neumann boundary conditions. Discrete & Continuous Dynamical Systems, 1999, 5 (3) : 585-598. doi: 10.3934/dcds.1999.5.585

2020 Impact Factor: 1.916

Metrics

  • PDF downloads (56)
  • HTML views (167)
  • Cited by (0)

Other articles
by authors

[Back to Top]