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September  2021, 41(9): 4041-4064. doi: 10.3934/dcds.2021027

On the dynamics of 3D electrified falling films

1. 

Laboratoire de Mathématiques et Modélisation d'Évry, University of Evry & Paris Saclay, Evry, France

2. 

Departamento de Matemáticas, Estadística y Computación, Universidad de Cantabria, Santander, Spain

* Corresponding author: Jiao He

Received  August 2020 Revised  December 2020 Published  January 2021

Fund Project: The first author is supported by the Sophie Germain post-doc program of the Fondation Mathématique Jacques Hadamard

In this article, we consider a non-local variant of the Kuramoto-Sivashinsky equation in three dimensions (2D interface). Besides showing the global wellposedness of this equation we also obtain some qualitative properties of the solutions. In particular, we prove that the solutions become analytic in the spatial variable for positive time, the existence of a compact global attractor and an upper bound on the number of spatial oscillations of the solutions. We observe that such a bound is particularly interesting due to the chaotic behavior of the solutions.

Citation: Jiao He, Rafael Granero-Belinchón. On the dynamics of 3D electrified falling films. Discrete & Continuous Dynamical Systems, 2021, 41 (9) : 4041-4064. doi: 10.3934/dcds.2021027
References:
[1]

J. C. Bronski and T. N. Gambill, Uncertainty estimates and ${L}^2$ bounds for the Kuramoto-Sivashinsky equation, Nonlinearity, 19 (2006), 2023-2039.  doi: 10.1088/0951-7715/19/9/002.  Google Scholar

[2]

J. Burczak and R. Granero-Belinchón, On a generalized doubly parabolic Keller–Segel system in one spatial dimension, Mathematical Models and Methods in Applied Sciences, 26 (2016), 111-160.   Google Scholar

[3]

B. I. CohenJ. KrommesW. Tang and and M. Rosenbluth, Non-linear saturation of the dissipative trapped-ion mode by mode coupling, Nuclear fusion, 16 (1976), 971-992.  doi: 10.1088/0029-5515/16/6/009.  Google Scholar

[4]

P. ColletJ.-P. EckmannH. Epstein and and J. Stubbe, Analyticity for the Kuramoto-Sivashinsky equation, Physica D: Nonlinear Phenomena, 67 (1993), 321-326.  doi: 10.1016/0167-2789(93)90168-Z.  Google Scholar

[5]

P. ColletJ.-P. EckmannH. Epstein and and J. Stubbe, A global attracting set for the Kuramoto-Sivashinsky equation, Communications in Mathematical Physics, 152 (1993), 203-214.  doi: 10.1007/BF02097064.  Google Scholar

[6]

A. V. Coward and P. Hall., On the nonlinear interfacial instability of rotating core-annual flow, Theoretical and Computational Fluid Dynamics, 5 (1993), 269-289.  doi: 10.1007/BF00271423.  Google Scholar

[7]

S. M. Cox and P. C. Matthews, Exponential time differencing for stiff systems, Journal of Computational Physics, 176 (2002), 430-455.  doi: 10.1006/jcph.2002.6995.  Google Scholar

[8]

C. Foias and R. Temam, Gevrey class regularity for the solutions of the Navier-Stokes equations, Journal of Functional Analysis, 87 (1989), 359-369.  doi: 10.1016/0022-1236(89)90015-3.  Google Scholar

[9]

A. L. Frenkel and K. Indireshkumar, Wavy film flows down an inclined plane: perturbation theory and general evolution equation for the film thickness, Physical Review E, 60 (1999), 4143-4157.  doi: 10.1103/PhysRevE.60.4143.  Google Scholar

[10]

L. Giacomelli and F. Otto, New bounds for the Kuramoto-Sivashinsky equation, Communications on Pure and Applied Mathematics, 58 (2005), 297-318.  doi: 10.1002/cpa.20031.  Google Scholar

[11]

A. Gonzalez and A. Castellanos, Nonlinear electrohydrodynamic waves on films falling down an inclined plane, Physical Review E, 53 (1996), 3573-3578.  doi: 10.1103/PhysRevE.53.3573.  Google Scholar

[12]

J. Goodman, Stability of the Kuramoto-Sivashinsky and related systems, Communications on Pure and Applied Mathematics, 47 (1994), 293-306.  doi: 10.1002/cpa.3160470304.  Google Scholar

[13]

R. Granero-Belinchón and J. K. Hunter, On a nonlocal analog of the Kuramoto-Sivashinsky equation, Nonlinearity, 28 (2015), 1103-1133.  doi: 10.1088/0951-7715/28/4/1103.  Google Scholar

[14]

Z. Grujić, Spatial analyticity on the global attractor for the Kuramoto-Sivashinsky equation, Journal of Dynamics and Differential Equations, 12 (2000), 217-228.  doi: 10.1023/A:1009002920348.  Google Scholar

[15]

J. S. Il'Yashenko, Global analysis of the phase portrait for the Kuramoto-Sivashinsky equation, Journal of Dynamics and Differential Equations, 4 (1992), 585-615.  doi: 10.1007/BF01048261.  Google Scholar

[16]

X. Ioakim and Y.-S. Smyrlis, Analyticity for Kuramoto-Sivashinsky type equations and related systems, Procedia IUTAM, 11 (2014), 69-80.  doi: 10.1016/j.piutam.2014.01.049.  Google Scholar

[17]

M. James and M. Wilczek, Vortex dynamics and lagrangian statistics in a model for active turbulence, The European Physical Journal E, 41 (2018), 21-26.  doi: 10.1140/epje/i2018-11625-8.  Google Scholar

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A.-K. Kassam, and L.-N. Trefethen, Fourth-order time-stepping for stiff PDEs, SIAM Journal on Scientific Computing, 26 (2005), 1214-1233. doi: 10.1137/S1064827502410633.  Google Scholar

[19]

Y. Kuramoto and T. Tsuzuki, Persistent propagation of concentration waves in dissipative media far from thermal equilibrium, Progress of Theoretical Physics, 55 (1976), 356-369.  doi: 10.1143/PTP.55.356.  Google Scholar

[20]

R. E. LaQueyS. MahajanP. Rutherford and and W. Tang, Nonlinear saturation of the trapped-ion mode, Physical Review Letters, 34 (1975), 391-394.  doi: 10.1103/physrevlett.34.391.  Google Scholar

[21]

Y. Lee and H. Chen, Nonlinear dynamical models of plasma turbulence, Physica Scripta, (1982), 41–47. doi: 10.1088/0031-8949/1982/T2A/005.  Google Scholar

[22]

D. Michelson and G. I. Sivashinsky, Nonlinear analysis of hydrodynamic instability in laminar flames-II. numerical experiments, Acta Astronautica, 4 (1977), 1207-1221.  doi: 10.1016/0094-5765(77)90097-2.  Google Scholar

[23]

B. NicolaenkoB. Scheurer and and R. Temam, Some global dynamical properties of the Kuramoto-Sivashinsky equations: nonlinear stability and attractors, Physica D: Nonlinear Phenomena, 16 (1985), 155-183.  doi: 10.1016/0167-2789(85)90056-9.  Google Scholar

[24]

F. C. Pinto, Nonlinear stability and dynamical properties for a Kuramoto-Sivashinsky equation in space dimension two, Discrete and Continuous Dynamical Systems - A, 5 (1999), 117-136.  doi: 10.3934/dcds.1999.5.117.  Google Scholar

[25]

F. C. Pinto, Analyticity and Gevrey class regularity for a Kuramoto-Sivashinsky equation in space dimension two, Applied mathematics letters, 14 (2001), 253-260.  doi: 10.1016/S0893-9659(00)00145-2.  Google Scholar

[26]

G. I. Sivashinsky, Nonlinear analysis of hydrodynamic instability in laminar flames-I. derivation of basic equations, Acta astronautica, 4 (1977), 1177-1206.  doi: 10.1016/0094-5765(77)90096-0.  Google Scholar

[27]

G. I. Sivashinsky and D. M. Michelson, On irregular wavy flow of a liquid film down a vertical plane, Progress of Theoretical Physics, 63 (1980), 2112-2114.  doi: 10.1143/PTP.63.2112.  Google Scholar

[28]

R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, 2$^{nd}$ edition, Applied Mathematical Sciences, Springer-Verlag, 1997. doi: 10.1007/978-1-4612-0645-3.  Google Scholar

[29]

R. TomlinD. Papageorgiou and and G. Pavliotis, Three-dimensional wave evolution on electrified falling films, Journal of Fluid Mechanics, 822 (2017), 54-79.  doi: 10.1017/jfm.2017.250.  Google Scholar

[30]

J. Topper and T. Kawahara, Approximate equations for long nonlinear waves on a viscous fluid, Journal of the Physical society of Japan, 44 (1978), 663-666.  doi: 10.1143/JPSJ.44.663.  Google Scholar

[31]

D. Tseluiko and D. Papageorgiou, Dynamics of an electrostatically modified Kuramoto-Sivashinsky–Korteweg–de Vries equation arising in falling film flows, Physical Review E, 82 (2010), 016322, 22 pp. doi: 10.1103/PhysRevE.82.016322.  Google Scholar

[32]

D. Tseluiko and D. T. Papageorgiou, A global attracting set for nonlocal Kuramoto-Sivashinsky equations arising in interfacial electrohydrodynamics, European Journal of Applied Mathematics, 17 (2006), 677-703.  doi: 10.1017/S0956792506006760.  Google Scholar

[33]

D. Tseluiko and D. T. Papageorgiou, Wave evolution on electrified falling films, Journal of Fluid Mechanics, 556 (2006), 361-386.  doi: 10.1017/S0022112006009712.  Google Scholar

show all references

References:
[1]

J. C. Bronski and T. N. Gambill, Uncertainty estimates and ${L}^2$ bounds for the Kuramoto-Sivashinsky equation, Nonlinearity, 19 (2006), 2023-2039.  doi: 10.1088/0951-7715/19/9/002.  Google Scholar

[2]

J. Burczak and R. Granero-Belinchón, On a generalized doubly parabolic Keller–Segel system in one spatial dimension, Mathematical Models and Methods in Applied Sciences, 26 (2016), 111-160.   Google Scholar

[3]

B. I. CohenJ. KrommesW. Tang and and M. Rosenbluth, Non-linear saturation of the dissipative trapped-ion mode by mode coupling, Nuclear fusion, 16 (1976), 971-992.  doi: 10.1088/0029-5515/16/6/009.  Google Scholar

[4]

P. ColletJ.-P. EckmannH. Epstein and and J. Stubbe, Analyticity for the Kuramoto-Sivashinsky equation, Physica D: Nonlinear Phenomena, 67 (1993), 321-326.  doi: 10.1016/0167-2789(93)90168-Z.  Google Scholar

[5]

P. ColletJ.-P. EckmannH. Epstein and and J. Stubbe, A global attracting set for the Kuramoto-Sivashinsky equation, Communications in Mathematical Physics, 152 (1993), 203-214.  doi: 10.1007/BF02097064.  Google Scholar

[6]

A. V. Coward and P. Hall., On the nonlinear interfacial instability of rotating core-annual flow, Theoretical and Computational Fluid Dynamics, 5 (1993), 269-289.  doi: 10.1007/BF00271423.  Google Scholar

[7]

S. M. Cox and P. C. Matthews, Exponential time differencing for stiff systems, Journal of Computational Physics, 176 (2002), 430-455.  doi: 10.1006/jcph.2002.6995.  Google Scholar

[8]

C. Foias and R. Temam, Gevrey class regularity for the solutions of the Navier-Stokes equations, Journal of Functional Analysis, 87 (1989), 359-369.  doi: 10.1016/0022-1236(89)90015-3.  Google Scholar

[9]

A. L. Frenkel and K. Indireshkumar, Wavy film flows down an inclined plane: perturbation theory and general evolution equation for the film thickness, Physical Review E, 60 (1999), 4143-4157.  doi: 10.1103/PhysRevE.60.4143.  Google Scholar

[10]

L. Giacomelli and F. Otto, New bounds for the Kuramoto-Sivashinsky equation, Communications on Pure and Applied Mathematics, 58 (2005), 297-318.  doi: 10.1002/cpa.20031.  Google Scholar

[11]

A. Gonzalez and A. Castellanos, Nonlinear electrohydrodynamic waves on films falling down an inclined plane, Physical Review E, 53 (1996), 3573-3578.  doi: 10.1103/PhysRevE.53.3573.  Google Scholar

[12]

J. Goodman, Stability of the Kuramoto-Sivashinsky and related systems, Communications on Pure and Applied Mathematics, 47 (1994), 293-306.  doi: 10.1002/cpa.3160470304.  Google Scholar

[13]

R. Granero-Belinchón and J. K. Hunter, On a nonlocal analog of the Kuramoto-Sivashinsky equation, Nonlinearity, 28 (2015), 1103-1133.  doi: 10.1088/0951-7715/28/4/1103.  Google Scholar

[14]

Z. Grujić, Spatial analyticity on the global attractor for the Kuramoto-Sivashinsky equation, Journal of Dynamics and Differential Equations, 12 (2000), 217-228.  doi: 10.1023/A:1009002920348.  Google Scholar

[15]

J. S. Il'Yashenko, Global analysis of the phase portrait for the Kuramoto-Sivashinsky equation, Journal of Dynamics and Differential Equations, 4 (1992), 585-615.  doi: 10.1007/BF01048261.  Google Scholar

[16]

X. Ioakim and Y.-S. Smyrlis, Analyticity for Kuramoto-Sivashinsky type equations and related systems, Procedia IUTAM, 11 (2014), 69-80.  doi: 10.1016/j.piutam.2014.01.049.  Google Scholar

[17]

M. James and M. Wilczek, Vortex dynamics and lagrangian statistics in a model for active turbulence, The European Physical Journal E, 41 (2018), 21-26.  doi: 10.1140/epje/i2018-11625-8.  Google Scholar

[18]

A.-K. Kassam, and L.-N. Trefethen, Fourth-order time-stepping for stiff PDEs, SIAM Journal on Scientific Computing, 26 (2005), 1214-1233. doi: 10.1137/S1064827502410633.  Google Scholar

[19]

Y. Kuramoto and T. Tsuzuki, Persistent propagation of concentration waves in dissipative media far from thermal equilibrium, Progress of Theoretical Physics, 55 (1976), 356-369.  doi: 10.1143/PTP.55.356.  Google Scholar

[20]

R. E. LaQueyS. MahajanP. Rutherford and and W. Tang, Nonlinear saturation of the trapped-ion mode, Physical Review Letters, 34 (1975), 391-394.  doi: 10.1103/physrevlett.34.391.  Google Scholar

[21]

Y. Lee and H. Chen, Nonlinear dynamical models of plasma turbulence, Physica Scripta, (1982), 41–47. doi: 10.1088/0031-8949/1982/T2A/005.  Google Scholar

[22]

D. Michelson and G. I. Sivashinsky, Nonlinear analysis of hydrodynamic instability in laminar flames-II. numerical experiments, Acta Astronautica, 4 (1977), 1207-1221.  doi: 10.1016/0094-5765(77)90097-2.  Google Scholar

[23]

B. NicolaenkoB. Scheurer and and R. Temam, Some global dynamical properties of the Kuramoto-Sivashinsky equations: nonlinear stability and attractors, Physica D: Nonlinear Phenomena, 16 (1985), 155-183.  doi: 10.1016/0167-2789(85)90056-9.  Google Scholar

[24]

F. C. Pinto, Nonlinear stability and dynamical properties for a Kuramoto-Sivashinsky equation in space dimension two, Discrete and Continuous Dynamical Systems - A, 5 (1999), 117-136.  doi: 10.3934/dcds.1999.5.117.  Google Scholar

[25]

F. C. Pinto, Analyticity and Gevrey class regularity for a Kuramoto-Sivashinsky equation in space dimension two, Applied mathematics letters, 14 (2001), 253-260.  doi: 10.1016/S0893-9659(00)00145-2.  Google Scholar

[26]

G. I. Sivashinsky, Nonlinear analysis of hydrodynamic instability in laminar flames-I. derivation of basic equations, Acta astronautica, 4 (1977), 1177-1206.  doi: 10.1016/0094-5765(77)90096-0.  Google Scholar

[27]

G. I. Sivashinsky and D. M. Michelson, On irregular wavy flow of a liquid film down a vertical plane, Progress of Theoretical Physics, 63 (1980), 2112-2114.  doi: 10.1143/PTP.63.2112.  Google Scholar

[28]

R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, 2$^{nd}$ edition, Applied Mathematical Sciences, Springer-Verlag, 1997. doi: 10.1007/978-1-4612-0645-3.  Google Scholar

[29]

R. TomlinD. Papageorgiou and and G. Pavliotis, Three-dimensional wave evolution on electrified falling films, Journal of Fluid Mechanics, 822 (2017), 54-79.  doi: 10.1017/jfm.2017.250.  Google Scholar

[30]

J. Topper and T. Kawahara, Approximate equations for long nonlinear waves on a viscous fluid, Journal of the Physical society of Japan, 44 (1978), 663-666.  doi: 10.1143/JPSJ.44.663.  Google Scholar

[31]

D. Tseluiko and D. Papageorgiou, Dynamics of an electrostatically modified Kuramoto-Sivashinsky–Korteweg–de Vries equation arising in falling film flows, Physical Review E, 82 (2010), 016322, 22 pp. doi: 10.1103/PhysRevE.82.016322.  Google Scholar

[32]

D. Tseluiko and D. T. Papageorgiou, A global attracting set for nonlocal Kuramoto-Sivashinsky equations arising in interfacial electrohydrodynamics, European Journal of Applied Mathematics, 17 (2006), 677-703.  doi: 10.1017/S0956792506006760.  Google Scholar

[33]

D. Tseluiko and D. T. Papageorgiou, Wave evolution on electrified falling films, Journal of Fluid Mechanics, 556 (2006), 361-386.  doi: 10.1017/S0022112006009712.  Google Scholar

Figure 1.  Initial data (16)
Figure 2.  Numerical solution of (17) for $ \beta = 2, \delta = 0.5, \epsilon = 1 $ at $ t = 40 $
Figure 2 at $ t = 0, 20, 40 $">Figure 3.  Numerical solution profile along $ x $-direction for $ y = 0 $ of (17) with $ \beta = 2, \delta = 0.5, \epsilon = 1 $ and the same initial data as in Figure 2 at $ t = 0, 20, 40 $
Figure 2 at $ t = 0, 20, 40 $">Figure 4.  Numerical solution profile along $ y $-direction for $ x = \pi/2 $ of (17) with $ \beta = 2, \delta = 0.5, \epsilon = 1 $ and the same initial data as in Figure 2 at $ t = 0, 20, 40 $
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