# American Institute of Mathematical Sciences

July  2015, 20(5): 1443-1460. doi: 10.3934/dcdsb.2015.20.1443

## On the Benilov-Vynnycky blow-up problem

 1 Institute of Mathematical Sciences, Claremont Graduate University, Claremont, CA 91711, United States 2 Department of Mathematical Sciences, Claremont McKenna College, Claremont, CA 91711, United States 3 Department of Mathematics, Pitzer College, Claremont, CA 91711, United States

Received  July 2013 Revised  June 2014 Published  May 2015

We study an initial-boundary value problem for a fourth-order parabolic partial differential equation with an unknown velocity. The equation originated from the linearization of a two-dimensional Couette flow model, that was recently proposed by Benilov and Vynnycky. In the case of a $180^{\circ}$-- contact angle between liquid and a moving plate Benilov and Vynnycky conjectured that the speed of the contact line blows up to infinity in finite time. In this paper we present numerical simulations and qualitative analysis of the model. We show that depending on the initial data and parameter values different long time behaviors of velocity can be observed. The speed of the contact line may blow up to infinity or converge to a constant.
Citation: Marina Chugunova, Chiu-Yen Kao, Sarun Seepun. On the Benilov-Vynnycky blow-up problem. Discrete & Continuous Dynamical Systems - B, 2015, 20 (5) : 1443-1460. doi: 10.3934/dcdsb.2015.20.1443
##### References:
 [1] M. J. Ablowitz and J. Villarroel, On the Kadomtsev-Petviashvili equation and associated constraints,, Stud. Appl. Math., 85 (1991), 195.   Google Scholar [2] E. S. Benilov, On the surface waves in a shallow channel with an uneven bottom,, Stud. Appl. Math., 87 (1992), 1.   Google Scholar [3] D. J. Benney and W. J. Timson, The rolling motion of a viscous fluid on and off a rigid surface,, Stud. Appl. Math, 63 (1980), 93.   Google Scholar [4] E. S. Benilov and M. Vynnycky, Contact lines with a $180^{\circ}$ contact angle,, J. Fluid Mech., 718 (2013), 481.   Google Scholar [5] B. B. Kadomtsev and V. I. Petviashvili, On the stability of solitary waves in weakly dispersing media,, Sov. Phys. Dokl., 15 (): 539.   Google Scholar [6] L. A. Ostrovskii, Nonlinear internal waves in the rotating ocean,, Okeanologiia, 18 (1978), 181.   Google Scholar [7] D. E. Pelinovsky and A. R. Giniyatullin, Finite-time singularities in the dynamical evolution of contact lines,, Bulletin of the Moscow State Regional University (Physics and Mathematics), 3 (2013), 14.   Google Scholar [8] D. E. Pelinovsky, A. R. Giniyatullin and Y. A. Panfilova, On solutions of the reduced model for the dynamical evolution of contact lines,, Transactions of Nizhni Novgorod State Technical University n.a. Alexeev N.4, 94 (2012), 45.   Google Scholar [9] D. E. Pelinovsky and C. Xu, On numerical modelling and the blow-up behavior of contact lines with a $180^{\circ}$ contact angle,, J. Engineer. Math., (2015).  doi: 10.1007/s10665-014-9763-9.  Google Scholar [10] J. Le Sommer, G. M. Reznik and V. Zeitlin, Nonlinear geostrophic adjustment of long-wave disturbances in the shallow-water model on the equatorial beta-plane,, Journal of Fluid Mechanics, 515 (2004), 135.  doi: 10.1017/S0022112004000229.  Google Scholar [11] M. Vynnycky and S. L. Mitchell, On the accuracy of a finite-difference method for parabolic partial differential equations with discontinuous boundary conditions,, Num. Heat Trans B, 64 (2013), 275.  doi: 10.1080/10407790.2013.797312.  Google Scholar [12] S. L. Mitchell and M. Vynnycky, On the numerical solution of two-phase Stefan problems with heat-flux boundary conditions,, J. Comp. Appl. Maths, 264 (2014), 49.  doi: 10.1016/j.cam.2014.01.003.  Google Scholar

show all references

##### References:
 [1] M. J. Ablowitz and J. Villarroel, On the Kadomtsev-Petviashvili equation and associated constraints,, Stud. Appl. Math., 85 (1991), 195.   Google Scholar [2] E. S. Benilov, On the surface waves in a shallow channel with an uneven bottom,, Stud. Appl. Math., 87 (1992), 1.   Google Scholar [3] D. J. Benney and W. J. Timson, The rolling motion of a viscous fluid on and off a rigid surface,, Stud. Appl. Math, 63 (1980), 93.   Google Scholar [4] E. S. Benilov and M. Vynnycky, Contact lines with a $180^{\circ}$ contact angle,, J. Fluid Mech., 718 (2013), 481.   Google Scholar [5] B. B. Kadomtsev and V. I. Petviashvili, On the stability of solitary waves in weakly dispersing media,, Sov. Phys. Dokl., 15 (): 539.   Google Scholar [6] L. A. Ostrovskii, Nonlinear internal waves in the rotating ocean,, Okeanologiia, 18 (1978), 181.   Google Scholar [7] D. E. Pelinovsky and A. R. Giniyatullin, Finite-time singularities in the dynamical evolution of contact lines,, Bulletin of the Moscow State Regional University (Physics and Mathematics), 3 (2013), 14.   Google Scholar [8] D. E. Pelinovsky, A. R. Giniyatullin and Y. A. Panfilova, On solutions of the reduced model for the dynamical evolution of contact lines,, Transactions of Nizhni Novgorod State Technical University n.a. Alexeev N.4, 94 (2012), 45.   Google Scholar [9] D. E. Pelinovsky and C. Xu, On numerical modelling and the blow-up behavior of contact lines with a $180^{\circ}$ contact angle,, J. Engineer. Math., (2015).  doi: 10.1007/s10665-014-9763-9.  Google Scholar [10] J. Le Sommer, G. M. Reznik and V. Zeitlin, Nonlinear geostrophic adjustment of long-wave disturbances in the shallow-water model on the equatorial beta-plane,, Journal of Fluid Mechanics, 515 (2004), 135.  doi: 10.1017/S0022112004000229.  Google Scholar [11] M. Vynnycky and S. L. Mitchell, On the accuracy of a finite-difference method for parabolic partial differential equations with discontinuous boundary conditions,, Num. Heat Trans B, 64 (2013), 275.  doi: 10.1080/10407790.2013.797312.  Google Scholar [12] S. L. Mitchell and M. Vynnycky, On the numerical solution of two-phase Stefan problems with heat-flux boundary conditions,, J. Comp. Appl. Maths, 264 (2014), 49.  doi: 10.1016/j.cam.2014.01.003.  Google Scholar
 [1] Juliana Fernandes, Liliane Maia. Blow-up and bounded solutions for a semilinear parabolic problem in a saturable medium. Discrete & Continuous Dynamical Systems - A, 2021, 41 (3) : 1297-1318. doi: 10.3934/dcds.2020318 [2] Michiel Bertsch, Danielle Hilhorst, Hirofumi Izuhara, Masayasu Mimura, Tohru Wakasa. A nonlinear parabolic-hyperbolic system for contact inhibition and a degenerate parabolic fisher kpp equation. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3117-3142. doi: 10.3934/dcds.2019226 [3] Takiko Sasaki. Convergence of a blow-up curve for a semilinear wave equation. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 1133-1143. doi: 10.3934/dcdss.2020388 [4] Tetsuya Ishiwata, Young Chol Yang. Numerical and mathematical analysis of blow-up problems for a stochastic differential equation. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 909-918. doi: 10.3934/dcdss.2020391 [5] Justin Holmer, Chang Liu. Blow-up for the 1D nonlinear Schrödinger equation with point nonlinearity II: Supercritical blow-up profiles. Communications on Pure & Applied Analysis, 2021, 20 (1) : 215-242. doi: 10.3934/cpaa.2020264 [6] Van Duong Dinh. Random data theory for the cubic fourth-order nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020284 [7] Youshan Tao, Michael Winkler. Critical mass for infinite-time blow-up in a haptotaxis system with nonlinear zero-order interaction. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 439-454. doi: 10.3934/dcds.2020216 [8] Alex H. Ardila, Mykael Cardoso. Blow-up solutions and strong instability of ground states for the inhomogeneous nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2021, 20 (1) : 101-119. doi: 10.3934/cpaa.2020259 [9] Manuel del Pino, Monica Musso, Juncheng Wei, Yifu Zhou. Type Ⅱ finite time blow-up for the energy critical heat equation in $\mathbb{R}^4$. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3327-3355. doi: 10.3934/dcds.2020052 [10] Daniele Bartolucci, Changfeng Gui, Yeyao Hu, Aleks Jevnikar, Wen Yang. Mean field equations on tori: Existence and uniqueness of evenly symmetric blow-up solutions. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3093-3116. doi: 10.3934/dcds.2020039 [11] Jun Zhou. Lifespan of solutions to a fourth order parabolic PDE involving the Hessian modeling epitaxial growth. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5581-5590. doi: 10.3934/cpaa.2020252 [12] Manuel de León, Víctor M. Jiménez, Manuel Lainz. Contact Hamiltonian and Lagrangian systems with nonholonomic constraints. Journal of Geometric Mechanics, 2020  doi: 10.3934/jgm.2021001 [13] Onur Şimşek, O. Erhun Kundakcioglu. Cost of fairness in agent scheduling for contact centers. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2021001 [14] Kihoon Seong. Low regularity a priori estimates for the fourth order cubic nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5437-5473. doi: 10.3934/cpaa.2020247 [15] Teresa D'Aprile. Bubbling solutions for the Liouville equation around a quantized singularity in symmetric domains. Communications on Pure & Applied Analysis, 2021, 20 (1) : 159-191. doi: 10.3934/cpaa.2020262 [16] Vo Van Au, Hossein Jafari, Zakia Hammouch, Nguyen Huy Tuan. On a final value problem for a nonlinear fractional pseudo-parabolic equation. Electronic Research Archive, 2021, 29 (1) : 1709-1734. doi: 10.3934/era.2020088 [17] Stanislav Nikolaevich Antontsev, Serik Ersultanovich Aitzhanov, Guzel Rashitkhuzhakyzy Ashurova. An inverse problem for the pseudo-parabolic equation with p-Laplacian. Evolution Equations & Control Theory, 2021  doi: 10.3934/eect.2021005 [18] Yohei Yamazaki. Center stable manifolds around line solitary waves of the Zakharov–Kuznetsov equation with critical speed. Discrete & Continuous Dynamical Systems - A, 2021  doi: 10.3934/dcds.2021008 [19] François Dubois. Third order equivalent equation of lattice Boltzmann scheme. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 221-248. doi: 10.3934/dcds.2009.23.221 [20] Kevin Li. Dynamic transitions of the Swift-Hohenberg equation with third-order dispersion. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2021003

2019 Impact Factor: 1.27