## Rocky Mountain Journal of Mathematics

### The Cauchy problem for the degenerate convective Cahn-Hilliard equation

#### Abstract

In this paper, we study the degenerate convective Cahn-Hilliard equation, which is a special case of the general convective Cahn-Hilliard equation with $M(u,\nabla u)=diag(0,1,\ldots ,1)$. We obtain the uniform a priori decay estimates of a solution by use of the long-short wave method and the frequency decomposition method. We prove the existence of the unique global classical solution with small initial data by establishing the uniform estimates of the solution. Decay estimates are also discussed.

#### Article information

Source
Rocky Mountain J. Math., Volume 48, Number 8 (2018), 2595-2623.

Dates
First available in Project Euclid: 30 December 2018

https://projecteuclid.org/euclid.rmjm/1546138823

Digital Object Identifier
doi:10.1216/RMJ-2018-48-8-2595

Mathematical Reviews number (MathSciNet)
MR3894995

Zentralblatt MATH identifier
06999276

Subjects
Primary: 35K25: Higher-order parabolic equations
Secondary: 35A09: Classical solutions 35K59: Quasilinear parabolic equations

#### Citation

Liu, Aibo; Liu, Changchun. The Cauchy problem for the degenerate convective Cahn-Hilliard equation. Rocky Mountain J. Math. 48 (2018), no. 8, 2595--2623. doi:10.1216/RMJ-2018-48-8-2595. https://projecteuclid.org/euclid.rmjm/1546138823

#### References

• R.A. Adams and J.F. Fournier, Sobolev spaces, Pure Appl. Math. 140 Elsevier/Academic Press, Amsterdam, 2003.
• J. Chen, Y. Li and W. Wang, Global classical solutions to the Cauchy problem of conservation laws with degenerate diffusion, J. Diff. Eqs. 260 (2016), 4657–4682.
• ––––, Global existence of solutions to the initial-boundary value problem of conservation laws with degenerate diffusion term, Nonlin. Anal. 78 (2013), 47–61.
• L. Duan, S.Q. Liu and H.J. Zhao, A note on the optimal temporal decay estimates of solutions to the Cahn-Hilliard equation, J. Math. Anal. Appl. 372 (2010), 666–678.
• A. Eden and V.K. Kalantarov, The convective Cahn-Hilliard equation, Appl. Math. Lett. 20 (2007), 455–461.
• ––––, $3$ D convective Cahn-Hilliard equation, Comm. Pure Appl. Anal. 6 (2007), 1075–1086.
• M. Heida, Existence of solutions for two types of generalized versions of the Chan-Hilliard equation, Appl. Math. 60 (2015), 51–90.
• K.H. Kwek, On the Cahn-Hilliard type equation, Ph.D. dissertation, Georgia Institute of Technology, Atlanta, 1991.
• T.T. Li and Y.M. Chen, Global classical solutions for nonlinear evolution equations, Pitman Mono. Surv. 45 (1992).
• C. Liu, On the convective Cahn-Hilliard equation, Bull. Polon. Acad. Sci. Math. 53 (2005), 299–314.
• ––––, On the convective Cahn-Hilliard equation with degenerate mobility, J. Math. Anal. Appl. 344 (2008), 124–144.
• W.K. Wang and Z.G. Wu, Optimal decay rate of solutions for Cahn-Hilliard equation with inertial term in multi-dimensions, J. Math. Anal. Appl. 387 (2012), 349–358.
• M.A. Zaks, A. Podolny, A.A. Nepomnyashchy and A.A. Golovin, Periodic stationary patterns governed by a convective Cahn-Hilliard equation, SIAM J. Appl. Math. 66 (2005), 700–720.
• X. Zhao and C. Liu, Optimal control for the convective Cahn-Hilliard equation in $2$ D case, Appl. Math. Optim. 70 (2014), 61–82.