Math @ Duke

Publications [#331396] of JianGuo Liu
Papers Published
 Liu, JG; Wang, J, A generalized Sz. Nagy inequality in higher dimensions and the critical thin film equation,
Nonlinearity, vol. 30 no. 1
(January, 2017),
pp. 3560, IOP Publishing [doi]
(last updated on 2019/06/17)
Abstract: © 2016 IOP Publishing Ltd and London Mathematical Society Printed in the UK. In this paper, we provide an alternative proof for the classical Sz. Nagy inequality in one dimension by a variational method and generalize it to higher dimensions d ≥ 1 J(h): = (∫ ℝd hdx) a1 ∫ ℝd ∇h 2 dx/(∫ ℝd h m+1 dx) a+1/m+1 ≥ β 0 , where m > 0 for d = 1, 2, 0 < m < d+2/d2 for d ≥ 3, and a = d+2(m+1)/md. The EulerLagrange equation for critical points of J(h) in the nonnegative radial decreasing function space is given by a free boundary problem for a generalized LaneEmden equation, which has a unique solution (denoted by h c ) and the solution determines the best constant for the above generalized Sz. Nagy inequality. The connection between the critical mass M c = ∫ Rdbl; h c dx = 2√2π/3 for the thinfilm equation and the best constant of the Sz. Nagy inequality in one dimension was first noted by Witelski et al (2004 Eur. J. Appl. Math. 15 22356). For the following critical thin film equation in multidimension d ≥ 2 h t + ∇ · (h ∇ Delta; h) + ∇ · (h ∇ h m ) = 0, x ϵ ℝ d , where m = 1 + 2/d, the critical mass is also given by M c := ∫ ℝd h c dx. A finite time blowup occurs for solutions with the initial mass larger than M c . On the other hand, if the initial mass is less than Mc and a global nonnegative entropy weak solution exists, then the second moment goes to infinity as t → ∞ or h(·, t k ) ⇀ 0 in L 1 (ℝ d ) for some subsequence t k → ∞. This shows that a part of the mass spreads to infinity.


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