A generalized Sz. Nagy inequality in higher dimensions and the critical thin film equation

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|h|dx)a-1 ∫ℝd |∇h|2 dx/(∫ℝd |h|m+1 dx)a+1/m+1 ≥ β0, where m > 0 for d = 1, 2, 0 < m < d+2/d-2 for d ≥ 3, and a = d+2(m+1)/md. The Euler-Lagrange equation for critical points of J(h) in the non-negative radial decreasing function space is given by a free boundary problem for a generalized Lane-Emden equation, which has a unique solution (denoted by hc) and the solution determines the best constant for the above generalized Sz. Nagy inequality. The connection between the critical mass Mc = ∫Rdbl; hc dx = 2√2π/3 for the thin-film 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. 15 223-56). For the following critical thin film equation in multi-dimension d ≥ 2 ht + ∇ · (h ∇ Delta; h) + ∇ · (h ∇ hm) = 0, x ϵ ℝd, where m = 1 + 2/d, the critical mass is also given by Mc:= ∫ℝd hc dx. A finite time blow-up occurs for solutions with the initial mass larger than Mc. On the other hand, if the initial mass is less than Mc and a global non-negative entropy weak solution exists, then the second moment goes to infinity as t → ∞ or h(·, tk) ⇀ 0 in L1(ℝd) for some subsequence tk → ∞. This shows that a part of the mass spreads to infinity.

Duke Scholars

Author Jian-Guo Liu Physics