A generalized Sz. Nagy inequality in higher dimensions and the critical thin film equation
© 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 |h|dx) a-1 ∫ 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/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 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 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 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 blow-up 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 non-negative 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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