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

作者：Liu, Jian-Guo; Wang, Jinhuan^*

来源：Nonlinearity, 2017, 30(1): 35-60.

DOI：10.1088/0951-7715/30/1/35

• long-wave instability
• free-surface evolution
• critical mass
• free boundary problem

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) := (integral(d)(R)vertical bar h vertical bar dx)(a-1) integral(d)(R) vertical bar del h vertical bar(2) dx/(integral(d)(R)vertical bar h vertical bar(m+1) dx)(a+1/m+1) >= beta(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 = integral(R) h(c) dx = 2 root 2 pi/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) + del . (h del Delta h) + del . (h del h(m)) = 0, x is an element of R-d,
where m = 1 + 2/d, the critical mass is also given by M-c := integral(d)(R) h(c) 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 -> infinity or h(., t(k)) -> 0 in L-1(R-d) for some subsequence t(k) -> infinity. This shows that a part of the mass spreads to infinity.

• 出版日期2017-1
• 单位辽宁大学