Ju l 2 01 0 New dissipated energy for the unstable thin film equation

The fluid thin film equation ht = −(h hxxx)x − a1 (h hx)x is known to conserve mass ∫ hdx, and in the case of a1 ≤ 0, to dissipate entropy ∫ h3/2−n dx (see [8]) and the L2-norm of the gradient ∫ hx dx (see [3]). For the special case of a1 = 0 a new dissipated quantity ∫ hα hx dx was recently discovered for positive classical solutions by Laugesen (see [15]). We extend it in two ways. First, we prove that Laugesen’s functional dissipates strong nonnegative generalized solutions. Second, we prove the full α-energy ∫ ( 1 2 h α hx − a1 hα+m−n+2 (α+m−n+1)(α+m−n+2) ) dx dissipation for strong nonnegative generalized solutions in the case of the unstable porous media perturbation a1 > 0 and the critical exponent m = n+ 2. 2000 MSC: 35K55, 35K35, 35Q35, 76D08 keywords: fourth-order degenerate parabolic equations, thin liquid films, energy, entropy


Introduction
It is well known that analysis of the existence, uniqueness and regularity of weak solutions for nonlinear evolution equations relies heavily on a priori estimates. Often, the physical energy or entropy which originate from the related model can provide non-increasing in time quantities. Unfortunately, it is far from obvious how to construct new non-increasing Lyapunov type functionals. A general algebraic approach to the construction of entropies in higher-order nonlinear PDEs can be found in [14] and can be applied to analyse thin film equations with stabilizing porus media type perturbations. In this paper, inspired by Laugesen's result [15] on dissipation, we prove that the energy functional introduced in [15] dissipates strong nonnegative generalized solutions. However, our method of the proof is only applicable to some subset of the Laugsen's dissipation region [15] (see the shaded area on Figure 1).
We study the longwave-unstable generalized thin film equation where h(x, t) gives the height of the evolving free-surface. The exponent n plays a stabilizing role due to fourth-order forward diffusion term and the exponent m plays a destabilizing role due to backward second-order diffusion term for the case when a 1 > 0. This class of equations originates from many physical/industrial applications involving air-fluid interface. For example: the case n = 1, m = 1 describes a thin jet in a Hele-Shaw cell [10], the case n = 3, m = −1 describes Van der Waals driven rupture of thin films [19], the case m = n = 3 describes shape of fluid droplets hanging from a ceiling [11], and the case n = 0, m = 1 describes solidification of a hyper-cooled melt (this is a modified Kuramoto-Sivashinsky equation) [4].
To prove that the nonnegativity property is preserved in nonlinear thin film equation h t = −(h n h xxx ) x for n ≥ 1 (case a 1 = 0) Bernis and Friedman [3] used set of dissipated and conserved quantities: mass conservation h dx = M, surface energy dissipation d dt h 2 x dx ≤ 0, and entropy dissipation d dt h 2−n dx ≤ 0. The new so-called β-entropy h 2−n+β dx was introduced by Bertozzi and Pugh [5] and independently and simultaneously by Beretta, Bertsch, Dal Passo [1] to extend this result to n > 0. They also successfully used this new entropy to obtain exponential with respect to the L ∞ -norm convergence toward the mean value steady state solution. To analyse this convergence rate in H 1 -norm for the special case n = 1, a 1 = 0 Carlen and Ulusoy [9] used the dissipated energy h α h 2 x dx constructed by Laugesen [15] for classical positive solutions. Exponential asymptotic convergence toward the mean value was also studied by Tudorascu in [18]. This list of connections between new properties of solutions in thin film PDEs proved by means of newly discovered dissipated quantities is far from complete.
In this paper we prove that there exists a subinterval I of −1 < α < 1 (I depends on n only) and a nonnegative strong generalized solution such that for any α ∈ I the full α-energy dissipates. For the unstable porus media perturbation case a 1 > 0 this dissipation is proven under the assumptions that the total mass of the solution is less than or equal to the critical one, m = n + 2 and domain Ω is unbounded or h is compactly supported. For the stable case a 1 ≤ 0 no such assumptions are needed.
We proceed as follows. First, we show the dissipation for the classical solutions of the regularized problem and then we take this dissipation to the limit. We prove dissipation of the full α-energy for positive classical solutions of the regularized problem for any value of the coefficient a 1 and without any additional assumptions about the total mass of the solution or its support. However our method of taking the dissipation to the limit due to the Bernis-Friedman method of regularization requires additional conditions for the case a 1 > 0.

6)
where P = Q T \ ({h = 0} ∪ {t = 0}) and h satisfies (2.1) in the following sense: Because the second term of (2.7) has an integral over P rather than over Q T , the generalized weak solution is "weaker" than a standard weak solution. Also note that the first term of (2.7) uses h t ∈ L 2 (0, T ; (H 1 (Ω)) ′ ); this is different from the definition of weak solution first introduced by Bernis and Friedman [3]; there, the first term was the integral of hφ t . The proof of the existence of generalized weak solutions follows the ideas of [3,1,5,6,7,17].
Let h ∈ L 2 (0, T loc ; H 2 (Ω)). (2.12) . Then the weak solution satisfies ] If the initial data from (a) also satisfies loc ≤ T loc such that the nonnegative generalized weak solution has the extra regularity There is nothing special about the time T loc in the Theorem 1. In the case a 1 > 0 and n/2 ≤ m < n + 2 (or m = n + 2 and M ≤ M c ), given a countable collection of times in [0, T loc ], one can construct a weak solution for which these bounds will hold at those times. Also, we note that the analogue of Theorem 4.2 in [3] also holds: there exists a nonnegative weak solution with the integral representation

Regularized Problem
Given δ, ε > 0, a regularized parabolic problem, similar to that of Bernis and Friedman [3], is considered: The δ > 0 in (3.10) makes the problem (3.7) regular (i.e. uniformly parabolic). The parameter ε is an approximating parameter which has the effect of increasing the degeneracy from f (h) ∼ |h| n to f ε (h) ∼ h s . The nonnegative initial data, h 0 , is approximated via h 0,δε ∈ C 4+γ (Ω), h 0,δε h 0,δ + ε θ for some 0 < θ < 2 2s−3 , The ε term in (3.11) "lifts" the initial data so that it will be positive even if δ = 0 and the δ is involved in smoothing the initial data from H 1 (Ω) to C 4+γ (Ω). (Ω × [0, T loc,δε ]). Although the solution h δε is initially positive, there is no guarantee that it will remain nonnegative. The goal is to take δ → 0, ε → 0 in such a way that 1) T loc,δε → T loc > 0, 2) the solutions h δε converge to a (nonnegative) limit, h, which is a generalized weak solution, and 3) h inherits certain a priori bounds. This is done by proving various a priori estimates for h δε that are uniform in δ and ε and hold on a time interval [0, T loc ] that is independent of δ and ε. As a result, {h δε } will be a uniformly bounded and equicontinuous (in the C   Figure 1, and
Note that, although we use the same convenient notations introduced in [15], the proof of Lemma 3.1 has essential differences from the proof of Theorem 1 of [15]. Indeed, we introduce new ideas in order to estimate the lower-order term in the equation (3.7). In particular, the new quantity N is introduced, the quantity R is modified, and so are the terms involving the regularization parameter ε in (3.19).
Proof of Lemma 3.1. To prove the bound (3.13), multiply (3.7) with ′ ε (h), integrate over Ω, use integration by parts, apply the periodic boundary conditions (3.8), to find 14) The equality (3.14) can be rewritten as where the quantities x , each represent half of an inner product in L 2 (Ω). We will need the following integration by parts formulas Here we use the auxiliary equality f ′ Our next step is to express (3.18) as the negative of a sum of squares to obtain the energy dissipation. To achieve this, we use (3.16) and (3.17) to deduce that for all κ ∈ R 1 , where where β ∈ (−1/2, 1) follows from (2.16). Similarly, we deal with the integral ε 2 ε,x dxdt is uniformly bounded then where the positive constant C is independent of ε. Letting ε → 0, we obtain (3.22) for 3 2 − n < α < 0 and 3 2 < n < 3. In view of the Lebesgue's theorem, we have if m > 0 and α > − 1 2 − m, due to h ∈ L ∞ (0, T ; H 1 (Ω)) and (2.16). Integrating (3.19) over the time interval, and letting ε → 0, in view of (3.22) and (3.24), we obtain (3.13) for some subinterval I for 0 ≤ α < 1 and 1 2 < n < 3 or for −1 < α < 0 and 3 2 < n < 3. Note that, the convergence on the left-hand side follows from Fatou's lemma and from the corresponding a priori estimate (see, for example, [3,5,7,17]).
Lemma A.2. ( [16]) If Ω ⊂ R N is a bounded domain with piecewisesmooth boundary, a > 1, b ∈ (0, a), d > 1, and 0 i < j, i, j ∈ N, then there exist positive constants d 1 and d 2 (d 2 = 0 if Ω is unbounded) depending only on Ω, d, j, b, and N such that the following inequality is valid for every v(x) ∈ W j,d (Ω) ∩ L b (Ω):