1. Introduction

The time evolution of thickness of a viscous liquid film spreading over a solid surface under the action of the surface tension and gravity can be described by lubrication models [1–5]. These models approximate the full Navier-Stokes system that describes the motion of the liquid flow. Thin films play an increasingly important role in a wide range of applications, for example, packaging, barriers, membranes, sensors, semiconductor devices, and medical implants [6–8].

In this paper we consider the dynamics of a viscous incompressible thin fluid film on the outer surface of a horizontal circular cylinder that is rotating around its axis in the presence of a gravitational field. The motion of the liquid film is governed by four physical effects: viscosity, gravity, surface tension, and centrifugal forces. These are reflected in the parameters: : the radius of the cylinder, : its rate of rotation (assumed constant), : the acceleration due to gravity, : the kinematic viscosity, : the fluid's density, and : the surface tension. These parameters yield three independent dimensionless numbers: the Reynolds number , , and the Weber number . The understanding of coating flow dynamics is important for industrial printing process where rotating cylinder transports the coating material in the form of liquid paint. The rotating thin fluid film can exhibit variety of different behaviour including: interesting pattern formations (“shark teeth” and “duck bill” patterns), fluid curtains, hydroplaning drops, and frontal avalanches [8–10]. As a result, the coating flow has been the subject of continuous study since the pioneering model was derived in 1977 by Moffatt (see [11]): The surface tension and inertial effects were neglected in (1.1). Here is the thickness of the fluid film, is a rotation angle, and is a time variable. The linear stability of rigidly rotating films on a rotating circular cylinder under three-dimensional disturbances was examined in [12, 13]. It was shown that the most unstable mode for thin film flows on the surface of a cylinder is the purely axial one that leads to so-called “ring instabilities”. During the past decade, coating and rimming problems attracted many researchers who analyzed different types of flow regime asymptotically [14–17] and numerically [18–20]. For a detailed review of a growing literature on different types of thin film flows please see [21] and references there in.

The coating flow is generated by viscous forces due to cylinder's surface motion relative to the fluid. There is no temperature gradient, hence the interface does not experience a shear stress. If the cylinder is fully coated there is only one free boundary where the liquid meets the surrounding air. Otherwise, there is also a free boundary (or contact line) where the air and liquid meet the cylinder's surface.

The asymptotic evolution equation for the thickness of the fluid film with the surface tension effect: was derived by Pukhnachev [22] in 1977. It is valid under the assumptions that the fluid film is thin and its slope is small . Later in 2009, taking into account inertial effects, Kelmanson [23] presented a more general model: He analyzed, asymptotically and numerically, diverse effects of inertia in both small- and large-surface-tension limits.

We should mention that all three lubrication approximation models described above were based on the assumption of the no-slip boundary condition. It is well known [24] that the combination of constant viscosity and no-slip boundary conditions at the liquid-solid interface yields a logarithmic divergence in the rate of dissipation at moving contact line, that is, an infinite energy is needed to make the droplet expand. The most common way to overcome this difficulty is to introduce effective slip conditions (see (2.1)) that indeed removes the force singularity at advancing contact lines (see [25]).

The main goal of our paper is to study waiting time phenomenon for the coating flows under an assumption of effective slip conditions, that is, we analyze (2.1) that is a modified version of (1.3). Our approach is based on now well-established nonlinear PDE analysis for degenerate higher order parabolic equations.

The sufficient conditions: for , for , (where and are some positive constants) on nonnegative initial data, for the occurrence of waiting time phenomena were derived by Dal Passo et al. [26] for the classic thin film equation: These results were based on an energy method developed in [27] for quasilinear parabolic equations. To the best of our knowledge, there is only one publication [28], where the waiting time phenomenon in the classic thin film equation (1.4) was discovered for for . The result was obtained by means of matching asymptotic methods and was supported by numerous numerical simulations. For more general nonlinear degenerate parabolic equations with nonlinear lower order terms the waiting time phenomenon was analyzed in [29–31].

It is well known [32] that the similarity solutions of the second order nonlinear parabolic equation: subject to prescribing appropriate initial data, demonstrate the existence of a waiting-time phenomena before the free boundary moves. The comparison theorem, that is not applicable in our case, then enabled a number of results to be obtained about the existence and length of waiting times for general initial data. Our approach is completely different and based on local entropy/energy functional estimates.

We also analyze speed of support propagation and obtain an upper bound on it for the modified version of (1.3) (see (2.1)). The first finite speed results for nonnegative generalized solutions of the classic thin film equation (1.4) were obtained in [33, 34] for the case and , respectively. For more general types of thin film equations the finite speed of support propagation phenomenon was studied in [35–39] (see also references there in).

The outline of our paper is as follows. We first prove for the long-time existence of a generalized weak solution and then prove that it can have an additional regularity in Section 2. In Sections 3 and 4 we show finite speed support propagation in the “slow” convection case (): for and waiting time phenomena for , accordingly. The general strategy is to use an extension of Stampacchia's lemma for a system of functional equations (see Lemma  3.1 [26], where this extension is proved for a single equation and Lemma  A.2 in [37], where this extension is proved for systems in the homogeneous case). This result to our knowledge is new and might be of independent interest. We leave as an open problem the “fast” convection case (): finite speed of support propagation and sufficient conditions for waiting time phenomenon.

2. Existence and Regularity of Solutions

We are interested in the existence of nonnegative generalized weak solutions to the following initial-boundary value problem: where ,  ,  ,  ,  ,    , and such that ote that (1.3) is a particular case of (2.1) that corresponds to and .

We consider a generalized weak solution in the following sense [40, 41].

Definition 2.1. A generalized weak solution of problem is a nonnegative function satisfying where . The solution satisfies (2.1) in the following sense: for all with ; for at all points of the lateral boundary where

Because the second term of (2.4) has an integral over rather than over , the generalized weak solution is “weaker” than a standard weak solution. Here, is short hand for . This short hand is used throughout: the time interval included in is to be inferred from the context it appears in.

A key object for proving additional properties of a generalized weak solution is an integral quantity introduced by Bernis and Friedman [42]: the “entropy” . The function is defined by where By construction, is a nonnegative convex function on . For , the linear part of is chosen to ensure that has a positive lower bound on . Also in the statement of Theorem 2.2 we use an “-entropy”, , where is a nonnegative convex function on . The linear part of is chosen to ensure that has a positive lower bound on if . If , the -entropy is the same as the entropy (2.7).

Theorem 2.2. (a) (Existence). Let and the nonnegative initial data , satisfy Then for any time there exists a nonnegative generalized weak solution, , on in the sense of the Definition 2.1. Furthermore, Let then the weak solution satisfies (b) (Regularity). If the initial data also satisfies for some ,   then the nonnegative generalized weak solution has the extra regularity and .

The theorem above was proved earlier in [41] for the case only. We note that the analogue of Theorem 4.2 in [42] also holds: there exists a nonnegative weak solution with the integral formulation If initial data satisfy finite -entropy condition, that is, then one can prove existence of nonnegative solutions with some additional regularity properties and use an integral formulation [43] to define them that is similar to that of (2.16) except that the second integral is replaced by the results of one more integration by parts (there are no terms). It is worth to mention that for the case the finite entropy assumption in Theorem 2.2 can be omitted because it does not impose any restriction on nonnegative initial data. One needs to impose finite entropy and finite -entropy conditions on initial data if only.

2.1. Regularized Problem

Given ,  , a regularized parabolic problem, similar to one that was studied by Bernis and Friedman [42] can be written as:

where The in (2.20) makes the problem (2.17) regular (i.e., uniformly parabolic). The parameter is an approximating parameter which has the effect of increasing the degeneracy from to . The nonnegative initial data, , is approximated via The term in (2.21) “lifts” the initial data so that they are smoothing from to . By Eĭdel’man [44, Theorem 6.3, p.302], the regularized problem has a unique classical solution for some time . For any fixed value of and , by Eĭdel’man [44, Theorem 9.3, p.316] if one can prove a uniform in time a priori bound for some longer time interval   ) and for all then Schauder-type interior estimates [44, Corollary 2, p.213] imply that the solution can be continued in time to be in .

Although the solution is initially positive, there is no guarantee that it will remain nonnegative. The goal is to take ,   in such a way that   , the solutions converge to a (nonnegative) limit, , which is a generalized weak solution, and    inherits certain a priori bounds. This is done by proving various a priori estimates for that are uniform in and and hold on a time interval that is independent of and . As a result, will be a uniformly bounded and equicontinuous (in the norm) family of functions in . Taking will result in a family of functions that are classical, positive, unique solutions to the regularized problem with . Taking will then result in the desired generalized weak solution . This last step is where the possibility of nonunique weak solutions arise; see [40] for simple examples of how such constructions applied to can result in two different solutions arising from the same initial data.

2.2. A Priori Estimates

Our first task is to derive a priori estimates for classical solutions of (2.17)–(2.21). The lemmas given in this section are proved in the Section 4.

We use an integral quantity based on a function chosen such that This is analogous to the “entropy” function first introduced by Bernis and Friedman [42].

Lemma 2.3. Let satisfy (2.21) and be built from a nonnegative function that satisfies the hypotheses of Theorem 2.2. Then there exist , and time such that if , , and is a solution of the problem (2.17)–(2.21) with initial data , then for any the following inequalities: hold. The energy (see (2.13)) satisfies The time and the constants are independent of and .

The proof of existence of ,  ,  ,  , and is constructive; how to find them and what quantities determine them are shown with details in Section 4.

Lemma 2.3 yields uniform-in--and- bounds for , , , and . However, these bounds are found in a different manner than in earlier work for the equation , for example. Although the inequality (2.24) is unchanged, the inequality (2.23) has an extra term involving . In the proof, this term was introduced to control additional, lower-order terms. This idea of a “blended” -entropy bound was first introduced by Shishkov and Taranets for long-wave stable thin film equations with convection [30].

The final a priori bounds for positive, classical solutions use the following functions, parameterized by for , where is given by (2.9). In the following lemma, we restrict ourselves to the case ; note that for such .

Lemma 2.4. Assume and are from Lemma 2.3, , and . Assume and that is a positive, classical solution of the problem (2.17)–(2.21) with initial data satisfying Lemma 2.3. If the initial data is built from which also satisfies then there exists such that holds for all and is independent of and is determined by ,  ,  ,  ,  ,   and . Here Furthermore, if then are uniformly bounded.

The -entropy, , was first introduced for in [45] and an a priori bound like that of Lemma 2.4 and regularity results like those of Theorem 2.2 were found simultaneously and independently in [40, 43].

The proof of existence starts from a construction of a classical solution on that satisfies the hypotheses of Lemma 2.3 if and . Taking the regularizing parameter, , to zero, one proves that there is a limit and that is a generalized weak solution. After that additional nonlinear estimates are required to analyze properties of the limit ; specifically to show that it is strictly positive, classical, and unique. Hence, the a priori bounds given by Lemmas 2.3 and 2.4 are applicable to . This allows us to take the approximating parameter, , to zero and to construct the desired nonnegative generalized weak solution of Theorems 2.2 (see, e.g., [41]).

2.3. Long-Time Existence of Solutions

Lemma 2.5. Let be a generalized solution of Theorem 2.2. Then where is defined in (2.13), , and

Proof of Lemma 2.5. By (2.13), The linear-in-time bound (2.14) on then implies Using the estimate (see [41, Lemma 4.1, page 1837]) and Young's inequality: Using this in (2.34), the desired bound (2.31) follows immediately.

This -estimate will be used to extend the short-time existence of a solution to the long-time existence result of Theorem 2.2 (see [41, Proof of Theorem 3, page 1838]).

3. Finite Speed of Support Propagation

Theorem 3.1. Let . Assume is nonnegative, and . Then the solution of Theorem 2.2 has finite speed of support propagation, that is, there exists a continuous nondecreasing function ,   such that for all .

In the following theorem, we find the explicit upper bounds of the for a solution of the corresponding Cauchy problem with a compactly supported nonnegative initial data . Note that the definition of generalized weak solution of the Cauchy problem is as Definition 2.1 except that is replaced by and the relation (2.6) is dropped. Using Lemma 2.5, we can show that the upper estimate of from Theorem 3.1 is independent on therefore the solution from Theorem 2.2 can be extended to be identically zero for and thus is a solution on the line for all . Performing a similar procedure in , we obtain a compactly supported nonnegative solution of the Cauchy problem for all and Theorem 2.2 holds with .

Theorem 3.2. Let . Assume is nonnegative, , and is a solution of the Cauchy problem. Then the following estimates:   for all if ,   for small enough time if , are valid. Here the constants depend on the parameters problem and initial data only.

3.1. Proof of Theorem 3.1 for the Case

The following lemma contains the local entropy estimate. The proof of Lemma 3.3 is similar to (A.16), (A.29), therefore it is omitted.

Lemma 3.3. Let such that , on , and . Assume that , and . Then there exist constants dependent on ,  ,  ,  , and , independent of , such that for all

Let , and let . For an arbitrary and we consider the families of sets We introduce a nonnegative cutoff function from the space with the following properties: Next we introduce our main cut-off functions such that for all and possess the following properties: for all ,  . Choosing , from (3.1) we arrive at for all , where and is enough small. We apply the Nirenberg-Gagliardo interpolation inequality (see Lemma B.2) in the region to a function with ,  ,  ,  ,  , and under the conditions: Integrating the resulted inequalities with respect to time and taking into account (3.5), we arrive at the following relations: where . These inequalities are true provided that Simple calculations show that inequalities (3.6) and (3.8) hold with some if and only if . The finite speed of propagations follows from (3.7) by applying Lemma B.3 with . Hence, where .

3.2. Proof of Theorem 3.2 for the Case

We can repeat the previous procedure from Section 3.1 for and we obtain instead of (3.7), and where Now we need to estimate . With that end in view, we obtain the following estimates: where , and depends on initial data only. Really, applying the Nirenberg-Gagliardo interpolation inequality (see Lemma B.2) in to a function with ,  ,  ,  ,  , and under the condition , we deduce that Integrating (3.14) with respect to time and taking into account the Hölder inequality (), we arrive at the following relations: From (3.15), due to (A.16) (as ) and (2.31), we find (3.13).