EXISTENCE OF WEAK SOLUTIONS FOR PARTICLE-LADEN FLOW WITH SURFACE TENSION

. We prove the existence of solutions for a coupled system modeling the ﬂow of a suspension of ﬂuid and negatively buoyant non-colloidal particles in the thin ﬁlm limit. The equations take the form of a fourth-order nonlinear degenerate parabolic equation for the ﬁlm height h coupled to a second- order degenerate parabolic equation for the particle density ψ . We prove the existence of physically relevant solutions, which satisfy the uniform bounds 0 (cid:54) ψ/h (cid:54) 1 and h (cid:62) 0.


1.
Introduction. In lubrication theory, the free surface height of a thin liquid film is governed by a degenerate fourth-order parabolic equation which in one dimension typically has the form where the coefficients f 0 , f 1 , f 2 depend on the relevant physics (e.g. f 0 = f 1 = f 2 = h 3 for the flow of a fluid driven by gravity down an incline) [17]. Equations of this type have been the subject of considerable theoretical study; the tools for analysis can provide insight into important phenomena such as instabilities in spreading films [7,9,18,4], and can be utilized to design efficient numerical schemes [20]. Bernis and Friedman [5] first demonstrated existence and positivity of solutions to the equation h t = −(h n h xxx ) x through the use of energy and entropy estimates.
In later work, Bertozzi and Pugh explicated the theory for the equation (1) with f 0 = 0, using different choices of regularization and entropy functions to study regularity, long-time behavior [3] and the growth of singularities [4].
There are a wide variety of problems in multiphase thin-film flows that lead to more complicated systems. Lubrication models of such flows reduce to coupled systems for the film height and a quantity tracking the second phase, whose complex dynamics have been the subject of considerable interest in recent research [11]. Here we consider one such model for gravity-driven suspension flow in one dimension that accounts for the non-uniform distribution of particles within the bulk of the fluid, proposed in [16] and recently extended to include surface tension in [15]. The model equations [15] for the film height h(x, t) and depth-integrated particle density ψ(x, t) have the form where φ = ψ/h is the depth-averaged concentration of particles that cannot exceed a maximum packing fraction φ m , normalized here so that φ m = 1. The fluxes vanish at the maximum packing fraction (i.e. f i (1) = g i (1) = g i (0) = 0), where flow of the suspension is completely inhibited by the particles. This adds an additional degeneracy into the equations (along with the standard degeneracy for thin films as h → 0), which has been studied in the related problem of non-linear diffusion equations for sedimenting particles [2]. The flux functions in the model equations (2) have a particular behavior in the dilute limit (φ → 0) and the high-concentration limit (φ → 1). In particular, for negatively buoyant particles, for constants a i , b i , c i , d i > 0 [15,19]. These fluxes arise from depth-integrating the suspension (f ) and particle (g) volume fluxes, which depend on the distribution of particles in the fluid depth. The exponents of φ in the dilute limit are a consequence of the particle accumulation towards the substrate of the fluid [15]. The suspension evolves as a settled layer of particles with a clear fluid layer above; the height of this layer can be shown to scale with φ 1/2 as φ → 0, so the fluxes for the particle transport in the lubrication model, which scale with the cube of the height, gain a factor of φ 3/2 . The quadratic decay in the high concentration limit is due to the singularity in the suspension viscosity µ ∼ (1 − φ) −2 , a law that captures the inhibiting of the flow near the maximum packing fraction [6].
The system (2) is closely related to the equations governing transport of insoluble surfactant on the fluid surface [11], for which the concentration satisfies an equation with a non-degenerate diffusion term. Existence and positivity of weak solutions was established in [13,1] using a finite element approach and studied for more general systems in later work by [8,14,10]. The techniques employed there are almost applicable to (2), but must be modified to account for a few key differences in the structure of the equations. First, we do not include the non-degenerate Brownian diffusion term for ψ, leaving only the degenerate diffusion term for the ψ equation which vanishes when φ = 0, φ = 1 or h = 0. Second, the fluxes depend on the ratio ψ/h of the conserved variables h and ψ and vanish when ψ/h 1, so it is critical to establish this bound.
Here, we are concerned with the existence of physically relevant solutions in the sense that h 0 and 0 ψ/h 1 when the initial data satisfies the same, with periodic boundary conditions. Under assumptions on the behavior of the flux coefficients f i and g i compatible with the properties (3) of the physical model, we prove existence of such solutions and the bound φ 1 when f i = g i = 0 for φ 1. In Section 2, the governing system and assumptions used in the existence result are introduced. In Section 3 the relevant notion of a weak solution is defined and the main result is stated, which is proven in Section 4.

2.
System and assumptions. Next, we assume that f 3 = g 3 = 0 for simplicity (due to the fourth order diffusion, this second order term is not important to the existence result). Let us consider the following system of equations: i = 0, 3, k = 0, 1, and initial conditions where Ω := (−a, a) ⊂ R 1 is bounded domain, and D i , f i , g i are continuous functions such that where a 2 , b 0 , b 2 satisfy the following restrictions:   . The assumptions (9)-(13) are motivated by the behavior of the fluxes in the physical model given by (3) (which also satisfy the assumptions here), but are modified slightly for the sake of the proof. Assumption (9) is a (not particularly restrictive) technical assumption on the growth rate of the fluxes that is easily satisfied by the physical model. Of particular note is (10), which is a bound on the size of the firstorder flux of particles in the dilute limit φ → 0; this matches the limiting behavior g 1 (φ) ∼ b 1 φ 3/2 for the physical model. The lower bound required on the diffusion 4982 ROMAN M. TARANETS AND JEFFREY T. WONG coefficient g 2 given by (11) is slightly more general than the limit g 2 ∼ b 2 φ 2 in the physical model.

4.2.
Galerkin approximation. Now we use a Galerkin approximation which transforms the system of partial differential equations into a system of ordinary differential equations. As basis functions for the finite dimensional space we select an L 2 -orthonormal basis of eigenfunctions which are solutions of the periodic boundary value problem: We make a Galerkin ansatz for h N δ (x, t) and ψ N δ (x, t) of the form According to (15) and (16) the functions a i (t) and b i (t) are subject to the following Galerkin equations which have to hold for j = 0, N : Due to (9)-(12), the right-hand side of this system is Lipschitz continuous on a j and b j . Thus, by the Picard-Lindelöf theorem a unique local solution of the system exists. Solvability for some T > 0 can be proved using a priori estimates (uniformly in N , ε and δ).
Next, we have to establish appropriate convergence properties. By (26) we have the following (uniformly in N , ε and δ) boundedness for all T T 0 (34) By (29) and the embedding theorem, we have Note that from (30) it follows and from (33), (36) we have By (37)  (Ω)), we can derive the following estimate for ψ N εδ :

4.3.2.
Positivity of h δ . Next, we show h δ > 0 for all δ < δ 0 . This allows us to extend the corresponding integrals in (52), (53) on all Q T . Multiplying (15) by G δε (h), we get d dt Choose α s 2 and s 3. Then Cβ(a 0 + δ) Integrating this inequality in time, taking into account (26), we deduce that for all T T 0 , where C 4 (T ) is independent of N, ε and δ < δ 0 . By Fatou's lemma, (44) and from the uniform (in N, ε, δ) bound of First of all, we show that h δ 0 in Q T0 when s 4. If this is not true, then there is a point (x 0 , t 0 ) ∈ Q T0 such that h δ (x 0 , t 0 ) < 0. Since convergence h δε to h δ is uniform as ε → 0 then there exist γ > 0 and ε 0 > 0 such that h δε (x, t 0 ) < −γ if |x − x 0 | < γ and ε < ε 0 . But for such x, by the monotone convergence theorem where A > max |h δε | for all small δ, ε. Hence, lim ε→0 Ω G δε (h δε )dx = ∞ and this is in contradiction with (54). Next, we show that h δ > 0 onΩ when s 8. Indeed, if h δ is not positive everywhere in Q T0 , then there exists a point (x 0 , t 0 ) in Q T0 such that h δ (x 0 , t 0 ) = 0. Then by the Hölder continuity of h δ ∈ C 1/2

Conclusions.
We have obtained an existence result for a coupled system of degenerate parabolic equations governing the height h and particle concentration ψ of a viscous suspension under the effect of surface tension. The solution satisfies the physical bounds h 0 and 0 ψ/h 1 corresponding to the boundedness of the particle concentration (which is bounded above by the maximum packing fraction for the suspension). The existence result depends on certain bounds on the flux coefficients, particularly on the degeneracy in the ψ-diffusion term as ψ → 0, that are consistent with the asymptotic results obtained for the physical system. The result established here may be useful in future study of this system, for example in developing numerical methods that preserve the bounds on the solution as done for other equations from lubrication theory [20] or in studying the growth of singularities and long-time behavior of advancing fronts.