Local strong solutions to a quasilinear degenerate fourth-order thin-film equation

We study the problem of existence and uniqueness of strong solutions to a degenerate quasilinear parabolic non-Newtonian thin-film equation. Originating from a non-Newtonian Navier–Stokes system, the equation is derived by lubrication theory and under the assumption that capillarity is the only driving force. The fluid’s shear-thinning rheology is described by the so-called Ellis constitutive law. For flow behaviour exponents α≥2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha \ge 2$$\end{document} the corresponding initial boundary value problem fits into the abstract setting of Amann (Function Spaces, Differential Operators and Nonlinear Analysis, Vieweg Teubner Verlag, Stuttgart, 1993). Due to a lack of regularity this is not true for flow behaviour exponents α∈(1,2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha \in (1,2)$$\end{document}. For this reason we prove an existence theorem for abstract quasilinear parabolic evolution problems with Hölder continuous dependence. This result provides existence of strong solutions to the non-Newtonian thin-film problem in the setting of fractional Sobolev spaces and (little) Hölder spaces. Uniqueness of strong solutions is derived by energy methods and by using the particular structure of the equation.


Introduction
This contribution is motivated by questions for existence and uniqueness of strong solutions to the degenerate quasilinear fourth-order evolution problem ⎧ ⎪ ⎨ ⎪ ⎩ u t + a u 3 1 + |buu xxx | α−1 u xxx x = 0, t>0, x ∈ Ω := (−l, l) u x = u xxx = 0, t > 0, x ∈ ∂Ω u(0, ·) = u 0 (·), x ∈ Ω, In this paper we treat the case α > 1, in which the fluid is said to be shearthinning. Moreover, we assume the fluid's dynamics to be driven by capillary forces only. In particular gravitational forces are neglected; c.f. Sect. 2 for a more detailed review of the derivation of the governing equations.
There is a rich literature on weak solutions of equations related to (1.1). The first pioneering results on the classical Newtonian thin-film equation u t + a(u n u xxx ) x = 0, n ∈ N, (1.2) go back to Bernis and Friedmann [8]. There the authors prove in particular the existence of global non-negative weak solutions, as well as positivity and uniqueness for n ≥ 4. Among others, also the works [9][10][11]15] can be mentioned in the context of global non-negative weak solutions. For contributions to the study of non-Newtonian fluids the reader shall be in particular referred to the works [25,26] of King, where non-Newtonian generalisations of (1.2) are investigated. In particular the author studies the doubly nonlinear equation describing for p = 2 the spreading of so-called power-law or Ostwald-de Waele fluids. In the work [6] Ansini and Giacomelli establish the existence of global non-negative weak solutions to (1.3) for p > 2 and p−1 2 < n < 2p − 1. In [5] the same authors verify the existence of travelling-wave solutions and study a class of quasi-self-similar solutions to (1.1). Moreover, in [33] the authors establish the existence of weak solutions to a non-Newtonian Stokes equation with a viscosity that depends on the fluid's shear rate and its pressure at the same time.
Uniqueness of weak solutions to the Newtonian thin-film equation (1.2) for n < 4 is still an open problem. Moreover, due to their weak regularity properties, weak solutions do usually not provide much control at the contact points, where the film height u tends to zero, i.e. where solid, liquid (and gas) meet . These are probably the reasons why existence and uniqueness of non-negative strong solutions of the free-boundary value problem associated to (1.2) have recently attracted much attention. The free boundary is the boundary ∂{u > 0} of the solution's support. The main difficulty in the analysis is again caused by the fact that the equation's parabolicity degenerates, as the film height u tends to zero. A rich amount of research has been dedicated to this problem in the last decade, mostly prescribing a constant slope at the free NoDEA Local strong solutions to a quasilinear degenerate Page 3 of 28 16 boundary; see [13,[16][17][18][19][22][23][24][27][28][29]36] for strong solutions in (weighted) Sobolev or Hölder spaces. In this paper we focus onpositive strong solutions, emanating from a positive initial film height. To construct such solutions it is convenient to give up the divergence form of (1.1), in order to emphasise the quasilinear structure. For sufficiently regular solutions, (1.1) is equivalent to (1.4) where A(u, u x , u xx , u xxx ) = au 3 In (1.4) the highest-order term u xxxx appears only linearly, while the nonlinearity |u xxx | α−1 u xxx is on the right-hand side. However, the coefficient function A contains the delicate nonlinearity |u xxx | α−1 .
Regarding the existence and uniqueness of strong solutions to (1.1), it turns out that there is a qualitative difference between flow behaviour exponents α ∈ (1, 2) and those larger than or equal 2. If we associate to (1.4) an abstract quasilinear Cauchy problem it turns out that in the latter case α ≥ 2 the operator A and the right-hand side F are Lipschitz continuous in an appropriate sense and the classical Hölder theory of Eidel'man [14, Thm. III.4.6.3] as well as the abstract theory of Amann [4, Thm. 12.1] are applicable and provide existence and uniqueness of strong solutions. For more details on the case α ≥ 2 we refer the reader to Remarks 4.3 and 5.3 below.
The situation is more delicate for flow behaviour exponents α ∈ (1, 2). In this regime the operator A and the right-hand side F are only (α − 1)-Hölder continuous, whence there is no hope to obtain existence and uniqueness by Banach's fixed point theorem. Instead we use compactness to prove an abstract existence result for quasilinear parabolic problems of fourth order with Hölder continuous coefficients in the spirit of [4,Thm. 12.1]. More precisely, the proof exploits the a-priori estimates and the smoothing properties of the corresponding abstract linear equation to obtain a solution for the quasilinear problem by a fixed-point argument.
Finally we apply this result to obtain existence of solutions to (1.1) in (fractional) Sobolev spaces as well as in (little) Hölder spaces.
In order to prove uniqueness of solutions to (1.1), we explicitly retain its divergence form. Indeed, we use the special structure of the equation to obtain uniqueness by energy methods. This idea has already been used in the pioneering work [8] of Bernis & Friedman on the (Newtonian) thin-film equation.
We close this introduction by briefly outlining the organisation of this work.
A derivation of the evolution problem is reviewed in Sect. 2. In Sect. 3 we recall an abstract well-posedness result of Amann [3] for linear parabolic problems. Section 4 is concerned with an existence result for abstract quasilinear parabolic equations.
In Sect. 5 this result is applied to the non-Newtonian thin-film equation for all α > 1 and in the regime of fractional Sobolev and little Hölder spaces, respectively. Uniqueness of strong solutions to the non-Newtonian thin-film equation for flow behaviour exponents α ∈ (1, 2) is proved in Sect. 6. There we also show that constants are the only possible steady state solutions of (1.1). Finally, Sect. 7 contains a result on the maximal existence time of solutions to (1.1).

Derivation of the non-Newtonian thin-film problem for an Ellis shear-thinning fluid
For the convenience of the reader in this section we review the derivation of (1.1). Lubrication approximation. Starting from the non-Newtonian Navier-Stokes equations with no-slip boundary condition, the lubrication approximation [20,34] leads-under the assumption of a positive film height-to the system of dimensionless equations for the velocity field (v, w) = (v(t, x, z), w(t, x, z)), the pressure p = p(t, x, z) and the film height u = u(t, x). Here, σ > 0 denotes the constant surface tension and μ : R → R the fluid's viscosity. The fluid domain Λ is defined by Λ := (x, z) ∈ R 2 ; 0 < x < l and 0 < z < u(t, x) .
Integrating (2.1) 2 from z to u and using the boundary condition (2.1) 5 , we obtain Moreover, an integration of (2.1) 1 from z to u, together with the boundary condition (2.1) 6 , yields Shear-thinning rheology: Ellis constitutive law. As for instance in [5,43], we use the Ellis constitutive law to describe the fluid's shear-thinning NoDEA Local strong solutions to a quasilinear degenerate Page 5 of 28 16 rheology. We thus introduce the shear stress τ : R → R implicitly by the relation and set Here, μ 0 denotes the viscosity at zero shear stress and τ * is the shear stress at which the viscosity is reduced by a factor 1/2. Shear-thinning fluids are characterised by an apparent viscosity that decreases with increasing shear stress. For α > 1 and τ * ∈ (0, ∞), the shear-thinning behaviour is reflected by the Ellis law. Moreover, it is observed in polymeric systems that, at rather low and/or rather high shear rates, the viscosity approaches a Newtonian plateau. For most polymers and polymer solutions α varies from 1 to 2 (see i.e. [7,32] whence the horizontal velocity is given by For x ∈ (0, l), this finally yields Hence, the evolution Eq. (2.1) 7 for the film's height u reads where the constants a and b are given by

A well-posedness result for abstract linear parabolic initial value problems
This section is concerned with the solvability of linear parabolic evolution problems. We mainly recall a basic well-posedness and regularity result for linear parabolic Cauchy problems. We start by introducing some notations and requirements.
if ω + A is an isomorphism from E 1 to E 0 , and We say that u is a solution of (3.1) (in E 0 ) if the following conditions are satisfied: (iii) the differential equation ( for some ρ ∈ (0, 1). Then the following holds true: (i) If u 0 ∈ E 0 , then the linear Cauchy problem (3.1) possesses a unique solution Moreover, assume that there are constants ω > 0, κ ≥ 1 and σ ∈ R such that Then we have in addition that, A proof of this theorem can be found in the book [3]. While part (i) and part (iv) are due to Amann, part (ii) goes back to Sobolevskii [38] and Tanabe [42]. Moreover, there is an analogous result by Acquistapace & Terreni [2] which is proved by different methods. As regards assertion (iii),

Abstract existence theorem for the quasilinear problem
In this section we prove an existence result for abstract quasilinear parabolic Cauchy problems, based on the theory for linear parabolic problems. This result will later be applied to the non-Newtonian thin-film equation in the setting of fractional Sobolev spaces and (little) Hölder spaces.
As in Sect. 3, let T > 0 be given and consider the quasilinear Cauchy where A and F are Hölder continuous in an appropriate sense, to be specified below.
The existence result for (4.1) is deduced from the uniform a-priori estimates for the corresponding linear equation and an application of the following fixed-point theorem [21,Cor. 11.2]. It is worthwhile to mention again that with Theorem 4.1 we may obtain an existence result for (4.1) by compactness of the solution operator for the corresponding linear problem. Since we do not require Lipschitz continuity we can in general not expect to get uniqueness.
is a densely injected Banach couple and that the injection is in addition compact. Let 0 < β < α ≤ 1 and σ ∈ (0, α−β). Let μ ∈ (0, 1) and assume that the maps Moreover, for all α ∈ (β + σ, α) the solution satisfies Proof. Fix α, β and R . We use a fixed-point argument in a suitable ball in the space C ρ ([0, T ]; E β ) for a suitable ρ > 0. The argument uses the estimate (3.7). Thus, we start with the following observation. There exist ω > 0, κ ≥ 1 and r 0 > 0 such that for all w ∈ E β we have the following implication: To see this, note first that the set Then U is open in E β and K ⊂ U . Since K is compact, it follows that U contains a 2r 0 neighbourhood of K for some r 0 > 0. Thus, (4.4) holds. Now let u 0 ∈ E α such that ||u 0 || Eα ≤ R . Byū 0 we denote the constant extension of u 0 on [0, T ]. We set up a fixed-point problem, suitable for an application of Theorem 4.1, as follows. We set Given v ∈B, the regularity assumptions on A and F imply (4.5) Thus, we are in a position to apply Theorem 3.1 to deduce existence of a unique solution u = Sv of the linear evolution problem in the sense that u = Sv satisfies Let now α ∈ (β + σ, α) and note that u 0 ∈ E α → E α . Then the solution enjoys in addition the regularity and by (3.7) we have the estimates where ν ≥ 0 is independent of T . In view of the uniform estimate (4.5) on F, the solution u = Sv satisfies in fact the uniform estimate where the constants C and ν are independent of T . In order to deduce from Theorem 4.1 the existence of a fixed point u = Su ∈ X, we are thus left with verifying that (i) S :B → X is continuous; (ii) S :B → X is compact; (iii) S preserves the ballB.
Since α − σ > β, we have E α −σ → E β and hence continuity of the embedding We know from the linear theory that S mapsB to Y . We show that Y is compactly embedded into X. Since α − σ > β, it follows from the compactness of the embedding E α −σ − → c E β and the Arzelà-Ascoli theorem that Hence, S is a compact operator fromB to C([0, T ]; E β ). In view of the interpolation estimate we see that S is even a compact operator fromB to X. (iii) S(B) ⊂B. To deduce that S has a fixed point, we are left with verifying that, for sufficiently small T > 0, the operator S maps the ballB ⊂ X into itself. Recall from (4.7) that, given v ∈B, we obtain the estimate for the solution u = Sv of the linear problem (4.6). This implies on the one hand that and on the other hand (recall that σ < α − β) (4.10) Combining (4.9) and (4.10), we find that Thus, for sufficiently small T > 0, the right-hand side of this inequality is less than or equal r 0 , which proves that S(B) ⊂B. allows one to get existence and uniqueness by a contraction argument. However, concerning existence of solutions to the non-Newtonian thin-film equation (1.1) we cover the case of flow behaviour exponents α ∈ (1, 2) by applying our abstract existence result Theorem 4.2, while we deduce uniqueness from energy estimates that use the structure of the particular equation.
In the remainder of this section we prove a result on the maximal existence time of solutions. We use the usual continuation argument to obtain a contradiction. However, some care is required in the formulation of the result since solutions may not be unique. We fix u 0 ∈ E α and set (4.11) and prove that the following holds true.
is a solution of (4.1) with u(0) = u 0 , then Proof. Fix γ ∈ (β, α]. Let T 0 = T 0 (γ, R) be the existence time in Theorem 4.2, with α replaced by γ and R replaced by R. We claim that the assertion of the theorem holds with Indeed, assume that there exists a solution u ∈ C([0, T ]; E α ) with Then by Theorem 4.2 there exists a solution Here we used Theorem 4.2 with α = γ and α = β+γ 2 . Thus, Thenũ ∈ C([0,t + T 0 ]; E α ) ∩ C((0,t + T 0 ]; E 1 ). The equatioṅ u + A(ũ)ũ = F(ũ) 16 Page 12 of 28 C. Lienstromberg and S. Müller NoDEA holds in (0,t) and in (t,t + T 0 ). Sinceũ ∈ C((0,t + T 0 ]; E 1 ), it follows thatu can be uniquely continued at t =t to a continuous function with values in E 0 . Indeed, thanks to the continuity of A and F we have for t >t 2 >T , this contradicts the definition ofT .

Existence of solutions to the non-Newtonian thin-film equation
In this section we apply the abstract existence result Theorem 4.2 to the non-Newtonian thin-film equation x ∈ Ω.
(5.1) We first introduce some notation. Using the identity whereb = σ/τ * , we may rewrite (5.1) in the following way in non-divergence form: and we use the abbreviations Constructing solutions of (5.1), respectively (5.2), naturally involves the following two challenges. First, to be able to apply the abstract existence result Theorem 4.2 we have to reformulate (5.2) as an abstract quasilinear Cauchy problem. In other words we have to choose a suitable Banach space E 0 in which we study the problem and we have to define the differential operator A properly. This means in particular that we have to define A such that its domain incorporates the first-order and third-order Neumann boundary conditions. Of course we need that A generates an analytic semigroup on E 0 and that A and F satisfy the required regularity properties. Moreover, (E 1 , E 0 ) has to be a densely and compactly injected Banach couple.
The second challenge we have to deal with is that the non-Newtonian thin-film equation is reasonable for positive film heights u only. Hence, in order to apply Theorem 4.2 we extend problem (5.2) in a way such that for positive initial data solutions of the extended problem coincide for a short time with solutions of the original problem.
To tackle the latter challenge we extend the coefficient map A to a globally defined, locally Hölder continuous function as follows. For v + = max(v, 0), we first introduce the map . Note that the function v → v + is locally Lipschitz continuous and hence, we still haveĀ ∈ C α−1 loc (R 4 ). Finally, let ε > 0 be given. To ensure parabolicity of the coefficient map, we set Summarising, we have that the maps A,Ā,Ā ε and F are locally (α − 1)-Hölder continuous on R 4 , in symbols (and analogously for A andĀ). That is, for all z, z ∈ R 4 with |z|, |z | ≤ R they satisfy (and analogously for A andĀ). As above, we finally introduce the notation The corresponding global version of (5.2) then reads x ∈ Ω. The task of setting up an appropriate framework for the abstract Cauchy problem in terms of function spaces is addressed in the following two subsections.

Solutions to (5.1) in fractional Sobolev spaces
In this section we study the problem of existence of solutions to (5.1), respectively (5.2), which are Hölder continuous in time and take values in Sobolev spaces of fractional order. Note that we consider only the case in which Ω ⊂ R is a bounded interval.
For k ∈ N and p ∈ [1, ∞) we denote by W k p (Ω) the usual Sobolev spaces with norm We then put and define the Sobolev-Slobodeckii or fractional Sobolev spaces by Here, [s] denotes the largest integer smaller than or equal to s. We now recall some important properties of these spaces, which are necessary to guarantee that we are in the setting of Theorem 4.2. It is wellknown that, for −∞ < s 0 < s 1 < ∞ and 0 < ρ < 1, the space W s p (Ω), s = (1−ρ)s 0 +ρs 1 , is the complex interpolation space between W s1 p (Ω) and W s0 p (Ω), in symbols W s p (Ω) = [W s0 p (Ω), W s1 p (Ω)] ρ . In order to take the (Neumann) boundary conditions of (5.1), respectively (5.2), into account, we further introduce the Banach spaces We can now apply the abstract existence result Theorem 4.2 to the non-Newtonian thin-film equation (5.2). More precisely, we prove the following theorem on the existence of solutions in Sobolev spaces of fractional order.
We can now prove the main result of this section.
Proof of Theorem 5.1. (i) Existence. Let p ∈ (1, ∞) and 1/p < s < r < 1. Then put σ = 3+s 4 and ρ = 3+r 4 . Suppose that u 0 ∈ W 4ρ p,B such that where ε, R > 0 are fixed. We first apply Theorem 4.2 to show that problem (5.11) possesses a solution for some time T ε > 0. To this end, note that in view of Lemma 5.2 we have the required Hölder continuity of the operatorĀ ε and the right-hand side;Ā (5.13) Note that here we used thatĀ ε is the composition of A with two Lipschitz continuous maps. Moreover, recall that the choice of σ = 3+s and thus finallyĀ In addition, the principal symbol a ε (x, ξ) of the operator A ε (v) satisfies the uniform Legendre-Hadamard condition i.e. −Ā ε (v) generates an analytic semigroup on L p (Ω). In virtue of (5.13) and (5.14) we may eventually apply Theorem 4.2 to conclude that there exists a positive time T ε and a solution It remains to show that the solution u ε is-at least for a short timealso a solution to (5.10). Indeed, by (5.15) we obtain , and hence u ε does also solve the original problem (5.10) on [0, T * ]. This completes the proof.

Solutions of (5.1) in (little) Hölder spaces
This section is devoted to the existence of classical solutions to the non-Newtonian thin-film equation (5.1), respectively (5.2). More precisely, we apply our abstract Theorem 4.2 in the setting of (little) Hölder spaces. Note again that we study the one-dimensional thin-film equation.
As in Sect. 5.1, we start by introducing the relevant notation and function spaces. Let Ω ⊂ R be an open and bounded interval. For k ∈ N and ρ ∈ (0, 1) we define the usual Hölder spaces by and We further introduce the so-called little-Hölder spaces We recall some important properties of these spaces. The space h ρ (Ω) is a closed subspace of C ρ (Ω) and hence a Banach space. If 0 < σ < 1, then h σ (Ω) is the closure of C ρ (Ω) in C σ (Ω) for all ρ ∈ (σ, ∞]. Furthermore, for 0 ≤ s 0 < s 1 and 0 < ρ < 1, the space h s (Ω), s = (1 − ρ)s 0 + ρs 1 is the real interpolation space between C s1 (Ω) and C s0 (Ω), in symbols In order to take the first-order and third-order Neumann boundary conditions of problem (5.1) into account, we further introduce for ρ ∈ (0, 1] the spaces The main result of this section may now be formulated as follows. Theorem 5.4. Let 3/4 < σ < ρ ≤ 1. Then, given an initial film height u 0 ∈ h 4ρ B (Ω), such that u 0 (x) > 0 for all x ∈Ω, for each α > 1 there exists a positive time T > 0 and a solution u of (5.1) on [0, T ] in the sense that ). In any case u satisfies The Proof of Theorem 5.4 is similar to the one in the setting of fractional Sobolev spaces.
A ε (v) ∈ H(C 4 B (Ω); C(Ω)). We obtain the assertion by following the lines of the proof of Theorem 5.1.

Uniqueness of solutions to (5.1) for flow behaviour exponents α ∈ (1, 2)
Recall from Sects. 5.1 and 5.2 that, for flow behaviour exponents α ≥ 2, we have Lipschitz continuity of the differential operator A as well as the righthand side F. For flow behaviour exponents α ∈ (1, 2) we get existence of solutions to (5.1) in fractional Sobolev and little Hölder spaces, respectively, by compactness of the solution operator for the linear problem, c.f. Theorems 5.1 and 5.4.
In this section we prove uniqueness of solutions to (5.1) by deriving an energy inequality for which we use the special structure of the equation. More precisely, we extend the approach used in [8] for the Newtonian thin-film equation to prove that, for α ∈ (1, 2), two positive strong solutions of (5.1) coincide if this is the case initially. For this purpose, observe that the energy decreases along smooth solutions of (5.1). Indeed, if u is a smooth solution of (5.1), then (6.1) To justify the energy inequality (6.1) for solutions in our regularity class and to apply a similar argument to the difference of two solutions we use the following fact.