Fourth-order Thin Film Equation Research Articles

Steady states of thin-film equations with van der Waals force with mass constraint

We consider steady states with mass constraint of the fourth-order thin-film equation with van der Waals force in a bounded domain which leads to a singular elliptic equation for the thickness with an unknown pressure term. By studying second-order nonlinear ordinary differential equation, \begin{equation*}h_{rr}+\frac{1}{r}h_{r}=\frac{1}{\alpha}h^{-\alpha}-p\end{equation*} we prove the existence of infinitely many radially symmetric solutions. Also, we perform rigorous asymptotic analysis to identify the blow-up limit when the steady state is close to a constant solution and the blow-down limit when the maximum of the steady state goes to the infinity.

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 alpha ge 2 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 alpha in (1,2). 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.

Towards optimal regularity for the fourth-order thin film equation in [formula omitted]: Graveleau-type focusing self-similarity

An approach to some “optimal” (more precisely, non-improvable) regularity of solutions of the thin film equationut=−∇⋅(|u|n∇Δu)inRN×R+,u(x,0)=u0(x)inRN, where n∈(0,2) is a fixed exponent, with smooth compactly supported initial data u0(x), in dimensions N≥2 is discussed. Namely, a precise exponent for the Hölder continuity with respect to the spatial radial variable |x| is obtained by construction of a Graveleau-type focusing self-similar solution. As a consequence, optimal regularity of the gradient ∇u in certain Lp spaces, as well as a Hölder continuity property of solutions with respect to x and t, are derived, which cannot be obtained by classic standard methods of integral identities–inequalities. Several profiles for the solutions in the cases n=0 and n>0 are also plotted. In general, we claim that, even for arbitrarily small n>0 and positive analytic initial data u0(x), the solutions u(x,t) cannot be better than Cx2−ε-smooth, where ε(n)=O(n) as n→0.

Well-posedness of the Cauchy problem for a fourth-order thin film equation via regularization approaches

This paper is devoted to some aspects of well-posedness of the Cauchy problem (the CP, for short) for a quasilinear degenerate fourth-order parabolic thin film equation (the TFE-4) (0.1)ut=−∇⋅(|u|n∇Δu)in RN×R+,u(x,0)=u0(x)in RN, where n>0 is a fixed exponent, with bounded smooth compactly supported initial data. Dealing with the CP (for, at least, n∈(0,32)) requires introducing classes of infinitely changing sign solutions that are oscillatory close to finite interfaces. The main goal of the paper is to detect proper solutions of the CP for the degenerate TFE-4 by uniformly parabolic analytic ε-regularizations at least for values of the parameter n sufficiently close to 0.Firstly, we study an analytic “homotopy” approach based on a priori estimates for solutions of uniformly parabolic analytic ε-regularization problems of the form ut=−∇⋅(ϕε(u)∇Δu)in RN×R+, where ϕε(u) for ε∈(0,1] is an analytic ε-regularization of the problem (0.1), such that ϕ0(u)=|u|n and ϕ1(u)=1, using a more standard classic technique of passing to the limit in integral identities for weak solutions. However, this argument has been demonstrated to be non-conclusive, basically due to the lack of a complete optimal estimate-regularity theory for these types of problems.Secondly, to resolve that issue more successfully, we study a more general similar analytic “homotopy transformation” in both the parameters, as ε→0+ and n→0+, and describe branching of solutions of the TFE-4 from the solutions of the notorious bi-harmonic equationut=−Δ2uin RN×R,u(x,0)=u0(x)in RN, which describes some qualitative oscillatory properties of CP-solutions of (0.1) for small n>0 providing us with the uniqueness of solutions for the problem (0.1) when n is close to 0.Finally, Riemann-like problems occurring in a boundary layer construction, that occur close to nodal sets of the solutions, as ε→0+, are discussed in other to get uniqueness results for the TFE-4 (0.1).

Symmetry properties for a generalised thin film equation

Symmetry properties are presented for a fourth-order parabolic equation written in conservation form. It was introduced in the literature as a generalisation of the fourth-order thin film equation. We derive equivalence transformations, Lie symmetries, potential symmetries, non-classical symmetries and potential non-classical symmetries. A chain of such equations is introduced. We conclude by presenting similar results for the third-order equation of this chain.

Velocity-Based Moving Mesh Methods for Nonlinear Partial Differential Equations

AbstractThis article describes a number of velocity-based moving mesh numerical methods for multidimensional nonlinear time-dependent partial differential equations (PDEs). It consists of a short historical review followed by a detailed description of a recently developed multidimensional moving mesh finite element method based on conservation. Finite element algorithms are derived for both mass-conserving and non mass-conserving problems, and results shown for a number of multidimensional nonlinear test problems, including the second order porous medium equation and the fourth order thin film equation as well as a two-phase problem. Further applications and extensions are referenced.

Local bifurcation-branching analysis of global and “blow-up” patterns for a fourth-order thin film equation

Countable families of global-in-time and blow-up similarity sign-changing patterns of the Cauchy problem for the fourth-order thin film equation (TFE-4) $$u_t=-\nabla \cdot \left(|u|^n \nabla \Delta u\right) \quad {\rm in} \quad \mathbb{R}^{N}\times\mathbb{R}_{+} \quad{\rm where}\quad n >0 ,$$are studied. The similarity solutions are of standard “forward” and “backward” forms $$\begin{array}{ll}u_{\pm}(x,t)\,=\,(\pm t)^{-\alpha}\, f(y), \quad y=x/(\pm t)^\beta,\\ \beta\,=\, \frac {1-\alpha n}{4}, \quad \pm t > 0, \quad {\rm where }\,f\, {\rm solve}\\ \quad\quad{\bf B}^{\pm}_{n} (\alpha,\,f)\,\equiv\, - \nabla \cdot \left(|f|^n \nabla \Delta f\right) \pm \beta y \cdot\nabla f\pm\alpha f=0 \quad {\rm in} \quad \mathbb{R}^{N}, \quad\quad(0.1)\end{array}$$and \({\alpha \in \mathbb{R}}\) is a parameter (a “nonlinear eigenvalue”). The sign “ + ”, i.e., t > 0, corresponds to global asymptotics as t → + ∞, while “−” (t < 0) yields blow-up limits t → 0− describing possible “micro-scale” (multiple zero) structures of solutions of the PDE. To get a countable set of nonlinear pairs {f γ , α γ }, a bifurcation-branching analysis is performed by using a homotopy path n → 0+ in (0.1), where \({{\bf B}_{0}^{\pm}\,(\alpha,f)}\) become associated with a pair {B, B*} of linear non-self-adjoint operators$${\bf B}=-\Delta^2 + \frac {1}{4}\, y \cdot \nabla + \frac {N}{4} \, I \quad {\rm and} \quad {\bf B}^* =- \Delta^2-\frac {1}{4}\, y\cdot\nabla\left({\rm so}\quad ({\bf B})^*_{L^2}={\bf B}^*\right),$$which are known to possess a discrete real spectrum, \({\sigma({\bf B})=\sigma({\bf B}^*)= \left\{ \lambda_ \gamma= - \frac{|\gamma|}{4}\right\}_{|\gamma| \ge 0}\,\, (\gamma\,{\rm is\, a\, multiindex\, in}\, \mathbb{R}^{N})}\). These operators occur after corresponding global and blow-up scaling of the classic bi-harmonic equation u t = − Δ2 u. This allows us to trace out the origin of a countable family of n-branches of nonlinear eigenfunctions by using simple or semisimple eigenvalues of the linear operators {B, B*} leading to important properties of oscillatory sign-changing nonlinear patterns of the TFE, at least, for small n > 0.

On continuous branches of very singular similarity solutions of a stable thin film equation. I – The Cauchy problem

We consider the fourth-order thin film equation, with a stable second-order diffusion term. For the first critical exponent, where N ≥ 1 is the space dimension, the Cauchy problem is shown to admit countable continuous branches of source-type self-similar very singular solutions of the form These solutions are inherently oscillatory in nature and will be shown in Part II to be the limit of appropriate free-boundary problem solutions. For p ≠ p0, the set of very singular solutions is shown to be finite and to be consisting of a countable family of branches (in the parameter p) of similarity profiles that originate at a sequence of critical exponents {pl, l ≥ 0}. At p = pl, these branches appear via a non-linear bifurcation mechanism from a countable set of similarity solutions of the second kind of the pure thin film equation Such solutions are detected by the ‘Hermitian spectral theory’, which allows an analytical n-branching approach. As such, a continuous path as n → 0+ can be constructed from the eigenfunctions of the linear rescaled operator for n = 0, i.e. for the bi-harmonic equation ut = −Δ2u. Numerics are used, wherever appropriate, to support the analysis.

Very Singular Solutions for Thin Film Equations with Absorption

The large-time behavior of weak nonnegative and sign changing solutions of the fourth-order thin film equation (TFE-4) with absorption where n ∈ (0, 3) and the absorption exponent p belongs to the subcritical range is studied. First, the standard free-boundary problem with zero-height, zero contact angle, and zero-flux conditions at the interface and bounded compactly supported initial data is considered. Very singular similarity solutions (VSSs) have the form Here f solves the quasi-linear degenerate elliptic equation that becomes an ODE for N = 1 or in the radial setting in . By a combination of analytical, asymptotic, and numerical methods, existence of various branches of similarity profiles f parameterized by p is established. Secondly, changing sign VSSs of the Cauchy problem are described. This study is motivated by the detailed VSS results for the second-order porous medium equation with absorption (u ≥ 0) which have been known since the 1980s.

Source-type solutions of the fourth-order unstable thin film equation

We consider the fourth-order thin film equation (TFE) with the unstable second-order diffusion term. We show that, for the first critical exponent where N ≥ 1 is the space dimension, the free-boundary problem the with zero contact angle and zero-flux conditions admits continuous sets (branches) of self-similar similarity solutions of the form For the Cauchy problem, we describe families of self-similar patterns, which admit a regular limit as n → 0+ and converge to the similarity solutions of the semilinear unstable limit Cahn-Hilliard equation studied earlier in [12]. Using both analytic and numerical evidence, we show that such solutions of the TFE are oscillatory and of changing sign near interfaces for all n ∈ (0,nh), where the value characterizes a heteroclinic bifurcation of periodic solutions in a certain rescaled ODE. We also discuss the cases p ⧧ = p0, the interface equation, and regular analytic approximations for such TFEs as an approach to the Cauchy problem.

ADI schemes for higher-order nonlinear diffusion equations

Alternating Direction Implicit (ADI) schemes are constructed for the solution of two-dimensional higher-order linear and nonlinear diffusion equations, particularly including the fourth-order thin film equation for surface tension driven fluid flows. First and second-order accurate schemes are derived via approximate factorization of the evolution equations. This approach is combined with iterative methods to solve nonlinear problems. Problems in the fluid dynamics of thin films are solved to demonstrate the effectiveness of the ADI schemes.