We consider the thin-film equation with linear mobility and a stabilizing second-order porous-medium type term modeling gravity. The model admits self-similar solutions, and our goal is to analyze their stability. We reformulate the problem in mass-Lagrangian coordinates and exploit the underlying gradient-flow structure of the equation with respect to a weighted $L^2$ inner product, where the weight is given by the self-similar source-type profile. This framework allows us to establish a coercivity result for the Hessian (the linearization around the self-similar solution) in a suitably weighted inner product. As a consequence, we prove the convergence of perturbations toward the self-similar profile at an algebraic rate of order $t^{-\frac 1 5}$, in arbitrary scales of weighted Sobolev norms. The analysis relies on maximal-regularity estimates for the linearized evolution, combined with appropriate estimates for the nonlinear terms. Notably, beyond perturbative regimes and in contrast to previous results for the thin-film equation (convergence to the Smyth-Hill profile) or the porous-medium equation (convergence to the Barenblatt-Pattle solution), our analysis does not rely on an explicit (algebraic) representation of the self-similar profile. Instead, it is based solely on a systematic use of the ordinary differential equation satisfied by the self-similar solution, together with a careful analysis of its boundary asymptotics. As a result, we expect that the approach developed here can serve as a flexible toolbox for the study of more general classes of equations and for the stability analysis of special solutions in future work.
We consider the thin-film equation with linear mobility and stabilizing gravity term according to h t + (hh yyy ) y -(h 3 h y ) y = 0 in {h > 0} (1.1a) for which we assume a zero contact-angle condition h y = 0 on ∂{h > 0}. (1.1b) In what follows, we prove well-posedness of compactly-supported solutions to (1.1) and analyze their intermediate time asymptotics as t → ∞. We therefore use the transformations ξ := e -s 5 y, h(t, y) = e -s 5 u(s, ξ), s := ln(t + 1), (1.2) which are motivated by the scaling of the equation t ∼ y 5 , and passage to the logarithmic time variable s.
We then have h = e -s 5 u, h t = e -s (e -s 5 u s ) -ξ 5 e -s (e -s 5 u) ξ -e -6 5 s 5 u = e -6 5 s u s -1 5 e -6 5 s (ξu) ξ , ∂ y = e -s 5 ∂ ξ , so that equation (1.1a) changes to
(ξu) ξ = 0 in {u > 0}.
(1.3a) and the boundary condition (1.1b) now reads u ξ = 0 on ∂{u > 0}.
(1.3b)
The self-similar source-type solution is a stationary solution to (1.3a), i.e., it satisfies u(s, ξ) = U (ξ), so that after one integration in ξ on assuming sufficient decay as ξ → ±∞ we have
(1.4)
One further integration leads to
where U (0) = U 0 and U ′′ (0) = U ′′ 0 . Equation (1.4) admits a nonnegative classical solution with compact support [-ℓ, ℓ] for some ℓ > 0, where U ′ = 0 on ∂{U > 0}, see Beretta’s work [4,Theorem 1.1]. This solution is even (cf. [4, §3-4]) and satisfies U > 0 in (-ℓ, ℓ) (cf. [4,Lemma 2.2]) and U ∈ C ∞ ([-ℓ, ℓ]). Note that smoothness on {U > 0} = (-ℓ, ℓ) is a consequence of a standard theory on ordinary differential equations (ODEs), and smoothness on [-ℓ, ℓ] follows from (1.5) and an iteration argument. We mention that a more refined boundary analysis of the self-similar solution also for nonlinear mobility exponents is provided in [29], while the unstable case has been analyzed in [39].
The crucial mathematical ingredient for proving well-posedness and stability of perturbations of U in a suitable set of coordinates is the coercivity of the Hessian when viewing the dynamics (1.3) as a gradient flow. This will first be motivated at a heuristic level (without introducing all necessary functional analysis) in §1.2-1.6. The functional-analytic setting we actually use, including a formal derivation of suitable linear estimates, is then presented in §1.7. Our rigorous results are stated in §2 and the proofs are provided in the subsequent sections §3-5.
We introduce the free energy functional
and the inner product (the metric) ⟨v, w⟩ u := {u>0} u -1 (∂ -1 ξ v)(∂ -1 ξ w) dξ, where v dξ = w dξ = 0.
(1.7)
We refer to the work of Otto [32] for details regarding the underlying geometry, and to the work of Benamou and Brenier [2] for connections between this metric and the equivalent Wasserstein distance. For monographs on these topics, see for instance [1,19,42,43].
We assume a zero contact-angle condition u ξ (1.3b) = 0 at ∂{u > 0} and obtain the first variation through integration by parts
From (1.5) it is immediate that DE [U ] = 0, that is, the self-similar solution u = U is a critical point of E . An elementary computation entails under the additional constraint v dξ = 0 that
This yields the gradient .8) This entails that the thin-film equation (1.3a) in self-similar variables is a gradient flow
where the gradient is taken with respect to the metric (1.7). Note that at this stage the above considerations are formal. However, rather than using the above gradient-flow formulation, in what follows we instead pass to mass-Lagrangian coordinates. The governing partial differential equation (PDE) can then be formulated as a weighted L 2 -gradient flow, where the weight is given by the self-similar profile U .
We introduce mass-Lagrangian coordinates (a convenient transform as the PDE (1.1a) is in divergence form and thus conserves mass) in conjunction with a logarithmic time scale and a suitable normalization that factors off the leading-order time asymptotics:
5 Y, and s := ln(t + 1).
(1.9) This transformation automatically conserves mass and we have Z = x for the self-similar solution. From now on, we assume Z x ≥ c > 0 for some c > 0 (which we will justify further below). Using (1.1a) we obtain from (1.9),
This results in
so that we obtain the PDE .11) where
and where in view of (1.8) we have introduced convenient abbreviations (∂ Z and Θ are reminiscent of ∂ ξ and u, respectively). Note that for Z = x, equation (1.11) transforms into
which is indeed (1.4). Hence, the self-similar solution Z = x is a stationary solution to (1.11), and we pass on to analyzing its stability in the sequel.
We transform the energy functional (1.6) into mass-Lagrangian coordinates. For that observe
= e -s 5 dy
(1.10)
= Z x dx =: dZ, where (1.9) and (1.10) constitute the mass-Langrangian transform. Then
Discarding constant terms, this motivates the definition of the energy functional
Provided that V, Z : [-ℓ, ℓ] → R are sufficiently regular, and Z x ≥ c > 0 for some constant c > 0, we can compute the differential through integration
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