We consider the thin-film equation for a class of free boundary conditions modelling friction at the contact line, as introduced by E and Ren. Our analysis focuses on formal long-time asymptotics of solutions in the perfect wetting regime. In particular, through the analysis of quasi-self-similar solutions, we characterize the profile and the spreading rate of solutions depending on the strength of friction at the contact line, as well as their (global or local) corrections, which are due to the dynamical nature of the free boundary conditions. These results are complemented with full transient numerical solutions of the free boundary problem. @en

We consider the thin-film equation ∂th+∇⋅(h2∇Δh)=0 in physical space dimensions (i.e., one dimension in time t and two lateral dimensions with h denoting the height of the film in the third spatial dimension), which corresponds to the lubrication approximation of the Navier–Stokes equations of a three-dimensional viscous thin fluid film with Navier-slip at the substrate. This equation can have a free boundary (the contact line), moving with finite speed, at which we assume a zero contact angle condition (complete-wetting regime). Previous results have focused on the 1+1-dimensional version, where it has been found that solutions are not smooth as a function of the distance to the free boundary. In particular, a well-posedness and regularity theory is more intricate than for the second-order counterpart, the porous-medium equation, or the thin-film equation with linear mobility (corresponding to Darcy dynamics in the Hele-Shaw cell). Here, we prove existence and uniqueness of classical solutions that are perturbations of an asymptotically stable traveling-wave profile. This leads to control on the free boundary and in particular its velocity.@en

We consider the stochastic thin-film equation with colored Gaussian Stratonovich noise in one space dimension and establish the existence of nonnegative weak (martingale) solutions. The construction is based on a Trotter–Kato-type decomposition into a deterministic and a stochastic evolution, which yields an easy to implement numerical algorithm. Compared to previous work, no interface potential has to be included, the initial data and the solution can have de-wetted regions of positive measure, and the Trotter–Kato scheme allows for a simpler proof of existence than in case of Itô noise.@en

We investigate the stability of traveling-pulse solutions to the stochastic FitzHugh–Nagumo equations with additive noise. Special attention is given to the effect of small noise on the classical deterministically stable fast traveling pulse. Our method is based on adapting the velocity of the traveling wave by solving a scalar stochastic ordinary differential equation (SODE) and tracking perturbations to the wave meeting a system of a scalar stochastic partial differential equation (SPDE) coupled to a scalar ordinary differential equation (ODE). This approach has been recently employed by Krüger and Stannat (Nonlinear Anal. 162 (2017) 197–223) for scalar stochastic bistable reaction–diffusion equations such as the Nagumo equation. A main difference in our situation of an SPDE coupled to an ODE is that the linearization has essential spectrum parallel to the imaginary axis and thus only generates a strongly continuous semigroup. Furthermore, the linearization around the traveling wave is not self-adjoint anymore, so that fluctuations around the wave cannot be expected to be orthogonal in a corresponding inner product. We demonstrate that this problem can be overcome by making use of Riesz instead of orthogonal spectral projections as recently employed in a series of papers by Hamster and Hupkes in case of analytic semigroups. We expect that our approach can also be applied to traveling waves and other patterns in more general situations such as systems of SPDEs with linearizations only generating a strongly continuous semigroup. This provides a relevant generalization as these systems are prevalent in many applications.@en

Consider the thin-film equation h t +(hh yyy ) y =0 with a zero contact angle at the free boundary, that is, at the triple junction where liquid, gas, and solid meet. Previous results on stability and well-posedness of this equation have focused on perturbations of equilibrium-stationary or self-similar profiles, the latter eventually wetting the whole surface. These solutions have their counterparts for the second-order porous-medium equation h t −(h m ) yy =0, where m>1 is a free parameter. Both porous-medium and thin-film equation degenerate as h↘0, but the porous-medium equation additionally fulfills a comparison principle while the thin-film equation does not. In this note, we consider traveling waves h=[Formula presented]x 3 +νx 2 for x≥0, where x=y−Vt and V,ν≥0 are free parameters. These traveling waves are receding and therefore describe de-wetting, a phenomenon genuinely linked to the fourth-order nature of the thin-film equation and not encountered in the porous-medium case as it violates the comparison principle. The linear stability analysis leads to a linear fourth-order degenerate-parabolic operator for which we prove maximal-regularity estimates to arbitrary orders of the expansion in x in a right-neighborhood of the contact line x=0. This leads to a well-posedness and stability result for the corresponding nonlinear equation. As the linearized evolution has different scaling as x↘0 and x→∞ the analysis is more intricate than in related previous works. We anticipate that our approach is a natural step towards investigating other situations in which the comparison principle is violated, such as droplet rupture. @en

Building on work of Prandtl and Alexander, we study logarithmic vortex spiral solutions of the two-dimensional incompressible Euler equations. We consider multi-branched spirals that are not symmetric, including mixtures of sheets and continuum vorticity. We find that non-trivial solutions allow only sheets, that there is a large variety of such solutions, but that only the Alexander spirals with three or more symmetric branches appear to yield convergent Biot–Savart integral.@en

Building on work of Prandtl and Alexander, we study logarithmic vortex spiral solutions of the two-dimensional incompressible Euler equations. We consider multi-branched spirals that are not symmetric, including mixtures of sheets and continuum vorticity. We find that non-trivial solutions allow only sheets, that there is a large variety of such solutions, but that only the Alexander spirals with three or more symmetric branches appear to yield convergent Biot–Savart integral.@en

We are interested in traveling-wave solutions to the thin-film equation with zero microscopic contact angle (in the sense of complete wetting without precursor) and inhomogeneous mobility h3+λ3?nhn, where h, λ, and n ϵ (3/2, 7/3) denote film height, slip parameter, and mobility exponent, respectively. Existence and uniqueness of these solutions have been established by Maria Chiricotto and the first of the authors in previous work under the assumption of subquadratic growth as h → ∞. In the present work we investigate the asymptotics of solutions as h ↘ 0 (the contact-line region) and h → ∞. As h ↘ 0 we observe, to leading order, the same asymptotics as for traveling waves or source-type self-similar solutions to the thin-film equation with homogeneous mobility hn and we additionally characterize corrections to this law. Moreover, as h → ∞ we identify, to leading order, the logarithmic Tanner profile, i.e. the solution to the corresponding unperturbed problem with λ = 0 that determines the apparent macroscopic contact angle. Besides higher-order terms, corrections turn out to affect the asymptotic law as h → ∞ only by setting the length scale in the logarithmic Tanner profile. Moreover, we prove that both the correction and the length scale depend smoothly on n. Hence, in line with the common philosophy, the precise modeling of liquid-solid interactions (within our model, the mobility exponent) does not affect the qualitative macroscopic properties of the film.@en

We are interested in a complete characterization of the contact-line singularity of thin-film flows for zero and nonzero contact angles. By treating the model problem of source-type self-similar solutions, we demonstrate that this singularity can be understood by the study of invariant manifolds of a suitable dynamical system. In particular, we prove regularity results for singular expansions near the contact line for a wide class of mobility exponents and for zero and nonzero dynamic contact angles. Key points are the reduction to center manifolds and identifying resonance conditions at equilibrium points. The results are extended to radially-symmetric source-type solutions in higher dimensions. Furthermore, we give dynamical systems proofs for the existence and uniqueness of self-similar droplet solutions in the nonzero dynamic contact-angle case.@en

We are interested in the thin-film equation with zero-contact angle and quadratic mobility, modeling the spreading of a thin liquid film, driven by capillarity and limited by viscosity in conjunction with a Navier-slip condition at the substrate. This degenerate fourth-order parabolic equation has the contact line as a free boundary. From the analysis of the self-similar source-type solution, one expects that the solution is smooth only as a function of two variables (x, xβ) (where x denotes the distance from the contact line) with β=13-14≈0.6514 irrational. Therefore, the well-posedness theory is more subtle than in case of linear mobility (coming from Darcy dynamics) or in case of the second-order counterpart (the porous medium equation).Here, we prove global existence and uniqueness for one-dimensional initial data that are close to traveling waves. The main ingredients are maximal regularity estimates in weighted L2-spaces for the linearized evolution, after suitable subtraction of a(t)+b(t)xβ-terms.@en

We are interested in the thin-film equation with zero-contact angle and quadratic mobility, modeling the spreading of a thin liquid film, driven by capillarity and limited by viscosity in conjunction with a Navier-slip condition at the substrate. This degenerate fourth-order parabolic equation has the contact line as a free boundary. From the analysis of the self-similar source-type solution, one expects that the solution is smooth only as a function of two variables (x, xβ) (where x denotes the distance from the contact line) with β=13-14≈0.6514 irrational. Therefore, the well-posedness theory is more subtle than in case of linear mobility (coming from Darcy dynamics) or in case of the second-order counterpart (the porous medium equation).Here, we prove global existence and uniqueness for one-dimensional initial data that are close to traveling waves. The main ingredients are maximal regularity estimates in weighted L2-spaces for the linearized evolution, after suitable subtraction of a(t)+b(t)xβ-terms.@en

We investigate compactly supported solutions for a thin-film equation with linear mobility in the regime of perfect wetting. This problem has already been addressed by Carrillo and Toscani, proving that the source-type self-similar profile is a global attractor of entropy solutions with compactly supported initial data. Here we study small perturbations of source-type self-similar solutions for the corresponding classical free boundary problem and set up a global existence and uniqueness theory within weighted L2-spaces under minimal assumptions. Furthermore, we derive asymptotics for the evolution of the solution, the free boundary, and the center of mass. As spatial translations are scaled out in our reference frame, the rate of convergence is higher than the one obtained by Carrillo and Toscani.@en

We study a problem related to the spin-coating process in which a fluid coats a rotating surface. Our interest lies in the contact-line region for which we propose a simplified travelling wave approximation. We construct solutions to this problem by a shooting method that matches solution branches in the contact-line region and in the interior of the droplet. Furthermore, we prove uniqueness and qualitative properties of the solution connected to the fourth-order nature of the equation, such as a global maximum in the film height close to the contact line, elevated from the average height of the film.@en

Estimates for sample paths of fast–slow stochastic ordinary differential equations have become a key mathematical tool relevant for theory and applications. In particular, there have been breakthroughs by Berglund and Gentz to prove sharp exponential error estimates. In this paper, we take the first steps in order to generalise this theory to fast–slow stochastic partial differential equations. In a simplified setting with a natural decomposition into low- and high-frequency modes, we demonstrate that for a short-time period the probability for the corresponding sample path to leave a neighbourhood around the stable slow manifold of the system is exponentially small as well.@en