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A linear sixth-order partial differential equation (PDE) of "parabolic" type describes the dynamics of thin liquid films beneath surfaces with elastic bending resistance when deflections from the equilibrium film height are small. On a finite domain, the associated sixth-order eigenvalue problem is self-adjoint for the boundary conditions corresponding to a thin film in a closed trough, and the eigenfunctions form a complete orthonormal set. Using these eigenfunctions, we derive the Green's function for the governing sixth-order PDE on a finite interval and compare it to the known infinite-line solution. Further, we propose a Galerkin spectral method based on the constructed sixth-order eigenfunctions and their derivative expansions. The system of ordinary differential equations for the time-dependent expansion coefficients is solved by standard numerical methods. The numerical approach is applied to versions of the governing PDE with a second-order spatial derivative (in addition to the sixth-order one), which arises from gravity acting on the film. In the absence of gravity, we demonstrate the self-similar intermediate asymptotics of initially localized disturbances on the film surface, at least until the disturbances "feel" the finite boundaries, and show that the derived Green's function is an attractor for such solutions. In the presence of gravity, we use the proposed Galerkin numerical method to demonstrate that self-similar behavior persists, albeit for shortened intervals of time, even for large values of the gravity-to-bending ratio. [ABSTRACT FROM AUTHOR]

A linear sixth-order partial differential equation (PDE) of "parabolic" type describes the dynamics of thin liquid films beneath surfaces with elastic bending resistance when deflections from the equilibrium film height are small. On a finite domain, the associated sixth-order eigenvalue problem is self-adjoint for the boundary conditions corresponding to a thin film in a closed trough, and the eigenfunctions form a complete orthonormal set. Using these eigenfunctions, we derive the Green's function for the governing sixth-order PDE on a finite interval and compare it to the known infinite-line solution. Further, we propose a Galerkin spectral method based on the constructed sixth-order eigenfunctions and their derivative expansions. The system of ordinary differential equations for the time-dependent expansion coefficients is solved by standard numerical methods. The numerical approach is applied to versions of the governing PDE with a second-order spatial derivative (in addition to the sixth-order one), which arises from gravity acting on the film. In the absence of gravity, we demonstrate the self-similar intermediate asymptotics of initially localized disturbances on the film surface, at least until the disturbances "feel" the finite boundaries, and show that the derived Green's function is an attractor for such solutions. In the presence of gravity, we use the proposed Galerkin numerical method to demonstrate that self-similar behavior persists, albeit for shortened intervals of time, even for large values of the gravity-to-bending ratio. [ABSTRACT FROM AUTHOR]

The concept of intermediate asymptotics for the solution of an evolution equation with initial data and a related solution obtained without initial conditions was introduced by G.N. Barenblatt and Ya.B. Zeldovich in the context of extending the concept of strict determinism in statistical physics and quantum mechanics. Here, according to V.P. Maslov, to axiomatize the mathematical theory, we need to know the conditions satisfied by the initial data of the problem. We show that the correct solvability of a problem without initial conditions for fractional differential equations in a Banach space is a necessary, but not sufficient, condition for intermediate asymptotics. Examples of intermediate asymptotics are given. [ABSTRACT FROM AUTHOR]

We study theoretically and numerically the bending-driven levelling of thin viscous films within the lubrication approximation. We derive Green's function of the linearized thin-film equation and further show that it represents a universal self-similar attractor at long times. As such, the rescaled perturbation of the film profile converges in time towards the rescaled Green's function, for any summable initial perturbation profile. In addition, for stepped axisymmetric initial conditions, we demonstrate the existence of another, short-term and one-dimensional-like self-similar regime. We also characterize the convergence time towards the long-term universal attractor in terms of the relevant physical and geometrical parameters, and provide the local hydrodynamic fields and global elastic energy in the universal regime as functions of time. Finally, we extend our analysis to the nonlinear thin-film equation through numerical simulations. [ABSTRACT FROM AUTHOR]

The present contribution investigates size effects on Wöhler's curve in accordance with dimensional analysis and intermediate asymptotics theory. These approaches provide a generalised equation able to interpret the specimen‐size effects on Wöhler's curve. Subsequently, using a different approach based on lacunar fractality concepts, analogous scaling laws are found for the coordinates of the limit‐points of Wöhler's curve, so that a theoretical explanation is provided to the decrement in fatigue resistance by increasing the specimen size. Eventually, the proposed models are compared with experimental data available in the Literature, which seem to confirm the advantage of applying fractal geometry to the problem. [ABSTRACT FROM AUTHOR]

The present contribution investigates the crack‐size effects on Paris' law in accordance with dimensional analysis and intermediate asymptotics theory, which makes it possible to obtain a generalised equation able to provide an interpretation to the various empirical power‐laws available in the Literature. Subsequently, within the framework of fractal geometry, scaling laws are determined for the coordinates of the limit‐points of Paris' curve so that a theoretical explanation is provided to the so‐called short cracks problem. Eventually, the proposed models are compared with experimental data available in the literature which seem to confirm the advantage of applying a fractal model to the fatigue problem. [ABSTRACT FROM AUTHOR]

A dimensionless model for the depth of penetration (DOP) of a projectile penetrating into a concrete target, based on the similarity theory involving intermediate asymptotics, complete similarity, and incomplete similarity is presented. The calculated numerical results are in good agreement with previous experimental data, including two sets of full-scale and twenty-four sets of sub-scale penetration of non-deformable projectiles into concrete targets. Moreover, compared with several empirical and semi-empirical DOP models, the new model is applicable within a relatively broader range, including the penetration of both sub-scale and full-scale projectiles. For the limitations of the validity, dimensionless parameters Π3 =ϕt/ϕ larger than 12, Π4 = (ϕ3fc)/(Mv0 2) smaller than 0.1, and the initial impact velocity of the projectile less than about 900 m/s to 1000 m/s are necessary for the model. [ABSTRACT FROM AUTHOR]

Cosserat continuum modelling of crack propagation in particulate materials with microrotations and in layered materials with sliding requires the analysis of the Cosserat characteristic lengths in their relation with the characteristic length of the material microstructure. The Cosserat lengths that are of the order of the microstructural characteristic length should be set to zero to derive the main asymptotic terms. When all Cosserat characteristic lengths are of the microscopic length the obtained continuum is termed the Small scale Cosserat continuum . Cracks of Modes I, II, III have conventional stress singularity, as well as the moment stress singularity. Moment stresses can cause bending failure of the bonds connecting the grains in the particulate material or layers in the layered material. Cases are identified when the bending stress induced by moment stress singularity exceeds tensile stresses induced by the conventional stress singularity by an order of magnitude. Then the main mechanism of crack propagation is bending failure produced by the moment stress, which explains, in a unified fashion the often-observed in-plane propagation of cracks of the classical modes. Another interesting case is when some Cosserat characteristic lengths are large. This gives what we term the Intermediate scale Cosserat continuum . An example is given by layered material with sliding layers that keep full contact during sliding. In this case the crack normal to the layering also propagates by bending failure produced by the moment stress singularity. [ABSTRACT FROM AUTHOR]

This paper collects results concerning global rates and large time asymptotics of a fractional fast diffusion equation on the Euclidean space, which is deeply related with a family of fractional Gagliardo–Nirenberg–Sobolev inequalities. Generically, self-similar solutions are not optimal for the Gagliardo–Nirenberg–Sobolev inequalities, in strong contrast with usual standard fast diffusion equations based on non-fractional operators. Various aspects of the stability of the self-similar solutions and of the entropy methods likecarré du champandRényi entropy powersmethods are investigated and raise a number of open problems. [ABSTRACT FROM AUTHOR]

The formation of elongated supra-molecular structures from protein building blocks generates functional intracellular filaments, but this process is also at the heart of many neurodegenerative conditions including Alzheimer’s and Parkinson’s diseases, where it occurs in an uncontrolled manner. When observed at appropriate concentration and time scales, the chemical kinetics of filamentous protein self-assembly exhibits the remarkable property of self-similarity: the dynamics appears similar as the observation scale changes. We discuss here how this property leads to crucial simplifications of the fundamental laws governing protein filament formation and the emergence of scaling laws that provide the basis for connecting microscopic events with macroscopic realisations of such processes. In particular, we review recent developments in the modelling of linear protein self-assembly phenomena in the light of the concepts of dimensional analysis and physical self-similarity. We show how these tools and concepts can be used to elucidate the nature of the scaling laws for filamentous protein self-assembly, which illuminate the ultimately simple mathematical and physical principles underlying this seemingly highly complex phenomenon, and are expected to guide further developments in the field of linear self-assembly. [ABSTRACT FROM PUBLISHER]

The problem of self-similar flat spreading of groundwater mound in porous stratum bounded by impermeable bed in the presence of constant in time and localized in space source of liquid has been considered. Basing on dimension analysis method, the problem was reduced to ordinary differential equations. One-dimensional and two-dimensional cylindrically symmetrical cases were investigated. Intermediate asymptotics for pressure head distribution was obtained, and limits to applicability of solutions were analyzed. [ABSTRACT FROM AUTHOR]

The paper considers scaling regularities in the deformation and failure of condensed matter (solids and liquids) as effects of a special type of critical phenomena-structural scaling transitions in mesodefect ensembles- and associated structural relaxation mechanisms. The scaling regularities in nonequilibrium processes of plasticity, failure, and turbulence are analyzed with the use of self-similar intermediate asymptotic solutions describing the collective behavior of mesodefects. The predicted role of defect modes in the self-similar response of condensed media is confirmed by original experiments on dynamic, fatigue, and shock loading over a wide range of load intensities. [ABSTRACT FROM AUTHOR]

Cosserat continuum for particulate materials has characteristic lengths commensurate with small particle sizes leading to small-scale Cosserat continuum – an asymptotics intermediate between the characteristic length and the crack length. We show that the main asymptotic term can be obtained from the classical crack problem by finding rotations from the displacements and, using the Cosserat constitutive equations finding the moment stresses. We obtained that Mode I, II stress singularities are the same in classical continuum; the moment stress has a stronger singularity (power 3/2). Bond bending and breakage caused by relative particle rotations constitutes the dominant mechanism of crack propagation. [ABSTRACT FROM AUTHOR]

Abstract: We consider a family of Gagliardo–Nirenberg–Sobolev interpolation inequalities which interpolate between Sobolevʼs inequality and the logarithmic Sobolev inequality, with optimal constants. The difference of the two terms in the interpolation inequalities (written with optimal constant) measures a distance to the manifold of the optimal functions. We give an explicit estimate of the remainder term and establish an improved inequality, with explicit norms and fully detailed constants. Our approach is based on nonlinear evolution equations and improved entropy–entropy production estimates along the associated flow. Optimizing a relative entropy functional with respect to a scaling parameter, or handling properly second moment estimates, turns out to be the central technical issue. This is a new method in the theory of nonlinear evolution equations, which can be interpreted as the best fit of the solution in the asymptotic regime among all asymptotic profiles. [Copyright &y& Elsevier]

We study the long-time behavior of non-negative, finite-energy solutions to the initial value problem for the Porous Medium Equation with variable density, i.e. solutions of the problem {ρ(x)αtu = Δ um, in Q := ℝn x ℝ+, u(x,0) = u0(x), in ℝn, where m > 1, u0 ∈ L¹ (ℝn, ρ (x)dx) and n ≥ 3. We assume that ρ (x) ~ C|x|-2 as |x| → ∞ in ℝn. Such a decay rate turns out to be critical. We show that the limit behavior can be described in terms of a family of source-type solutions of the associated singular equation |x|-2 ut = Δum . The latter have a self-similar structure and exhibit a logarithmic singularity at the origin.Δ [ABSTRACT FROM AUTHOR]

Abstract: This letter is devoted to results on intermediate asymptotics for the heat equation. We study the convergence towards a stationary solution in self-similar variables. By assuming the equality of some moments of the initial data and of the stationary solution, we get improved convergence rates using entropy/entropy-production methods. We establish the equivalence of the exponential decay of the entropies with new, improved functional inequalities in restricted classes of functions. This letter is the counterpart in a linear framework of a recent work on fast diffusion equations; see Bonforte et al. (2009) . The results extend to the case of a Fokker–Planck equation with a general confining potential. [ABSTRACT FROM AUTHOR]

The goal of this paper is to state the optimal decay rate for solutions of the nonlinear fast diffusion equation and, in self-similar variables, the optimal convergence rates to Barenblatt self-similar profiles and their generalizations. It relies an the identification of the optimal constants in some related Hardy-Poincaré inequalities and concludes a long series of papers devoted to generalized entropies, functional inequalities, and rates for nonlinear diffusion equations. [ABSTRACT FROM AUTHOR]

Abstract: The Keller–Segel system describes the collective motion of cells that are attracted by a chemical substance and are able to emit it. In its simplest form, it is a conservative drift-diffusion equation for the cell density coupled to an elliptic equation for the chemo-attractant concentration. This paper deals with the rate of convergence towards a unique stationary state in self-similar variables, which describes the intermediate asymptotics of the solutions in the original variables. Although it is known that solutions globally exist for any mass less 8π, a smaller mass condition is needed in our approach for proving an exponential rate of convergence in self-similar variables. [Copyright &y& Elsevier]

A periodic perturbation of a Gaussian measure modifies the sharp constants in Poincaré and logarithmic Sobolev inequalities in the homogenization limit, that is, when the period of a periodic perturbation converges to zero. We use variational techniques to determine the homogenized constants and get optimal convergence rates towards equilibrium of the solutions of the perturbed diffusion equations. The study of these sharp constants is motivated by the study of the stochastic Stokes' drift. It also applies to Brownian ratchets and molecular motors in biology. We first establish a transport phenomenon. Asymptotically, the center of mass of the solution moves with a constant velocity, which is determined by a doubly periodic problem. In the reference frame attached to the center of mass, the behavior of the solution is governed at large scale by a diffusion with a modified diffusion coefficient. Using the homogenized logarithmic Sobolev inequality, we prove that the solution converges in self-similar variables attached to the center of mass to a stationary solution of a Fokker--Planck equation modulated by a periodic perturbation with fast oscillations, with an explicit rate. We also give an asymptotic expansion of the traveling diffusion front corresponding to the stochastic Stokes' drift with given potential flow. [ABSTRACT FROM AUTHOR]

Complex spatiotemporal structures develop during the process of aging glasses after cooling and of rejuvenating glasses on heating. The key to understanding these structures is the interplay between the activated reconfiguration events that generate mobility and the transport of mobility. These effects are both accounted for by combining the random first-order transition theory of activated events with mode coupling theory in an inhomogeneous setting. The predicted modifications by mobility transport of the time course of the aging regime are modest. In contrast, the rejuvenation process is strongly affected through the propagation of fronts of enhanced mobility originating from the initial reconfiguration events. The structures in a rejuvenating glass resemble flames. An analysis along the lines of combustion theory provides an estimate of the front propagation speed. Heterogeneous rejuvenation naturally should occur for glasses with free surfaces. The analogy with combustion also provides a way of looking at the uptake of diluents by glasses described by case II and super case II diffusion. [ABSTRACT FROM AUTHOR]

Abstract: We study the linear stability properties of the Barenblatt–Pattle (B–P) self-similar solutions of the porous medium equation which models flow including viscous and porous media gravity currents. Grundy and McLaughlin [R.E. Grundy, R. McLaughlin, Eigenvalues of the Barenblatt–Pattle similarity solution in nonlinear diffusion, Proc. Roy. Soc. London Ser. A 383 (1982) 89–100] have shown that, in both planar and axisymmetric geometries, the B–P solutions are linearly stable to symmetric perturbations. Using a new technique that eliminates singularities in the linear stability analysis, we extend their result and establish that the axisymmetric B–P solution is linearly stable to asymmetric perturbations. This suggests that the axisymmetric B–P solution provides the intermediate asymptotics of gravity currents that evolve from a wide range of initial distributions including those that are not axisymmetric. We use the connection between the perturbation eigenfunctions and the symmetry transformations of the B–P solution to demonstrate that the leading order rate of decay of the perturbations can be maximised by redefining the volume, time and space variables. We show that, in general, radially symmetric perturbations decay faster than asymmetric perturbations of equal amplitude. These theoretical predictions are confirmed by numerical results. [Copyright &y& Elsevier]

The Keller-Segel system describes the collective motion of cells which are attracted by a chemical substance and are able to emit it. In its simplest form it is a conservative drift-diffusion equation for the cell density coupled to an elliptic equation for the chemo-attractant concentration. It is known that, in two space dimensions, for small initial mass, there is global existence of solutions and for large initial mass blow-up occurs. In this paper we complete this picture and give a detailed proof of the existence of weak solutions below the critical mass, above which any solution blows-up in finite time in the whole Euclidean space. Using hypercontractivity methods, we establish regularity results which allow us to prove an inequality relating the free energy and its time derivative. For a solution with sub-critical mass, this allows us to give for large times an "intermediate asymptotics" description of the vanishing. In self-similar coordinates, we actually prove a convergence result to a limiting self-similar solution which is not a simple reflect of the diffusion. [ABSTRACT FROM AUTHOR]

In this paper the dynamics of a system of spherical particles that fill a small volume fraction of the space and that evolves in a concentration field is discussed. Corrections to the Lifshitz–Slyozov–Wagner (LSW) model that take into account the stochastic character of the problem are computed. It is proved, under suitable smallness assumptions for the volume fraction filled by the particles, that the effect of these corrections does not modify much the dynamics of the self-similar solutions of the LSW system of equations. [ABSTRACT FROM AUTHOR]

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