Asymptotic Analysis

We study the well-posedness and asymptotic behavior of solutions to the Cauchy problem of a three-dimensional sixth-order Cahn–Hilliard equation arising in oil-water-surfactant mixtures. First, by using the pure energy method and a standard continuity argument, we prove that there exists a unique global strong solution provided that the H 2 -norm of the initial data is sufficiently small. Moreover, we establish suitable negative Sobolev norm estimates and obtain the optimal decay rates of the higher-order spatial derivatives of the strong solution.

This work addresses the mathematical analysis – by means of asymptotic analysis – of a penalisation strategy for the full discretisation of elastic wave propagation problems in quasi-incompressible media that has been recently developed by the authors. We provide a convergence analysis of the solution to the continuous version of the penalised problem towards its formal limit when the penalisation parameter tends to infinity. Moreover, as a fundamental intermediate step we provide an asymptotic analysis of the convergence of solutions to quasi-incompressible problems towards solutions to purely incompressible problems when the incompressibility parameter tends to infinity. Finally, we further detail the regularity assumptions required to guarantee that the mentioned convergence holds.

We investigate the boundary layers of a singularly perturbed reaction-diffusion equation in a 3D channel domain. The equation is supplemented with a Robin boundary condition especially when the smooth function on the boundary, appearing in the Robin boundary condition, depends on the perturbation parameter. By constructing an explicit function, called corrector, which describes behavior of the perturbed solution near the boundary, we obtain an asymptotic expansion of the perturbed solution as the sum of the corresponding limit solution and the corrector, and show the convergence in L 2 of the perturbed solution to the limit solution as the perturbation parameter tends to zero.

We consider recurrent diffusive random walks on a strip. We present constructive conditions on Green functions of finite sub-domains which imply a Central Limit Theorem with polynomial error bound, a Local Limit Theorem, and mixing of environment viewed by the particle process. Our conditions can be verified for a wide class of environments including independent environments, quasiperiodic environments, and environments which are asymptotically constant at infinity. The conditions presented deal with a fixed environment, in particular, no stationarity conditions are imposed.

We consider a sinh-Poisson type equation with variable intensities and Dirichlet boundary condition on a pierced domain Δ u + ρ ( V 1 ( x ) e u − V 2 ( x ) e − τ u ) = 0 in  Ω ϵ : = Ω ∖ ⋃ i = 1 m B ( ξ i , ϵ i ) ‾ u = 0 on  ∂ Ω ϵ , where ρ > 0 , V 1 , V 2 > 0 are smooth potentials in Ω, τ > 0 , Ω is a smooth bounded domain in R 2 and B ( ξ i , ϵ i ) is a ball centered at ξ i ∈ Ω with radius ϵ i > 0 , i = 1 , … , m . When ρ > 0 is small enough and m 1 ∈ { 1 , … , m − 1 } , there exist radii ϵ = ( ϵ 1 , … , ϵ m ) small enough such that the problem has a solution which blows-up positively at the points ξ 1 , … , ξ m 1 and negatively at the points ξ m 1 + 1 , … , ξ m as ρ → 0 . The result remains true in cases m 1 = 0 with V 1 ≡ 0 and m 1 = m with V 2 ≡ 0 , which are Liouville type equations.

This paper investigates the initial boundary value problem for a class of fourth order nonlinear damped wave equations modeling longitudinal motion of an elasto-plastic bar. By applying a suitable potential well-convexity method, we derive the global existence, asymptotic behavior and finite time blow up for the considered problem with more generalized nonlinear functions at subcritical initial energy level. Further for arbitrarily positive initial energy we give some sufficient conditions ensuring finite time blow up.

We consider in this work small random perturbations (of multiplicative noise type) of the gradient flow. We prove that under mild conditions, when the potential function is a Morse function with additional strong saddle condition, the perturbed gradient flow converges to the neighborhood of local minimizers in O ( ln ( ε − 1 ) ) time on the average, where ε is the scale of the random perturbation. Under a change of time scale, this indicates that for the diffusion process that approximates the stochastic gradient method, it takes (up to logarithmic factor) only a linear time of inverse stepsize to evade from all saddle points. This can be regarded as a manifestation of fast convergence of the discrete-time stochastic gradient method, the latter being used heavily in modern statistical machine learning.

In this paper we prove the existence of approximate controls for certain classes of parabolic problems with non-smooth coefficients and discuss as examples the problem of approximate controllability for the heat flow in heterogeneous media such as, periodic composites, perforated domains or periodic microstructures separated by rough interfaces.