Asymptotic Analysis - Volume 99, issue 1-2

Authors: Fukao, Takeshi

Article Type: Research Article

Abstract: This paper examines the well-posedness of the Stefan problem with a dynamic boundary condition. To show the existence of the weak solution, the original problem is approximated by the limit of an equation and a dynamic boundary condition of Cahn–Hilliard-type. Using this Cahn–Hilliard approach, the enthalpy formulation of the Stefan problem is characterized by the asymptotic limit of a fourth-order system that has a double-well structure. The main result obtained for the Stefan problem can also be applied to a wider class of degenerate parabolic equations by setting the nonlinearity to give the general maximal monotone graph.

Keywords: Stefan problem, dynamic boundary condition, weak solution, Cahn–Hilliard system

DOI: 10.3233/ASY-161373

Citation: Asymptotic Analysis, vol. 99, no. 1-2, pp. 1-21, 2016

Authors: Griso, Georges | Migunova, Anastasia | Orlik, Julia

Article Type: Research Article

Abstract: The elasticity problem for two domains separated by a heterogeneous layer of the thickness ε is considered. The layer has an ε -periodic structure, ε ≪ 1 , including a multiple cracks and the contact between the structural components. The inclusions are surrounded by cracks and can have rigid displacements. The contacts are described by the Signorini and Tresca-friction conditions. In order to obtain preliminary estimates, a modification of the Korn inequality for the ε -dependent periodic layer is performed. An asymptotic analysis with respect to ε → 0 is provided and the …limit elasticity problem is obtained, together with the transmission condition across the interface. The periodic unfolding method is used to study the limit behavior. Show more

Keywords: Korn inequality for contact with inclusions, unfolding on the oscillating interface, homogenization of contact, linear elasticity

DOI: 10.3233/ASY-161374

Citation: Asymptotic Analysis, vol. 99, no. 1-2, pp. 23-52, 2016

Authors: Matthies, Karsten | Uecker, Hannes

Article Type: Research Article

Abstract: We consider a cubic nonlinear wave equation with highly oscillating cubic coefficient and wave packet initial data. Using a regularization step of the initial data, we give a low regularity justification of the Nonlinear Schrödinger equation as the envelope equation.

Keywords: Klein–Gordon equation, nonlinear Schrödinger equation, justification, wave packet, highly oscillating periodic media, amplitude formalism, low regularity

DOI: 10.3233/ASY-161375

Citation: Asymptotic Analysis, vol. 99, no. 1-2, pp. 53-65, 2016

Authors: Mohan, Manil T. | Sritharan, Sivaguru S.

Article Type: Research Article

Abstract: In this work we prove the existence and uniqueness of pathwise solutions up to a stopping time to the stochastic Euler equations perturbed by additive and multiplicative Lévy noise in two and three dimensions. The existence of a unique maximal solution is also proved.

Keywords: Euler equations of fluid dynamics, Lévy process, commutator estimates

DOI: 10.3233/ASY-161376

Citation: Asymptotic Analysis, vol. 99, no. 1-2, pp. 67-103, 2016

Authors: Colombini, Ferruccio | Petkov, Vesselin | Rauch, Jeffrey

Article Type: Research Article

Abstract: Let V ( t ) = e t G b , t ⩾ 0 , be the semigroup generated by Maxwell’s equations in an exterior domain Ω ⊂ R 3 with dissipative boundary condition E tan − γ ( x ) ( ν ∧ B tan ) = 0 , γ ( x ) > 0 , ∀ x ∈ Γ = ∂ Ω . We prove that if γ ( x …) is nowhere equal to 1, then for every 0 < ϵ ≪ 1 and every N ∈ N the eigenvalues of G b lie in the region Λ ϵ ∪ R N , where Λ ϵ = { z ∈ C : | Re z | ⩽ C ϵ ( | Im z | 1 2 + ϵ + 1 ) , Re z < 0 } , R N = { z ∈ C : | Im z | ⩽ C N ( | Re z | + 1 ) − N , Re z < 0 } . Show more

Keywords: Maxwell’s equations, asymptotically disappearing solutions, location of the eigenvalues

DOI: 10.3233/ASY-161377

Citation: Asymptotic Analysis, vol. 99, no. 1-2, pp. 105-124, 2016

Authors: Pham, Du | Nguyen, Phuong

Article Type: Research Article

Abstract: In this work, we consider the stochastic version of the diffusion equations with polynomial reaction terms forced by a multiplicative white noise. We establish the existence and uniqueness of a maximal pathwise solution for a limited period of time. The proof relies on the Skorohod representation theorem, the Gyöngy–Krylov theorem and stopping time arguments.

Keywords: stochastic quasi-linear parabolic equations, martingale and pathwise solutions, local problems

DOI: 10.3233/ASY-161378

Citation: Asymptotic Analysis, vol. 99, no. 1-2, pp. 125-161, 2016