Asymptotic Analysis - Volume 110, issue 3-4

Authors: Delgadillo, Ricardo | Lu, Jianfeng | Yang, Xu

Article Type: Research Article

Abstract: Propagation of high-frequency wave in periodic media is a challenging problem due to the existence of multiscale characterized by short wavelength, small lattice constant and large physical domain size. Conventional computational methods lead to extremely expensive costs, especially in high dimensions. In this paper, based on Bloch decomposition and asymptotic analysis in the phase space, we derive the frozen Gaussian approximation for high-frequency wave propagation in periodic media and establish its converge to the true solution. The formulation leads to efficient numerical algorithms, which are presented in a companion paper [SIAM J. Sci. Comput. 38 (2016), A2440–A2463].

Keywords: Frozen Gaussian approximation, high-freqency wave propagation, lattice potential, Bloch decomposition

DOI: 10.3233/ASY-181479

Citation: Asymptotic Analysis, vol. 110, no. 3-4, pp. 113-135, 2018

Authors: Khrabustovskyi, Andrii | Post, Olaf

Article Type: Research Article

Abstract: Let Δ Ω ε be the Dirichlet Laplacian in the domain Ω ε : = Ω ∖ ( ⋃ i D i ε ) . Here Ω ⊂ R n and { D i ε } i is a family of tiny identical holes (“ice pieces”) distributed periodically in R n with period ε . We denote by cap …( D i ε ) the capacity of a single hole. It was known for a long time that − Δ Ω ε converges to the operator − Δ Ω + q in strong resolvent sense provided the limit q : = lim ε → 0 cap ( D i ε ) ε − n exists and is finite. In the current contribution we improve this result deriving estimates for the rate of convergence in terms of operator norms. As an application, we establish the uniform convergence of the corresponding semi-groups and (for bounded Ω) an estimate for the difference of the k th eigenvalue of − Δ Ω ε and − Δ Ω ε + q . Our proofs relies on an abstract scheme for studying the convergence of operators in varying Hilbert spaces developed previously by the second author. Show more

Keywords: Crushed ice problem, homogenization, norm resolvent convergence, operator estimates, varying Hilbert spaces

DOI: 10.3233/ASY-181480

Citation: Asymptotic Analysis, vol. 110, no. 3-4, pp. 137-161, 2018

Authors: Cherednichenko, K. | Dondl, P. | Rösler, F.

Article Type: Research Article

Abstract: For several different boundary conditions (Dirichlet, Neumann, Robin), we prove norm-resolvent convergence for the operator − Δ in the perforated domain Ω ∖ ⋃ i ∈ 2 ε Z d B a ε ( i ) , a ε ≪ ε , to the limit operator − Δ + μ ι on L 2 ( Ω ) , where μ ι ∈ C …is a constant depending on the choice of boundary conditions. This is an improvement of previous results [Progress in Nonlinear Differential Equations and Their Applications 31 (1997 ), 45–93; in: Proc. Japan Acad. , 1985 ], which show strong resolvent convergence. In particular, our result implies Hausdorff convergence of the spectrum of the resolvent for the perforated domain problem. Show more

Keywords: Perforated domain, homogenisation, norm-resolvent convergence, analysis of PDE

DOI: 10.3233/ASY-181481

Citation: Asymptotic Analysis, vol. 110, no. 3-4, pp. 163-184, 2018

Authors: Robinson, James C. | Rodrigo, José L. | Skipper, Jack W.D.

Article Type: Research Article

Abstract: We study weak solutions of the incompressible Euler equations on T 2 × R + ; we use test functions that are divergence free and have zero normal component, thereby obtaining a definition that does not involve the pressure. We prove energy conservation under the assumptions that u ∈ L 3 ( 0 , T ; L 3 ( T 2 × R + ) ) , lim | y | → 0 …1 | y | ∫ 0 T ∫ T 2 ∫ x 3 > | y | ∞ | u ( x + y ) − u ( x ) | 3 d x d t = 0 , and an additional continuity condition near the boundary: for some δ > 0 we require u ∈ L 3 ( 0 , T ; C 0 ( T 2 × [ 0 , δ ] ) ) . We note that all our conditions are satisfied whenever u ( x , t ) ∈ C α , for some α > 1 / 3 , with Hölder constant C ( x , t ) ∈ L 3 ( T 2 × R + × ( 0 , T ) ) . Show more

Keywords: Onsager’s conjecture, Euler equations, energy conservation

DOI: 10.3233/ASY-181482

Citation: Asymptotic Analysis, vol. 110, no. 3-4, pp. 185-202, 2018

Authors: Pohjola, Valter

Article Type: Research Article

Abstract: We prove that the Dirichlet eigenvalues and Neumann boundary data of the corresponding eigenfunctions of the operator − Δ + q , determine the potential q , when q ∈ L n / 2 ( Ω , R ) and n ⩾ 3 . We also consider the case of incomplete spectral data, in the sense that the above spectral data is unknown for some finite number of eigenvalues. In this case we prove that the potential q is uniquely determined for q ∈ L p …( Ω , R ) with p = n / 2 , for n ⩾ 4 and p > n / 2 , for n = 3 . Show more

Keywords: Inverse spectral problem, Borg–Levinson theorem

DOI: 10.3233/ASY-181484

Citation: Asymptotic Analysis, vol. 110, no. 3-4, pp. 203-226, 2018