Asymptotic Analysis - Volume 106, issue 2

Authors: Lteif, Ralph | Israwi, Samer

Article Type: Research Article

Abstract: The Green–Naghdi type model in the Camassa–Holm regime derived in [Comm. Pure Appl. Anal. 14 (6) (2015) 2203–2230], describe the propagation of medium amplitude internal waves over medium amplitude topography variations. It is fully justified in the sense that it is well-posed, consistent with the full Euler system and converges to the latter with corresponding initial data. In this paper, we generalize this result by constructing a fully justified coupled asymptotic model in a more complex physical case of variable topography. More precisely, we are interested in specific bottoms wavelength of characteristic order λ b = …λ / α where λ is a characteristic horizontal length (wave-length of the interface). We assume a slowly varying topography with large amplitude (β α = O ( μ ) , where β characterizes the shape of the bottom). In addition, our system permits the full justification of any lower order, well-posed and consistent model. We apply the procedure to scalar models driven by simple unidirectional equations in the Camassa–Holm and long wave regimes and under some restrictions on the topography variations. We also show that wave breaking of solutions to such equations occurs in the Camassa–Holm regime with slow topography variations and for a specific set of parameters. Show more

Keywords: Green–Naghdi equations, Camassa–Holm regime, variable topography, asymptotic models, full justification

DOI: 10.3233/ASY-171440

Citation: Asymptotic Analysis, vol. 106, no. 2, pp. 61-98, 2018

Authors: Belhachmi, Z. | Ben Abda, A. | Meftahi, B. | Meftahi, H.

Article Type: Research Article

Abstract: We consider the two-dimensional steady-state heat conduction on a bounded domain Ω containing a heat source at an unknown location ω ⊂ Ω . We are interested in determining the locations ω allowing a suitable thermal environment. The resulting shape optimization problem consists of a geometric control of either the maximum of the temperature or its L 2 mean oscillations in Ω. We derive the topological asymptotic expansion of the considered shape functionals with respect to the insertion of small circular coated inclusion characterized by a discontinuous thermal conductivity and their …radius. We propose a reconstruction algorithm based on this topological gradient to identify the locations. Some numerical simulations are presented to show the efficiency of the algorithm. Show more

Keywords: Topological optimization, asymptotic analysis, coated inclusion, heat conduction

DOI: 10.3233/ASY-171441

Citation: Asymptotic Analysis, vol. 106, no. 2, pp. 99-119, 2018

Authors: Kukavica, Igor | Tuffaha, Amjad

Article Type: Research Article

Abstract: We establish a priori estimates for local-in-time existence of solutions to the water wave model consisting of the 3D incompressible Euler equations on a domain with a free surface, without surface tension, under minimal regularity assumptions on the initial data and the Rayleigh–Taylor sign condition. The initial data are allowed to be rotational and they are assumed to belong to H 2.5 + δ , where δ > 0 is arbitrary.

Keywords: Euler equations, free boundary, waves, local existence, Taylor condition

DOI: 10.3233/ASY-171459

Citation: Asymptotic Analysis, vol. 106, no. 2, pp. 121-145, 2018