Asymptotic Analysis - Volume 7, issue 4

Authors: Frank, L.S.

Article Type: Research Article

Abstract: Quasi-stationary surface water waves propagating along a distinguished space direction in an infinite channel of finite depth are considered under the assumption that the liquid is inviscid and incompressible. Besides the gravitation effect also the capillary phenomenon is taken into consideration. After rescaling the mathematical problem is reduced to a singularly perturbed pseudodifferential equation on the free boundary of the liquid. Depending on the values of two basic dimensionless parameters, the rescaled speed of propagation and the parameter characterizing the surface tension (capillarity), the existence of different kind of quasi-stationary waves (periodic, quasi-periodic and solitary) is established for this exact …mathematical model and their asymptotic behavior is investigated when the natural dimensionless small parameter (the ratio of the depth of the undisturbed liquid's layer and the wave's length scale) vanishes. Show more

DOI: 10.3233/ASY-1993-7401

Citation: Asymptotic Analysis, vol. 7, no. 4, pp. 233-238, 1993

Authors: Jurkat, Wolfgang B.

Article Type: Research Article

Abstract: New summability methods, similar to Lindelöf's, are introduced which sum “most” asymptotic power series which occur in applications.

DOI: 10.3233/ASY-1993-7402

Citation: Asymptotic Analysis, vol. 7, no. 4, pp. 239-250, 1993

Authors: Eizenberg, A.

Article Type: Research Article

Abstract: A Hamilton–Jacobi equation, with a convex Hamiltonian such that the corresponding Langrangean vanishes along the orbits of certain dynamical systems, is considered. Small elliptic and inhomogeneous perturbations of such HJE are studied in bounded domains. In the problem at hand the HJE may have more than one viscosity solution under zero boundary conditions, which brings up the question of uniqueness conditions. It is found that under the uniqueness conditions the problem can be reduced to the linear case by a completely new method (notice that the logarithmic transformation, generally speaking, cannot be applied in the considered problem).

DOI: 10.3233/ASY-1993-7403

Citation: Asymptotic Analysis, vol. 7, no. 4, pp. 251-285, 1993

Authors: Penzel, F. | Speck, F.-O.

Article Type: Research Article

Abstract: The topic of asymptotic expansion of solutions of pseudodifferential equations in the spirit of Eskin's work is extended to a more general situation. Taylor expansion of Fourier symbol matrix functions is replaced by a series of generalized invertible operators, which act on vector Sobolev spaces. The fractional orders of these spaces are obtained from the jumps of the lifted symbol matrix at infinity in a situation which is most interesting for applications. Asymptotic and regularity results for the solutions of corresponding systems of equations are direct consequences.

DOI: 10.3233/ASY-1993-7404

Citation: Asymptotic Analysis, vol. 7, no. 4, pp. 287-300, 1993

Authors: Miersemann, Erich

Article Type: Research Article

Abstract: In a recent paper on the circular capillary tube we proved the asymptotic correctness of a formal expansion of the rise height given by Laplace in 1806. Here we extend this result to tubes of more general cross section. We prove the existence of an asymptotic expansion of the rise height with respect to the (small) Bond number under the main assumption that the zero gravity solution exists. The proof is completely based on the comparison principle of Concus and Finn. As examples, the annulus and the regular n-gon will be considered. In the case of the annulus the …expansion is uniform with respect to the boundary contact angle which is a consequence of the special nonlinearity of the problem. Show more

DOI: 10.3233/ASY-1993-7405

Citation: Asymptotic Analysis, vol. 7, no. 4, pp. 301-309, 1993

Article Type: Other

Citation: Asymptotic Analysis, vol. 7, no. 4, pp. 311-311, 1993