Abstract wave equations with acoustic boundary conditions

We define an abstract setting to treat wave equations equipped with time-dependent acoustic boundary conditions on bounded domains of ${\bf R}^n$. We prove a well-posedness result and develop a spectral theory which also allows to prove a conjecture proposed in (Gal-Goldstein-Goldstein, J. Evol. Equations 3 (2004), 623-636). Concrete problems are also discussed.Comment: 21 pages. A few minor bugs appearing in the journal version have been fixe

Thin waveguides with Robin boundary conditions

We consider the Laplace operator in a thin three dimensional tube with a Robin type condition on its boundary and study, asymptotically, the spectrum of such operator as the diameter of the tube's cross section becomes infinitesimal. In contrast with the Dirichlet condition case, we evidence different behaviors depending on a symmetry criterium for the fundamental mode in the cross section. If that symmetry condition fails, then we prove the localization of lower energy levels in the vicinity of the minimum point of a suitable function on the tube's axis depending on the curvature and the rotation angle. In the symmetric case, the behavior of lower energy modes is shown to be ruled by a one dimensional Sturm-Liouville problem involving an effective potential given in explicit form

Results for a turbulent system with unbounded viscosities: weak formulations, existence of solutions, boundedness, smoothness'

We consider a circulation system arising in turbulence modelling in fluid dynamics with unbounded eddy viscosities. Various notions of weak solutions are considered and compared. We establish existence and regularity results. In particular we study the boundedness of weak solutions. We also establish an existence result for a classical solutio

Numerical scheme based on the spectral method for calculating nonlinear hyperbolic evolution equations

High-precision numerical scheme for nonlinear hyperbolic evolution equations is proposed based on the spectral method. The detail discretization processes are discussed in case of one-dimensional Klein-Gordon equations. In conclusion, a numerical scheme with the order of total calculation cost $O(N \log 2N)$ is proposed. As benchmark results, the relation between the numerical precision and the discretization unit size are demonstrated.Comment: To appear in the proceedings of ICCM2020. Figure is modified from the original versio

Analysis of segregated boundary-domain integral equations for variable-coefficient problems with cracks

This is the pre-print version of the article. The official published version can be obtained from the link below - Copyright @ 2011 Wiley-BlackwellSegregated direct boundary-domain integral equation (BDIE) systems associated with mixed, Dirichlet and Neumann boundary value problems (BVPs) for a scalar “Laplace” PDE with variable coefficient are formulated and analyzed for domains with interior cuts (cracks). The main results established in the paper are the BDIE equivalence to the original BVPs and invertibility of the BDIE operators in the corresponding Sobolev spaces.This work was supported by the International Joint Project Grant - 2005/R4 ”Boundary- Domain Integral Equations: Formulation, Analysis, Localisation” of the Royal Society, UK, and the grant ”Mathematical analysis of localised boundary-domain integral equations for BVPs with variable coefficients” of the EPSRC, UK

Generalized solutions and distributional shadows for Dirac equations

We discuss the application of recent results on generalized solutions to the Cauchy problem for hyperbolic systems to Dirac equations with external fields. In further analysis we focus on the question of existence of associated distributional limits and derive their explicit form in case of free Dirac fields with regularizations of initial values corresponding to point-like probability densities