Related papers: On the analyticity of the Dirichlet-Neumann operat…
We put forward a conjecture about an universal asymptotical behaviour of the symbol of the Dirichlet-to-Neumann operator (considered as a pseudodifferential operator) in the 2D exterior problem for the Hemholtz equation. The conjecture is…
We consider ionic electrodiffusion in fluids, described by the Nernst-Planck-Navier-Stokes system. We prove that the system has global smooth solutions for arbitrary smooth data: arbitrary positive Dirichlet boundary conditions for the…
It is well known that the Fourier series Dirichlet-to-Neumann (DtN) boundary condition can be used to solve the Helmholtz equation in unbounded domains. In this work, applying such DtN boundary condition and using the finite element method,…
Stokes waves are steady periodic water waves on the free surface of an infinitely deep irrotational two dimensional flow under gravity without surface tension. They can be described in terms of solutions of the Euler-Lagrange equation of a…
This paper is concerned with the inverse scattering problem involving the time-domain elastic wave equations in a bounded $d$-dimensional domain. First, an explicit reconstruction formula for the density is established by means of the…
The author proves the existence of strong solutions of the Dirichlet problem for the nonstationary Stokes system in polygonal domain. Here, the solutions are elements of weighted Sobolev spaces, where the weight function is a power of the…
In recent papers (arXiv:2407.16507, arXiv:2408.05158) we presented results suggesting the existence of a new class of time-periodic solutions to the defocusing cubic wave equation on a one-dimensional interval with Dirichlet boundary…
We consider Stokes' conjecture concerning the shape of the extremal two-dimensional water wave. By new geometric methods including a nonlinear frequency formula, we prove Stokes' conjecture in the original variables. Our results do not rely…
In this paper, we consider a variational formulation for the Dirichlet problem of the wave equation with zero boundary and initial conditions, where we use integration by parts in space and time. To prove unique solvability in a subspace of…
This paper recalls some classical motivations in fluid dynamics leading to a partial differential equation which is prescribed on a domain whose boundary possesses two connected components, one endowed with a Dirichlet datum, and the other…
We present a way of defining the Dirichlet-to-Neumann operator on general Hilbert spaces using a pair of operators for which each one's adjoint is formally the negative of the other. In particular, we define an abstract analogue of trace…
In this article we study two classical linear water wave problems, i) normal modes of infinite straight channels of bounded constant cross-section, and ii) trapped longitudinal modes in domains with unbounded constant cross-section. Both…
Periodic travelling waves at the free surface of an incompressible inviscid fluid in two dimensions under gravity are numerically computed for an arbitrary vorticity distribution. The fluid domain over one period is conformally mapped from…
We consider discrete nonlinear Schr\"odinger equations of n sites with periodic boundary conditions. These equations have n branches of standing waves that bifurcate from zero. Traveling waves appear as a symmetry-breaking from the standing…
We study the boundary value problem for the stationary Navier--Stokes system in two dimensional exterior domain. We prove that any solution of this problem with finite Dirichlet integral is uniformly bounded. Also we prove the existence…
A general method for the derivation of asymptotic nonlinear shallow water and deep water models is presented. Starting from a general dimensionless version of the water-wave equations, we reduce the problem to a system of two equations on…
A unidirectional reduction of the deep-water surface gravity wave problem is derived in physical space using real variables. By employing a near-identity canonical transformation, cubic interactions are eliminated from the Hamiltonian, with…
Stokes perturbative solution of the nonlinear (boundary value dependent) surface gravity wave problem is known to provide results of reasonable accuracy to engineers in estimating the phase speed and amplitudes of such nonlinear waves. The…
A high-frequency asymptotics of the symbol of the Dirichlet-to-Neumann map, treated as a periodic pseudodifferential operator, in 2D diffraction problems is discussed. Numerical results support a conjecture on a universal limit shape of the…
In this paper, we consider the problem of nonlinear (in particular, saturated) stabilization of the high-dimensional wave equation with Dirichlet boundary conditions. The wave dynamics are subject to a dissipative nonlinear velocity…