Infinitely many solitary waves in three space dimensions
Infinitely narrow soliton solutions to systems of conservation laws.
Influence of bottom topography on long water waves
We focus here on the water waves problem for uneven bottoms in the long-wave regime, on an unbounded two or three-dimensional domain. In order to derive asymptotic models for this problem, we consider two different regimes of bottom topography, one for small variations in amplitude, and one for strong variations. Starting from the Zakharov formulation of this problem, we rigorously compute the asymptotic expansion of the involved Dirichlet-Neumann operator. Then, following the global strategy...
Inf-sup stable nonconforming finite elements of higher order on quadrilaterals and hexahedra
We present families of scalar nonconforming finite elements of arbitrary order with optimal approximation properties on quadrilaterals and hexahedra. Their vector-valued versions together with a discontinuous pressure approximation of order form inf-sup stable finite element pairs of order r for the Stokes problem. The well-known elements by Rannacher and Turek are recovered in the case r=1. A numerical comparison between conforming and nonconforming discretisations will be given. Since higher order...
Inhomogeneous boundary value problems for the von Kármán equations. II
Inhomogeneous boundary value problems for the von Kármán equations. I
Initial boundary value problem for generalized Zakharov equations
This paper considers the existence and uniqueness of the solution to the initial boundary value problem for a class of generalized Zakharov equations in dimensions, and proves the global existence of the solution to the problem by a priori integral estimates and the Galerkin method.
Initial boundary value problem for the mKdV equation on a finite interval
We analyse an initial-boundary value problem for the mKdV equation on a finite interval by expressing the solution in terms of the solution of an associated matrix Riemann-Hilbert problem in the complex -plane. This RH problem is determined by certain spectral functions which are defined in terms of the initial-boundary values at and . We show that the spectral functions satisfy an algebraic “global relation” which express the implicit relation between all boundary values in terms of spectral...
Initial-boundary value problem for equations of generalized Newtonian incompressible fluid.
Initial-boundary value problem for generalized Stokes equations
The paper is concerned with the solvability theory of the generalized Stokes equations arising in the study of the motion of non-Newtonian fluids.
Initial-boundary value problems describing mobile carrier transport in semiconductor devices
Initial-boundary value problems for nonlinear pseudoparabolic equations in a critical case.
Initial-boundary value problems for quasilinear dispersive equations posed on a bounded interval.
Initial-boundary value problems for second order systems of partial differential equations
We develop a well-posedness theory for second order systems in bounded domains where boundary phenomena like glancing and surface waves play an important role. Attempts have previously been made to write a second order system consisting of n equations as a larger first order system. Unfortunately, the resulting first order system consists, in general, of more than 2n equations which leads to many complications, such as side conditions which must be satisfied by the solution of the larger first order...
Initial-boundary value problems for second order systems of partial differential equations∗
We develop a well-posedness theory for second order systems in bounded domains where boundary phenomena like glancing and surface waves play an important role. Attempts have previously been made to write a second order system consisting of n equations as a larger first order system. Unfortunately, the resulting first order system consists, in general, of more than 2n equations which leads to many complications, such as side conditions which must...
Injectivity of the spherical mean operator on the conical manifolds of spheres.
Instability of radial standing waves of Schrödinger equation on the exterior of a ball.
Instability of symmetric stationary states for some nonlinear Schrödinger equations with an external magnetic field
Instability of the stationary solutions of generalized dissipative Boussinesq equation
In this work we study the generalized Boussinesq equation with a dissipation term. We show that, under suitable conditions, a global solution for the initial value problem exists. In addition, we derive sufficient conditions for the blow-up of the solution to the problem. Furthermore, the instability of the stationary solutions of this equation is established.
Integrability and Einstein's equations
1. Introduction. In recent years, there has been considerable interest in Oxford and elsewhere in the connections between Einstein's equations, the (anti-) self-dual Yang-Mills (SDYM) equations, and the theory of integrable systems. The common theme running through this work is that, to a greater or lesser extent, all three areas involve questions that can be addressed by twistor methods. In this paper, I shall review progress, with particular emphasis on the known and potential applications in...