In this paper, we establish the local well-posedness for the Cauchy problem of the simplified version of hydrodynamic flow of nematic liquid crystals (1.1) in R3 for any initial data (u0, d0) having small L3 uloc -norm of (u0,∇d0). Here L3uloc(R 3) is the space of uniformly locally L3-integrable functions. For any initial data (u0, d0) with small ‖(u0,∇d0)‖L3(R3), we show that there exists a un...

A well-posedness theory for the initial value problem for hydroelastic waves in two spatial dimensions is presented. This problem, which arises in numerous applications, describes the evolution of a thin elastic membrane in a two-dimensional potential flow. We use a model for the elastic sheet that accounts for bending stresses and membrane tension, but which neglects the mass of the membrane. ...

In this paper we carry out a theoretical analysis of two regularization methods for nonlinear ill-posed problems. The rst is a penalty method called Tikhonov regularization, in which one solves an unconstrained optimization problem while the second is based on a constrained optimization problem. For each method we examine the well-posedness of the respective optimization problem. We then show s...

Abstract. Because of their sparsity enhancing properties, `1 penalty terms have recently received much attention in the field of inverse problems. Also, it has been shown that certain properties of the linear operator A to be inverted imply that `1-regularization is equivalent to `0-regularization, which tries to minimise the number of non-zero coefficients. In the context of compressed sensing...

In this paper, we introduce the concepts of extended well-posedness for quasivariational inequalities and establish some characterizations. We show that the extended well-posedness is equivalent to the existence and uniqueness of solutions under suitable conditions. In addition, the corresponding concepts of extended well-posedness in the generalized sense are introduced and investigated for qu...

where u is the velocity and p is the pressure, with inital data u(x, 0) = u0(x). Existence of weak solutions has been shown by Leray. Uniqueness (and regularity) of weak solutions is unknown and both are among the major open questions in applied analysis. Under stronger assumptions there exist local and/or global smooth solutions. One version of this has been shown by Kato for initial data in L...

Article history: Received 4 April 2008 Revised 11 January 2009 Available online 28 February 2009

1.1. Presentation of the problem. The water-waves problem for an ideal liquid consists of describing the motion of the free surface and the evolution of the velocity field of a layer of perfect, incompressible, irrotational fluid under the influence of gravity. In this paper, we restrict our attention to the case when the surface is a graph parameterized by a function ζ(t,X), where t denotes th...