In this paper, we consider in R the Cauchy problem for nonlinear Schrödinger equation with initial data in Sobolev space W s,p for p < 2. It is well known that this problem is ill posed. However, We show that after a linear transformation by the linear semigroup the problem becomes locally well posed in W s,p for 2n n+1 < p < 2 and s > n(1− 1 p ). Moreover, we show that in one space dimension, ...

In this paper, we consider in R the Cauchy problem for nonlinear Schrödinger equation with initial data in Sobolev space W s,p for p < 2. It is well known that this problem is ill posed. However, We show that after a linear transformation by the linear semigroup the problem becomes locally well posed in W s,p for 2n n+1 < p < 2 and s > n(1− 1 p ). Moreover, we show that in one space dimension, ...

We propose time regularizations for ill posed evolution equations of the type of the Perona-Malik equation of image processing, prove that they are well posed, and give numerical evidence for their superiority to the widely used space regularizations.

We study the initial value problem (IVP) associated to some canonical dispersive equations. Our main concern is to establish the minimal regularity property required in the data which guarantees the local well-posedness of the problem. Measuring this regularity in the classical Sobolev spaces, we show ill-posedness results for Sobolev index above the value suggested by the scaling argument.

The paper is concerned with the solution of nonlinear ill-posed problems by methods that utilise the second derivative. A general predictor{corrector approach is developed; one which avoids solving quadratic equations during the iteration process. Combining regularisation of each iteration step with an adequate stopping condition leads to a general regularisation scheme for nonlinear equations....

In this paper, we consider distribution solutions to the aggregation equation ρt + div(ρu) = 0, u = −∇V ∗ρ in R where the density ρ concentrates on a co-dimension one manifold. We show that an evolution equation for the manifold itself completely determines the dynamics of such solutions. We refer to such solutions aggregation sheets. When the equation for the sheet is linearly well-posed, we s...

This is a study concerning the identification of the heterogeneous flexural rigidity of a beam governed by the steady-state Euler-Bernoulli fourth order ordinary differential equation. We use the method of Variational Imbedding (MVI) to deal with the inverse problem for the coefficient identification from over-posed data. The method is identifying the coefficient by approximating it with a piec...

In this article, we prove that the initial value problem associated with the Korteweg-de Vries equation is well-posed in weighted Sobolev spaces X s,θ, for s ≥ 2θ ≥ 2 and the initial value problem associated with the nonlinear Schrödinger equation is well-posed in weighted Sobolev spaces X s,θ, for s ≥ θ ≥ 1. Persistence property has been proved by approximation of the solutions and using a pri...