In this paper, we introduce the concept of generalized -contractivityof a pair of maps w.r.t. another pair. We establish a common fixed point result fortwo pairs of self-mappings, when one of these pairs is generalized -contractionw.r.t. the other and study the well-posedness of their fixed point problem. Inparticular, our fixed point result extends the main result of a recent paper ofQingnian ...

We discuss a moving boundary problem arising from a model of gas ionization in the case of negligible electron diffusion and suitable initial data. It describes the time evolution of an ionization front. Mathematically, it can be considered as a system of transport equations with different characteristics for positive and negative charge densities. We show that only advancing fronts are possibl...

In this article, we prove the local well-posedness, for arbitrary initial data with certain regularity assumptions, of the equations of a Viscoelastic Fluid of Johnson-Segalman type with a free surface. More general constitutive laws can be easily managed in the same way. The geometry is defined by a solid fixed bottom and an upper free boundary submitted to surface tension. The proof relies on...

The paper develops a model of traffic flow near an intersection, where drivers seeking to enter a congested road wait in a buffer of limited capacity. Initial data comprise the vehicle density on each road, together with the percentage of drivers approaching the intersection who wish to turn into each of the outgoing roads. If the queue sizes within the buffer are known, then the initial-bounda...

The main results of this paper are concerned with the “bad” behavior of the KP-I equation with respect to a Picard iteration scheme applied to the associated integral equation, for data in usual or anisotropic Sobolev spaces. This leads to some kind of ill-posedness of the corresponding Cauchy problem: the flow map cannot be of class C2 in any Sobolev space.

This paper concerns transonic shocks in compressible inviscid flow passing a twodimensional variable-area duct for the complete steady Euler system. The flow is supersonic at the entrance of the duct, whose boundaries are slightly curved. The condition of impenetrability is posed on the boundaries. After crossing a nearly flat shock front, which passes through a fixed point on the boundary of t...

The Cauchy problem for the higher order equations in the mKdV hierarchy is investigated with data in the spaces b H s (R) defined by the norm ‖v0‖ b Hr s (R) := ‖〈ξ〉 b v0‖Lr′ ξ , 〈ξ〉 = (1 + ξ) 1 2 , 1 r + 1 r = 1. Local well-posedness for the jth equation is shown in the parameter range 2 ≥ r > 1, s ≥ 2j−1 2r . The proof uses an appropriate variant of the Fourier restriction norm method. A coun...

A slightly modified variant of the cubic periodic one-dimensional nonlinear Schrödinger equation is shown to be well-posed, in a relatively weak sense, in certain function spaces wider than L. Solutions are constructed as sums of infinite series of multilinear operators applied to initial data; no fixed point argument or energy inequality are used.