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...

We consider lexicographic vector equilibrium problems in metric spaces. Sufficient conditions for a family of such problems to be (uniquely) well-posed at the reference point are established. As an application, we derive several results on well-posedness for a class of variational inequalities.

We give rigorous results on the analytical properties of multilane traffic flow models based on hyperbolic balance laws. 2000 Mathematics Subject Classification: 35L65, 35L67, 90B20

We define entropy weak solutions and establish well-posedness for the Cauchy problem for the formal equation ∂tu(t, x) + ∂x u 2 (t, x) = −λu(t, x) δ0(x), which can be seen as two Burgers equations coupled in a non-conservative way through the interface located at x = 0. This problem appears as an important auxiliary step in the theoretical and numerical study of the one-dimensional particle-in-...

In this talk, we consider the recent results on well-posedness and small asymptotic analysis on the quantum hydrodynamic model which takes the form of Euler-Poisson with dispersion.

We consider an acoustic wave system with discontinuous coe cients and nonsmooth inputs. Existence, uniqueness and continuous dependence on input data of weak solutions are established.

The Kuramoto-Sivashinsky equation arises in a variety of applications, among which are modeling reaction-diffusion systems, flame-propagation and viscous flow problems. It is considered here, as a prototype to the larger class of generalized Burgers equations: those consist of quadratic nonlinearity and arbitrary linear parabolic part. We show that such equations are well-posed, thus admitting ...

In this paper, we prove the local well-posedness of the Ericksen-Leslie system, and the global well-posednss for small initial data under the physical constrain condition on the Leslie coefficients, which ensures that the energy of the system is dissipated. Instead of the GinzburgLandau approximation, we construct an approximate system with the dissipated energy based on a new formulation of th...

We consider the cubic and quintic Gross-Pitaevskii (GP) hierarchy in d dimensions, for focusing and defocusing interactions. We introduce new higher order conserved energy functionals that allow us to prove global existence and uniqueness of solutions for defocusing GP hierarchies, with arbitrary initial data in the energy space. Moreover, we prove generalizations of the Sobolev and Gagliardo-N...