Related papers: Well-posedness for a dispersive system of the Whit…

We regard the Cauchy problem for a particular Whitham-Boussinesq system modelling surface waves of an inviscid incompressible fluid layer. The system can be seen as a weak nonlocal dispersive perturbation of the shallow water system. The…

The initial-value problem for a particular bidirectional Whitham system modelling surface water waves is under consideration. This system was recently introduced in [4]. It is numerically shown to be stable and a good approximation to the…

We study the Cauchy problem for one-dimensional dispersive system of Boussinesq type which models weakly nonlinear long wave surface waves. We establish the local well-posedness and ill-posedness of solutions to the system. We also provide…

This article represents a first step towards understanding the well-posedness for the dispersive Hunter-Saxton equation. This problem arises in the study of nematic liquid crystals, and although the equation has formal similarities with the…

In this note, we prove local-in-time well-posedness for a fully dispersive Boussinesq system arising in the context of free surface water waves in two and three spatial dimensions. Those systems can be seen as a weak nonlocal dispersive…

Consideration is given to three different full dispersion Boussinesq systems arising as asymptotic models in the bi-directional propagation of weakly nonlinear surface waves in shallow water. We prove that, under a non-cavitation condition…

This paper is concerned with a two dimensional Whitham-Boussinesq system modelling surface waves of an inviscid incompressible fluid layer. We prove that the associated Cauchy problem is well-posed for initial data of low regularity, with…

We consider the initial-value problem for the bidirectional Whitham equation, a system which combines the full two-way dispersion relation from the incompressible Euler equations with a canonical shallow-water nonlinearity. We prove local…

In this paper we propose a new approach to prove the local well-posedness of the Cauchy problem associated with strongly non resonant dispersive equations. As an example we obtain unconditional well-posedness of the Cauchy problem below $…

A family of Boussinesq systems has been proposed to describe the bi-directional propagation of small amplitude long waves on the surface of shallow water. In this paper, we investigate the well-posedness and boundary stabilization of the…

These notes are devoted to the notion of well-posedness of the Cauchy problem for nonlinear dispersive equations. We present recent methods for proving ill-posedness type results for dispersive PDE's. The common feature in the analysis is…

In this paper, we consider the Cauchy problem for $(abcd)$-Boussinesq system posed on one- and two-dimensional Euclidean spaces. This model, initially introduced by Bona, Chen, and Saut, describes a small-amplitude waves on the surface of…

We consider in this paper the well-posedness for the Cauchy problem associated to two-dimensional dispersive systems of Boussinesq type which model weakly nonlinear long wave surface waves. We emphasize the case of the {\it strongly…

In this paper, we are concerned with the well-posedness and large time behavior of Cauchy problem for 3D incompressible Navier-Stokes-Cahn-Hilliard equations. First, using Banach fixed point theorem, we establish the local well-posedness of…

The present paper deals with the Cauchy problem of a multi-dimensional non-conservative viscous compressible two-fluid system. We first study the well-posedness of the model in spaces with critical regularity indices with respect to the…

In this paper we study the well-posedness of the Cauchy problem for first order hyperbolic systems with constant multiplicities and with low regularity coefficients depending just on the time variable. We consider Zygmund and log-Zygmund…

We investigate a class of scalar conservation laws on manifolds driven by multiplicative Gaussian (Ito) noise. The Cauchy problem defined on a Riemannian manifold is shown to be well-posed. We prove existence of generalized kinetic…

The paper is devoted to investigating the well-posedness, stability and large-time behavior near the hydrostatic balance for the 2D Boussinesq equations with partial dissipation. More precisely, the global well-posedness is obtained in the…

We study the well-posedness of the Hele-Shaw-Cahn-Hilliard system modeling binary fluid flow in porous media with arbitrary viscosity contrast but matched density between the components. For initial data in $H^s, s>\frac{d}{2}+1$, the…

This paper is dedicated to the study of a one-dimensional congestion model, consisting of two different phases. In the congested phase, the pressure is free and the dynamics is incompressible, whereas in the non-congested phase, the fluid…