Related papers: Well-posedness for a two-dimensional dispersive mo…
This work focuses on the mathematical analysis of the Cauchy problem associated with a two-dimensional equation describing the dynamics of a thin fluid film flowing down an inclined flat plate under the influence of gravity and an electric…
We consider the Cauchy problem associated with the modified Zakharov-Kuznetsov equation over $\mathbb{R}^2$. Taking into consideration the associated dispersive effects, we introduce, for $s,a\ge 0$, a two-parameter space…
We study a two fluid system which models the motion of a charged fluid with Rayleigh friction, and in the presence of an electro-magnetic field satisfying Maxwell's equations. We study the well-posdness of the system in both space…
We study the Cauchy problem for a generalized derivative nonlinear Schr\"odinger equation with the Dirichlet boundary condition. We establish the local well-posedness results in the Sobolev spaces $H^1$ and $H^2$. Solutions are constructed…
This paper is concerned with the Cauchy problem of the one-dimensional free surface equation of shallow water wave, we obtain local well-posedness of the free surface equation of shallow water wave in Sobolev spaces. In addition, we also…
In the first part of this work we study the local well-posedness of dispersive equations in the weighted spaces $H^s(\mathbb{R})\cap L^2(|x|^{2b}dx)$. We then apply our results for several dispersive models such as the Hirota-Satsuma…
We study the well-posedness in weighted Sobolev spaces, for the initial value problem (IVP) associated with the dissipative Benjamin-Ono (dBO) equation. We establish persistence properties of the solution flow in the weighted Sobolev spaces…
We study a class of two-dimensional non-linear Schr\"odinger equations with point-like singular perturbation and Hartree non-linearity. The point-like singular perturbation of the free Laplacian induces appropriate perturbed Sobolev spaces…
In this paper we consider the Cauchy problem for the nonlinear wave equation (NLW) with quadratic derivative nonlinearities in two space dimensions. Following Gr\"{u}nrock's result in 3D, we take the data in the Fourier-Lebesgue spaces…
In this paper we analyse the well-posedness of the Cauchy problem for a rather general class of hyperbolic systems with space-time dependent coefficients and with multiple characteristics of variable multiplicity. First, we establish a…
This paper is concerned with the Cauchy problem of the modified Kawahara equation (posed on $\mathbb T$), which is well-known as a model of capillary-gravity waves in an infinitely long canal over a flat bottom in a long wave regime…
In this paper, we consider the Cauchy problem for the generalized KP-II equation \begin{eqnarray*} u_{t}-|D_{x}|^{\alpha}u_{x}+\partial_{x}^{-1}\partial_{y}^{2}u+\frac{1}{2}\partial_{x}(u^{2})=0,\alpha\geq4. \end{eqnarray*} The goal of this…
The aim of this paper is to investigate well-posedness of the Cauchy problem for the degenerate Zakharov system. Local well-posedness holds for anisotropic Sobolev data by applying $U^2, V^2$ type spaces. We give the Schr\"odinger initial…
This paper is concerned with the Cauchy problem of the modified Zakharov-Kuznetsov equation on $\mathbb{R}^d$. If $d=2$, we prove the sharp estimate which implies local in time well-posedness in the Sobolev space $H^s(\mathbb{R}^2)$ for $s…
We show the local in time well-posedness of the Cauchy problem for the Kadomtsev-Petviashvili II equation for initial data in the non-isotropic Sobolev space H^{s_1,s_2}(R^2) with s_1 > -1/2 and s_2 \geq 0. On the H^{s_1,0}(R^2) scale this…
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…
In this paper, we study local well-posedness theory of the Cauchy problem for Schr\"{o}dinger-KdV system in Sobolev spaces $H^{s_1}\times H^{s_2}$. We obtain the local well-posedness when $s_1\geq 0$, $\max\{-3/4,s_1-3\}\leq s_2\leq…
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…
In recent work the authors proposed a broad global well-posedness conjecture for cubic defocusing dispersive equations in one space dimension, and then proved this conjecture in two cases, namely for one dimensional semilinear and…
The Cauchy problem for the Kadomtsev-Petviashvili-II equation (u_t+u_{xxx}+uu_x)_x+u_{yy}=0 is considered. A small data global well-posedness and scattering result in the scale invariant, non-isotropic, homogeneous Sobolev space \dot…