Related papers: An improved bilinear estimate for Benjamin-Ono typ…
In this paper we consider the initial value problem of the Benjamin equation $$ \partial_{t}u+\nu \H(\partial^2_xu) +\mu\partial_{x}^{3}u+\partial_xu^2=0, $$ where $u:\R\times [0,T]\mapsto \R$, and the constants $\nu,\mu\in \R,\mu\neq0$. We…
The goal of this paper is three-fold. Firstly, we prove that the Cauchy problem for generalized KP-I equation \begin{eqnarray*}…
In this paper, we study the Cauchy problem of the 3-dimensional (3D) generalized incompressible Navier-Stokes equations (gNS) in Triebel-Lizorkin space $\dot{F}^{-\alpha,r}_{q_\alpha}(\mathbb{R}^3)$ with…
We study global well-posedness for the Kadomtsev-Petviashvili II equation in three space dimensions with small initial data. The crucial points are new bilinear estimates and the definition of the function spaces. As by-product we obtain…
We study the Cauchy problem to the KP-I equation posed on $\R^2$. We prove that it is $C^0$ locally well-posed in $H^{s,0}(\R\times \R)$ for $s>1/2$, which improves the previous results in \cite{GPW,GMo}.
We prove estimates for solutions of the Cauchy problem for the inhomogeneous wave equation on $\R^{1+n}$ in a class of Banach spaces whose norms only depend on the size of the space-time Fourier transform. The estimates are local in time,…
The Euler-Korteweg equations are a modification of the Euler equations that takes into account capillary effects. In the general case they form a quasi-linear system that can be recast as a degenerate Schr\"odinger type equation. Local…
We consider the long time well-posedness of the Cauchy problem with large Sobolev data for a class of nonlinear Schr\"odinger equations (NLS) on $\mathbb{R}^2$ with power nonlinearities of arbitrary odd degree. Specifically, the method in…
The Cauchy Problem for the modified Zakharov-Kuznetsov equation in three space dimensions is shown to be locally well-posed in $H^s(\R^3)$ for $s > \frac12$. Combined with the conservation of mass and energy this result implies global…
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…
We consider local well-posedness for the Maxwell-Chern-Simons-Higgs system in Lorenz gauge for data with minimal regularity assumptions in Fourier-Lebesgue spaces $\widehat{H}^{s,r}$ , where $\|u\|_{\widehat{H}^{s,r}} := \| \langle \xi…
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…
The 1D Cauchy problem for the Zakharov system is shown to be locally well-posed for low regularity Schr\"odinger data u_0 \in \hat{H^{k,p}} and wave data (n_0,n_1) \in \hat{H^{l,p}} \times \hat{H^{l-1,p}} under certain assumptions on the…
The Cauchy problem for the L^2-critical boson star equation with initial data of low regularity in spatial dimension d=3 is studied. Local well-posedness in H^s for s > 1/4 is proved. Moreover, for radial initial data, local well-posedness…
We prove the local well posedness of the Benjamin-Ono equation and the generalized Benjamin-Ono equation in $ H^1(\T) $. This leads to a global well-posedness result in $ H^1(\T)$ for the Benjamin-Ono equation.
We prove that the Cauchy problem for the Dirac-Klein-Gordon system of equations in 1+3 dimensions is locally well-posed in a range of Sobolev spaces for the Dirac spinor and the meson field. The result contains and extends the earlier known…
In this paper, we investigate the Cauchy problem for the shallow water type equation \[ u_{t}+\partial_{x}^{3}u + \frac{1}{2}\partial_{x}(u^{2})+\partial_{x} (1-\partial_{x}^{2})^{-1}\left[u^{2}+\frac{1}{2}u_{x}^{2}\right]=0,x\in {\mathbf…
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…
We consider the Cauchy problem for the spatially inhomogeneous non-cutoff Boltzmann equation with polynomially decaying initial data in the velocity variable. We establish short-time existence for any initial data with this decay in a fifth…
We consider the mass-subcritical Hartree equation with a homogeneous kernel, in the space of square integrable functions whose Fourier transform is integrable. We prove a global well-posedness result in this space. On the other hand, we…